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Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.
Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.
There are 5 main steps in hypothesis testing:
Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.
Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.
After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.
The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.
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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.
There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).
If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.
Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.
Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .
Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.
In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.
In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).
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The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .
In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.
In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.
However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.
If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”
These are superficial differences; you can see that they mean the same thing.
You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.
If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
Methodology
Research bias
Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.
A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.
A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).
Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.
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Hypothesis testing is the act of testing a hypothesis or a supposition in relation to a statistical parameter. Analysts implement hypothesis testing in order to test if a hypothesis is plausible or not.
In data science and statistics , hypothesis testing is an important step as it involves the verification of an assumption that could help develop a statistical parameter. For instance, a researcher establishes a hypothesis assuming that the average of all odd numbers is an even number.
In order to find the plausibility of this hypothesis, the researcher will have to test the hypothesis using hypothesis testing methods. Unlike a hypothesis that is ‘supposed’ to stand true on the basis of little or no evidence, hypothesis testing is required to have plausible evidence in order to establish that a statistical hypothesis is true.
Perhaps this is where statistics play an important role. A number of components are involved in this process. But before understanding the process involved in hypothesis testing in research methodology, we shall first understand the types of hypotheses that are involved in the process. Let us get started!
In data sampling, different types of hypothesis are involved in finding whether the tested samples test positive for a hypothesis or not. In this segment, we shall discover the different types of hypotheses and understand the role they play in hypothesis testing.
Alternative Hypothesis (H1) or the research hypothesis states that there is a relationship between two variables (where one variable affects the other). The alternative hypothesis is the main driving force for hypothesis testing.
It implies that the two variables are related to each other and the relationship that exists between them is not due to chance or coincidence.
When the process of hypothesis testing is carried out, the alternative hypothesis is the main subject of the testing process. The analyst intends to test the alternative hypothesis and verifies its plausibility.
The Null Hypothesis (H0) aims to nullify the alternative hypothesis by implying that there exists no relation between two variables in statistics. It states that the effect of one variable on the other is solely due to chance and no empirical cause lies behind it.
The null hypothesis is established alongside the alternative hypothesis and is recognized as important as the latter. In hypothesis testing, the null hypothesis has a major role to play as it influences the testing against the alternative hypothesis.
(Must read: What is ANOVA test? )
The Non-directional hypothesis states that the relation between two variables has no direction.
Simply put, it asserts that there exists a relation between two variables, but does not recognize the direction of effect, whether variable A affects variable B or vice versa.
The Directional hypothesis, on the other hand, asserts the direction of effect of the relationship that exists between two variables.
Herein, the hypothesis clearly states that variable A affects variable B, or vice versa.
A statistical hypothesis is a hypothesis that can be verified to be plausible on the basis of statistics.
By using data sampling and statistical knowledge, one can determine the plausibility of a statistical hypothesis and find out if it stands true or not.
(Related blog: z-test vs t-test )
Now that we have understood the types of hypotheses and the role they play in hypothesis testing, let us now move on to understand the process in a better manner.
In hypothesis testing, a researcher is first required to establish two hypotheses - alternative hypothesis and null hypothesis in order to begin with the procedure.
To establish these two hypotheses, one is required to study data samples, find a plausible pattern among the samples, and pen down a statistical hypothesis that they wish to test.
A random population of samples can be drawn, to begin with hypothesis testing. Among the two hypotheses, alternative and null, only one can be verified to be true. Perhaps the presence of both hypotheses is required to make the process successful.
At the end of the hypothesis testing procedure, either of the hypotheses will be rejected and the other one will be supported. Even though one of the two hypotheses turns out to be true, no hypothesis can ever be verified 100%.
(Read also: Types of data sampling techniques )
Therefore, a hypothesis can only be supported based on the statistical samples and verified data. Here is a step-by-step guide for hypothesis testing.
First things first, one is required to establish two hypotheses - alternative and null, that will set the foundation for hypothesis testing.
These hypotheses initiate the testing process that involves the researcher working on data samples in order to either support the alternative hypothesis or the null hypothesis.
Once the hypotheses have been formulated, it is now time to generate a testing plan. A testing plan or an analysis plan involves the accumulation of data samples, determining which statistic is to be considered and laying out the sample size.
All these factors are very important while one is working on hypothesis testing.
As soon as a testing plan is ready, it is time to move on to the analysis part. Analysis of data samples involves configuring statistical values of samples, drawing them together, and deriving a pattern out of these samples.
While analyzing the data samples, a researcher needs to determine a set of things -
Significance Level - The level of significance in hypothesis testing indicates if a statistical result could have significance if the null hypothesis stands to be true.
Testing Method - The testing method involves a type of sampling-distribution and a test statistic that leads to hypothesis testing. There are a number of testing methods that can assist in the analysis of data samples.
Test statistic - Test statistic is a numerical summary of a data set that can be used to perform hypothesis testing.
P-value - The P-value interpretation is the probability of finding a sample statistic to be as extreme as the test statistic, indicating the plausibility of the null hypothesis.
The analysis of data samples leads to the inference of results that establishes whether the alternative hypothesis stands true or not. When the P-value is less than the significance level, the null hypothesis is rejected and the alternative hypothesis turns out to be plausible.
As we have already looked into different aspects of hypothesis testing, we shall now look into the different methods of hypothesis testing. All in all, there are 2 most common types of hypothesis testing methods. They are as follows -
The frequentist hypothesis or the traditional approach to hypothesis testing is a hypothesis testing method that aims on making assumptions by considering current data.
The supposed truths and assumptions are based on the current data and a set of 2 hypotheses are formulated. A very popular subtype of the frequentist approach is the Null Hypothesis Significance Testing (NHST).
The NHST approach (involving the null and alternative hypothesis) has been one of the most sought-after methods of hypothesis testing in the field of statistics ever since its inception in the mid-1950s.
A much unconventional and modern method of hypothesis testing, the Bayesian Hypothesis Testing claims to test a particular hypothesis in accordance with the past data samples, known as prior probability, and current data that lead to the plausibility of a hypothesis.
The result obtained indicates the posterior probability of the hypothesis. In this method, the researcher relies on ‘prior probability and posterior probability’ to conduct hypothesis testing on hand.
On the basis of this prior probability, the Bayesian approach tests a hypothesis to be true or false. The Bayes factor, a major component of this method, indicates the likelihood ratio among the null hypothesis and the alternative hypothesis.
The Bayes factor is the indicator of the plausibility of either of the two hypotheses that are established for hypothesis testing.
(Also read - Introduction to Bayesian Statistics )
To conclude, hypothesis testing, a way to verify the plausibility of a supposed assumption can be done through different methods - the Bayesian approach or the Frequentist approach.
Although the Bayesian approach relies on the prior probability of data samples, the frequentist approach assumes without a probability. A number of elements involved in hypothesis testing are - significance level, p-level, test statistic, and method of hypothesis testing.
(Also read: Introduction to probability distributions )
A significant way to determine whether a hypothesis stands true or not is to verify the data samples and identify the plausible hypothesis among the null hypothesis and alternative hypothesis.
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Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.
A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.
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Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.
Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.
The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as \(H_{0}\). Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.
The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as \(H_{1}\) or \(H_{a}\). For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.
In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, \(\alpha\) or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.
All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.
Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:
We will learn more about these test statistics in the upcoming section.
Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.
A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:
The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.
The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.
One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.
Right Tailed Hypothesis Testing
The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:
\(H_{0}\): The population parameter is ≤ some value
\(H_{1}\): The population parameter is > some value.
If the test statistic has a greater value than the critical value then the null hypothesis is rejected
Left Tailed Hypothesis Testing
The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:
\(H_{0}\): The population parameter is ≥ some value
\(H_{1}\): The population parameter is < some value.
The null hypothesis is rejected if the test statistic has a value lesser than the critical value.
In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:
\(H_{0}\): the population parameter = some value
\(H_{1}\): the population parameter ≠ some value
The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.
Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:
The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.
Step 1: This is an example of a right-tailed test. Set up the null hypothesis as \(H_{0}\): \(\mu\) = 100.
Step 2: The alternative hypothesis is given by \(H_{1}\): \(\mu\) > 100.
Step 3: As this is a one-tailed test, \(\alpha\) = 100% - 95% = 5%. This can be used to determine the critical value.
1 - \(\alpha\) = 1 - 0.05 = 0.95
0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.
Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.
z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
\(\mu\) = 100, \(\overline{x}\) = 112.5, n = 30, \(\sigma\) = 15
z = \(\frac{112.5-100}{\frac{15}{\sqrt{30}}}\) = 4.56
Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.
Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.
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Important Notes on Hypothesis Testing
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What is hypothesis testing.
Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.
The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.
The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.
The formula for a one sample z test in hypothesis testing is z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) and for two samples is z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).
The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.
When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.
To get the alpha level in a two tail hypothesis testing divide \(\alpha\) by 2. This is done as there are two rejection regions in the curve.
Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.
A hypothesis is an assumption or idea, specifically a statistical claim about an unknown population parameter. For example, a judge assumes a person is innocent and verifies this by reviewing evidence and hearing testimony before reaching a verdict.
Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data.
To test the validity of the claim or assumption about the population parameter:
Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.
Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing.
One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.
There are two types of one-tailed test:
A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.
Example: H 0 : [Tex]\mu = [/Tex] 50 and H 1 : [Tex]\mu \neq 50 [/Tex]
To delve deeper into differences into both types of test: Refer to link
In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.
Null Hypothesis is True | Null Hypothesis is False | |
---|---|---|
Null Hypothesis is True (Accept) | Correct Decision | Type II Error (False Negative) |
Alternative Hypothesis is True (Reject) | Type I Error (False Positive) | Correct Decision |
Step 1: define null and alternative hypothesis.
State the null hypothesis ( [Tex]H_0 [/Tex] ), representing no effect, and the alternative hypothesis ( [Tex]H_1 [/Tex] ), suggesting an effect or difference.
We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.
Select a significance level ( [Tex]\alpha [/Tex] ), typically 0.05, to determine the threshold for rejecting the null hypothesis. It provides validity to our hypothesis test, ensuring that we have sufficient data to back up our claims. Usually, we determine our significance level beforehand of the test. The p-value is the criterion used to calculate our significance value.
Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.
The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.
There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.
We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.
T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.
In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.
Comparing the test statistic and tabulated critical value we have,
Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.
We can also come to an conclusion using the p-value,
Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.
At last, we can conclude our experiment using method A or B.
To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .
When population means and standard deviations are known.
[Tex]z = \frac{\bar{x} – \mu}{\frac{\sigma}{\sqrt{n}}}[/Tex]
T test is used when n<30,
t-statistic calculation is given by:
[Tex]t=\frac{x̄-μ}{s/\sqrt{n}} [/Tex]
Chi-Square Test for Independence categorical Data (Non-normally distributed) using:
[Tex]\chi^2 = \sum \frac{(O_{ij} – E_{ij})^2}{E_{ij}}[/Tex]
Let’s examine hypothesis testing using two real life situations,
Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.
Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.
If the evidence suggests less than a 5% chance of observing the results due to random variation.
Using paired T-test analyze the data to obtain a test statistic and a p-value.
The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.
t = m/(s/√n)
then, m= -3.9, s= 1.8 and n= 10
we, calculate the , T-statistic = -9 based on the formula for paired t test
The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.
thus, p-value = 8.538051223166285e-06
Step 5: Result
Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.
Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.
Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.
We will implement our first real life problem via python,
import numpy as np from scipy import stats # Data before_treatment = np . array ([ 120 , 122 , 118 , 130 , 125 , 128 , 115 , 121 , 123 , 119 ]) after_treatment = np . array ([ 115 , 120 , 112 , 128 , 122 , 125 , 110 , 117 , 119 , 114 ]) # Step 1: Null and Alternate Hypotheses # Null Hypothesis: The new drug has no effect on blood pressure. # Alternate Hypothesis: The new drug has an effect on blood pressure. null_hypothesis = "The new drug has no effect on blood pressure." alternate_hypothesis = "The new drug has an effect on blood pressure." # Step 2: Significance Level alpha = 0.05 # Step 3: Paired T-test t_statistic , p_value = stats . ttest_rel ( after_treatment , before_treatment ) # Step 4: Calculate T-statistic manually m = np . mean ( after_treatment - before_treatment ) s = np . std ( after_treatment - before_treatment , ddof = 1 ) # using ddof=1 for sample standard deviation n = len ( before_treatment ) t_statistic_manual = m / ( s / np . sqrt ( n )) # Step 5: Decision if p_value <= alpha : decision = "Reject" else : decision = "Fail to reject" # Conclusion if decision == "Reject" : conclusion = "There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different." else : conclusion = "There is insufficient evidence to claim a significant difference in average blood pressure before and after treatment with the new drug." # Display results print ( "T-statistic (from scipy):" , t_statistic ) print ( "P-value (from scipy):" , p_value ) print ( "T-statistic (calculated manually):" , t_statistic_manual ) print ( f "Decision: { decision } the null hypothesis at alpha= { alpha } ." ) print ( "Conclusion:" , conclusion )
T-statistic (from scipy): -9.0 P-value (from scipy): 8.538051223166285e-06 T-statistic (calculated manually): -9.0 Decision: Reject the null hypothesis at alpha=0.05. Conclusion: There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.
In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05.
Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.
Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.
Populations Mean = 200
Population Standard Deviation (σ): 5 mg/dL(given for this problem)
As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.
The test statistic is calculated by using the z formula Z = [Tex](203.8 – 200) / (5 \div \sqrt{25}) [/Tex] and we get accordingly , Z =2.039999999999992.
Step 4: Result
Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL
import scipy.stats as stats import math import numpy as np # Given data sample_data = np . array ( [ 205 , 198 , 210 , 190 , 215 , 205 , 200 , 192 , 198 , 205 , 198 , 202 , 208 , 200 , 205 , 198 , 205 , 210 , 192 , 205 , 198 , 205 , 210 , 192 , 205 ]) population_std_dev = 5 population_mean = 200 sample_size = len ( sample_data ) # Step 1: Define the Hypotheses # Null Hypothesis (H0): The average cholesterol level in a population is 200 mg/dL. # Alternate Hypothesis (H1): The average cholesterol level in a population is different from 200 mg/dL. # Step 2: Define the Significance Level alpha = 0.05 # Two-tailed test # Critical values for a significance level of 0.05 (two-tailed) critical_value_left = stats . norm . ppf ( alpha / 2 ) critical_value_right = - critical_value_left # Step 3: Compute the test statistic sample_mean = sample_data . mean () z_score = ( sample_mean - population_mean ) / \ ( population_std_dev / math . sqrt ( sample_size )) # Step 4: Result # Check if the absolute value of the test statistic is greater than the critical values if abs ( z_score ) > max ( abs ( critical_value_left ), abs ( critical_value_right )): print ( "Reject the null hypothesis." ) print ( "There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL." ) else : print ( "Fail to reject the null hypothesis." ) print ( "There is not enough evidence to conclude that the average cholesterol level in the population is different from 200 mg/dL." )
Reject the null hypothesis. There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL.
Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.
1. what are the 3 types of hypothesis test.
There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.
Null Hypothesis ( [Tex]H_o [/Tex] ): No effect or difference exists. Alternative Hypothesis ( [Tex]H_1 [/Tex] ): An effect or difference exists. Significance Level ( [Tex]\alpha [/Tex] ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.
Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.
Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.
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Priya ranganathan.
1 Department of Anesthesiology, Critical Care and Pain, Tata Memorial Hospital, Mumbai, Maharashtra, India
2 Department of Surgical Oncology, Tata Memorial Centre, Mumbai, Maharashtra, India
The second article in this series on biostatistics covers the concepts of sample, population, research hypotheses and statistical errors.
Ranganathan P, Pramesh CS. An Introduction to Statistics: Understanding Hypothesis Testing and Statistical Errors. Indian J Crit Care Med 2019;23(Suppl 3):S230–S231.
Two papers quoted in this issue of the Indian Journal of Critical Care Medicine report. The results of studies aim to prove that a new intervention is better than (superior to) an existing treatment. In the ABLE study, the investigators wanted to show that transfusion of fresh red blood cells would be superior to standard-issue red cells in reducing 90-day mortality in ICU patients. 1 The PROPPR study was designed to prove that transfusion of a lower ratio of plasma and platelets to red cells would be superior to a higher ratio in decreasing 24-hour and 30-day mortality in critically ill patients. 2 These studies are known as superiority studies (as opposed to noninferiority or equivalence studies which will be discussed in a subsequent article).
A sample represents a group of participants selected from the entire population. Since studies cannot be carried out on entire populations, researchers choose samples, which are representative of the population. This is similar to walking into a grocery store and examining a few grains of rice or wheat before purchasing an entire bag; we assume that the few grains that we select (the sample) are representative of the entire sack of grains (the population).
The results of the study are then extrapolated to generate inferences about the population. We do this using a process known as hypothesis testing. This means that the results of the study may not always be identical to the results we would expect to find in the population; i.e., there is the possibility that the study results may be erroneous.
A clinical trial begins with an assumption or belief, and then proceeds to either prove or disprove this assumption. In statistical terms, this belief or assumption is known as a hypothesis. Counterintuitively, what the researcher believes in (or is trying to prove) is called the “alternate” hypothesis, and the opposite is called the “null” hypothesis; every study has a null hypothesis and an alternate hypothesis. For superiority studies, the alternate hypothesis states that one treatment (usually the new or experimental treatment) is superior to the other; the null hypothesis states that there is no difference between the treatments (the treatments are equal). For example, in the ABLE study, we start by stating the null hypothesis—there is no difference in mortality between groups receiving fresh RBCs and standard-issue RBCs. We then state the alternate hypothesis—There is a difference between groups receiving fresh RBCs and standard-issue RBCs. It is important to note that we have stated that the groups are different, without specifying which group will be better than the other. This is known as a two-tailed hypothesis and it allows us to test for superiority on either side (using a two-sided test). This is because, when we start a study, we are not 100% certain that the new treatment can only be better than the standard treatment—it could be worse, and if it is so, the study should pick it up as well. One tailed hypothesis and one-sided statistical testing is done for non-inferiority studies, which will be discussed in a subsequent paper in this series.
There are two possibilities to consider when interpreting the results of a superiority study. The first possibility is that there is truly no difference between the treatments but the study finds that they are different. This is called a Type-1 error or false-positive error or alpha error. This means falsely rejecting the null hypothesis.
The second possibility is that there is a difference between the treatments and the study does not pick up this difference. This is called a Type 2 error or false-negative error or beta error. This means falsely accepting the null hypothesis.
The power of the study is the ability to detect a difference between groups and is the converse of the beta error; i.e., power = 1-beta error. Alpha and beta errors are finalized when the protocol is written and form the basis for sample size calculation for the study. In an ideal world, we would not like any error in the results of our study; however, we would need to do the study in the entire population (infinite sample size) to be able to get a 0% alpha and beta error. These two errors enable us to do studies with realistic sample sizes, with the compromise that there is a small possibility that the results may not always reflect the truth. The basis for this will be discussed in a subsequent paper in this series dealing with sample size calculation.
Conventionally, type 1 or alpha error is set at 5%. This means, that at the end of the study, if there is a difference between groups, we want to be 95% certain that this is a true difference and allow only a 5% probability that this difference has occurred by chance (false positive). Type 2 or beta error is usually set between 10% and 20%; therefore, the power of the study is 90% or 80%. This means that if there is a difference between groups, we want to be 80% (or 90%) certain that the study will detect that difference. For example, in the ABLE study, sample size was calculated with a type 1 error of 5% (two-sided) and power of 90% (type 2 error of 10%) (1).
Table 1 gives a summary of the two types of statistical errors with an example
Statistical errors
(a) Types of statistical errors | |||
: Null hypothesis is | |||
True | False | ||
Null hypothesis is actually | True | Correct results! | Falsely rejecting null hypothesis - Type I error |
False | Falsely accepting null hypothesis - Type II error | Correct results! | |
(b) Possible statistical errors in the ABLE trial | |||
There is difference in mortality between groups receiving fresh RBCs and standard-issue RBCs | There difference in mortality between groups receiving fresh RBCs and standard-issue RBCs | ||
Truth | There is difference in mortality between groups receiving fresh RBCs and standard-issue RBCs | Correct results! | Falsely rejecting null hypothesis - Type I error |
There difference in mortality between groups receiving fresh RBCs and standard-issue RBCs | Falsely accepting null hypothesis - Type II error | Correct results! |
In the next article in this series, we will look at the meaning and interpretation of ‘ p ’ value and confidence intervals for hypothesis testing.
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Hypothesis testing in statistics involves testing an assumption about a population parameter using sample data. Learners can download Hypothesis Testing PDF to get instant access to all information!
What exactly is hypothesis testing, and how does it work in statistics? Can I find practical examples and understand the different types from this blog?
Hypothesis Testing : Ever wonder how researchers determine if a new medicine actually works or if a new marketing campaign effectively drives sales? They use hypothesis testing! It is at the core of how scientific studies, business experiments and surveys determine if their results are statistically significant or just due to chance.
Hypothesis testing allows us to make evidence-based decisions by quantifying uncertainty and providing a structured process to make data-driven conclusions rather than guessing. In this post, we will discuss hypothesis testing types, examples, and processes!
Table of Contents
Hypothesis testing is a statistical method used to evaluate the validity of a hypothesis using sample data. It involves assessing whether observed data provide enough evidence to reject a specific hypothesis about a population parameter.
Hypothesis testing in data science is a statistical method used to evaluate two mutually exclusive population statements based on sample data. The primary goal is to determine which statement is more supported by the observed data.
Hypothesis testing assists in supporting the certainty of findings in research and data science projects. This statistical inference aids in making decisions about population parameters using sample data. For those who are looking to deepen their knowledge in data science and expand their skillset, we highly recommend checking out Master Generative AI: Data Science Course by Physics Wallah .
Also Read: What is Encapsulation Explain in Details
The hypothesis testing procedure in data science involves a structured approach to evaluating hypotheses using statistical methods. Here’s a step-by-step breakdown of the typical procedure:
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Hypothesis testing is a fundamental concept in statistics that aids analysts in making informed decisions based on sample data about a larger population. The process involves setting up two contrasting hypotheses, the null hypothesis and the alternative hypothesis, and then using statistical methods to determine which hypothesis provides a more plausible explanation for the observed data.
Once these hypotheses are established, analysts gather data from a sample and conduct statistical tests. The objective is to determine whether the observed results are statistically significant enough to reject the null hypothesis in favor of the alternative.
Hypothesis testing is a cornerstone in statistical analysis, providing a framework to evaluate the validity of assumptions or claims made about a population based on sample data. Within this framework, several specific tests are utilized based on the nature of the data and the question at hand. Here’s a closer look at the three fundamental types of hypothesis tests:
The z-test is a statistical method primarily employed when comparing means from two datasets, particularly when the population standard deviation is known. Its main objective is to ascertain if the means are statistically equivalent.
A crucial prerequisite for the z-test is that the sample size should be relatively large, typically 30 data points or more. This test aids researchers and analysts in determining the significance of a relationship or discovery, especially in scenarios where the data’s characteristics align with the assumptions of the z-test.
The t-test is a versatile statistical tool used extensively in research and various fields to compare means between two groups. It’s particularly valuable when the population standard deviation is unknown or when dealing with smaller sample sizes.
By evaluating the means of two groups, the t-test helps ascertain if a particular treatment, intervention, or variable significantly impacts the population under study. Its flexibility and robustness make it a go-to method in scenarios ranging from medical research to business analytics.
The Chi-Square test stands distinct from the previous tests, primarily focusing on categorical data rather than means. This statistical test is instrumental when analyzing categorical variables to determine if observed data aligns with expected outcomes as posited by the null hypothesis.
By assessing the differences between observed and expected frequencies within categorical data, the Chi-Square test offers insights into whether discrepancies are statistically significant. Whether used in social sciences to evaluate survey responses or in quality control to assess product defects, the Chi-Square test remains pivotal for hypothesis testing in diverse scenarios.
Also Read: Python vs Java: Which is Best for Machine learning algorithm
Hypothesis testing is a fundamental concept in statistics used to make decisions or inferences about a population based on a sample of data. The process involves setting up two competing hypotheses, the null hypothesis H 0 and the alternative hypothesis H 1.
Through various statistical tests, such as the t-test, z-test, or Chi-square test, analysts evaluate sample data to determine whether there’s enough evidence to reject the null hypothesis in favor of the alternative. The aim is to draw conclusions about population parameters or to test theories, claims, or hypotheses.
In research, hypothesis testing serves as a structured approach to validate or refute theories or claims. Researchers formulate a clear hypothesis based on existing literature or preliminary observations. They then collect data through experiments, surveys, or observational studies.
Using statistical methods, researchers analyze this data to determine if there’s sufficient evidence to reject the null hypothesis. By doing so, they can draw meaningful conclusions, make predictions, or recommend actions based on empirical evidence rather than mere speculation.
R, a powerful programming language and environment for statistical computing and graphics, offers a wide array of functions and packages specifically designed for hypothesis testing. Here’s how hypothesis testing is conducted in R:
Hypothesis testing is an integral part of statistics and research, offering a systematic approach to validate hypotheses. Leveraging R’s capabilities, researchers and analysts can efficiently conduct and interpret various hypothesis tests, ensuring robust and reliable conclusions from their data.
Yes, data scientists frequently engage in hypothesis testing as part of their analytical toolkit. Hypothesis testing is a foundational statistical technique used to make data-driven decisions, validate assumptions, and draw conclusions from data. Here’s how data scientists utilize hypothesis testing:
Let’s delve into some common examples of hypothesis testing and provide solutions or interpretations for each scenario.
Scenario : A coffee shop owner believes that the average waiting time for customers during peak hours is 5 minutes. To test this, the owner takes a random sample of 30 customer waiting times and wants to determine if the average waiting time is indeed 5 minutes.
Hypotheses :
Solution : Using a t-test (assuming population variance is unknown), calculate the t-statistic based on the sample mean, sample standard deviation, and sample size. Then, determine the p-value and compare it with a significance level (e.g., 0.05) to decide whether to reject the null hypothesis.
Scenario : An e-commerce company wants to determine if changing the color of a “Buy Now” button from blue to green increases the conversion rate.
Solution : Split website visitors into two groups: one sees the blue button (control group), and the other sees the green button (test group). Track the conversion rates for both groups over a specified period. Then, use a chi-square test or z-test (for large sample sizes) to determine if there’s a statistically significant difference in conversion rates between the two groups.
The formula for hypothesis testing typically depends on the type of test (e.g., z-test, t-test, chi-square test) and the nature of the data (e.g., mean, proportion, variance). Below are the basic formulas for some common hypothesis tests:
Z-Test for Population Mean :
Z=(σ/n)(xˉ−μ0)
T-Test for Population Mean :
t= (s/ n ) ( x ˉ −μ 0 )
s = Sample standard deviation
Chi-Square Test for Goodness of Fit :
χ2=∑Ei(Oi−Ei)2
Also Read: Full Form of OOPS
While you can perform hypothesis testing manually using the above formulas and statistical tables, many online tools and software packages simplify this process. Here’s how you might use a calculator or software:
When using any calculator or software, always ensure you understand the underlying assumptions of the test, interpret the results correctly, and consider the broader context of your research or analysis.
What are the key components of a hypothesis test.
The key components include: Null Hypothesis (H0): A statement of no effect or no difference. Alternative Hypothesis (H1 or Ha): A statement that contradicts the null hypothesis. Test Statistic: A value computed from the sample data to test the null hypothesis. Significance Level (α): The threshold for rejecting the null hypothesis. P-value: The probability of observing the given data, assuming the null hypothesis is true.
The significance level (often denoted as α) is the probability threshold used to determine whether to reject the null hypothesis. Commonly used values for α include 0.05, 0.01, and 0.10, representing a 5%, 1%, or 10% chance of rejecting the null hypothesis when it's actually true.
The choice between one-tailed and two-tailed tests depends on your research question and hypothesis. Use a one-tailed test when you're specifically interested in one direction of an effect (e.g., greater than or less than). Use a two-tailed test when you want to determine if there's a significant difference in either direction.
The p-value is a probability value that helps determine the strength of evidence against the null hypothesis. A low p-value (typically ≤ 0.05) suggests that the observed data is inconsistent with the null hypothesis, leading to its rejection. Conversely, a high p-value suggests that the data is consistent with the null hypothesis, leading to no rejection.
No, hypothesis testing cannot prove a hypothesis true. Instead, it helps assess the likelihood of observing a given set of data under the assumption that the null hypothesis is true. Based on this assessment, you either reject or fail to reject the null hypothesis.
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S.3 hypothesis testing.
In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail.
The general idea of hypothesis testing involves:
Every hypothesis test — regardless of the population parameter involved — requires the above three steps.
Is normal body temperature really 98.6 degrees f section .
Consider the population of many, many adults. A researcher hypothesized that the average adult body temperature is lower than the often-advertised 98.6 degrees F. That is, the researcher wants an answer to the question: "Is the average adult body temperature 98.6 degrees? Or is it lower?" To answer his research question, the researcher starts by assuming that the average adult body temperature was 98.6 degrees F.
Then, the researcher went out and tried to find evidence that refutes his initial assumption. In doing so, he selects a random sample of 130 adults. The average body temperature of the 130 sampled adults is 98.25 degrees.
Then, the researcher uses the data he collected to make a decision about his initial assumption. It is either likely or unlikely that the researcher would collect the evidence he did given his initial assumption that the average adult body temperature is 98.6 degrees:
In statistics, we generally don't make claims that require us to believe that a very unusual event happened. That is, in the practice of statistics, if the evidence (data) we collected is unlikely in light of the initial assumption, then we reject our initial assumption.
Criminal trial analogy section .
One place where you can consistently see the general idea of hypothesis testing in action is in criminal trials held in the United States. Our criminal justice system assumes "the defendant is innocent until proven guilty." That is, our initial assumption is that the defendant is innocent.
In the practice of statistics, we make our initial assumption when we state our two competing hypotheses -- the null hypothesis ( H 0 ) and the alternative hypothesis ( H A ). Here, our hypotheses are:
In statistics, we always assume the null hypothesis is true . That is, the null hypothesis is always our initial assumption.
The prosecution team then collects evidence — such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, and handwriting samples — with the hopes of finding "sufficient evidence" to make the assumption of innocence refutable.
In statistics, the data are the evidence.
The jury then makes a decision based on the available evidence:
In statistics, we always make one of two decisions. We either "reject the null hypothesis" or we "fail to reject the null hypothesis."
Did you notice the use of the phrase "behave as if" in the previous discussion? We "behave as if" the defendant is guilty; we do not "prove" that the defendant is guilty. And, we "behave as if" the defendant is innocent; we do not "prove" that the defendant is innocent.
This is a very important distinction! We make our decision based on evidence not on 100% guaranteed proof. Again:
We merely state that there is enough evidence to behave one way or the other. This is always true in statistics! Because of this, whatever the decision, there is always a chance that we made an error .
Let's review the two types of errors that can be made in criminal trials:
Jury Decision | Truth | ||
---|---|---|---|
Not Guilty | Guilty | ||
Not Guilty | OK | ERROR | |
Guilty | ERROR | OK |
Table S.3.2 shows how this corresponds to the two types of errors in hypothesis testing.
Decision | |||
---|---|---|---|
Null Hypothesis | Alternative Hypothesis | ||
Do not Reject Null | OK | Type II Error | |
Reject Null | Type I Error | OK |
Note that, in statistics, we call the two types of errors by two different names -- one is called a "Type I error," and the other is called a "Type II error." Here are the formal definitions of the two types of errors:
There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!
Recall that it is either likely or unlikely that we would observe the evidence we did given our initial assumption. If it is likely , we do not reject the null hypothesis. If it is unlikely , then we reject the null hypothesis in favor of the alternative hypothesis. Effectively, then, making the decision reduces to determining "likely" or "unlikely."
In statistics, there are two ways to determine whether the evidence is likely or unlikely given the initial assumption:
In the next two sections, we review the procedures behind each of these two approaches. To make our review concrete, let's imagine that μ is the average grade point average of all American students who major in mathematics. We first review the critical value approach for conducting each of the following three hypothesis tests about the population mean $\mu$:
: = 3 | : > 3 | |
: = 3 | : < 3 | |
: = 3 | : ≠ 3 |
Upon completing the review of the critical value approach, we review the P -value approach for conducting each of the above three hypothesis tests about the population mean \(\mu\). The procedures that we review here for both approaches easily extend to hypothesis tests about any other population parameter.
Home Blog Data Science Hypothesis Testing in Data Science [Types, Process, Example]
In day-to-day life, we come across a lot of data lot of variety of content. Sometimes the information is too much that we get confused about whether the information provided is correct or not. At that moment, we get introduced to a word called “Hypothesis testing” which helps in determining the proofs and pieces of evidence for some belief or information.
Hypothesis testing is an integral part of statistical inference. It is used to decide whether the given sample data from the population parameter satisfies the given hypothetical condition. So, it will predict and decide using several factors whether the predictions satisfy the conditions or not. In simpler terms, trying to prove whether the facts or statements are true or not.
For example, if you predict that students who sit on the last bench are poorer and weaker than students sitting on 1st bench, then this is a hypothetical statement that needs to be clarified using different experiments. Another example we can see is implementing new business strategies to evaluate whether they will work for the business or not. All these things are very necessary when you work with data as a data scientist. If you are interested in learning about data science, visit this amazing Data Science full course to learn data science.
It is important to know how and where we can use hypothesis testing techniques in the field of data science. Data scientists predict a lot of things in their day-to-day work, and to check the probability of whether that finding is certain or not, we use hypothesis testing. The main goal of hypothesis testing is to gauge how well the predictions perform based on the sample data provided by the population. If you are interested to know more about the applications of the data, then refer to this D ata Scien ce course in India which will give you more insights into application-based things. When data scientists work on model building using various machine learning algorithms, they need to have faith in their models and the forecasting of models. They then provide the sample data to the model for training purposes so that it can provide us with the significance of statistical data that will represent the entire population.
Hypothesis testing is widely used when we need to compare our results based on predictions. So, it will compare before and after results. For example, someone claimed that students writing exams from blue pen always get above 90%; now this statement proves it correct, and experiments need to be done. So, the data will be collected based on the student's input, and then the test will be done on the final result later after various experiments and observations on students' marks vs pen used, final conclusions will be made which will determine the results. Now hypothesis testing will be done to compare the 1st and the 2nd result, to see the difference and closeness of both outputs. This is how hypothesis testing is done.
In the whole data science life cycle, hypothesis testing is done in various stages, starting from the initial part, the 1st stage where the EDA, data pre-processing, and manipulation are done. In this stage, we will do our initial hypothesis testing to visualize the outcome in later stages. The next test will be done after we have built our model, once the model is ready and hypothesis testing is done, we will compare the results of the initial testing and the 2nd one to compare the results and significance of the results and to confirm the insights generated from the 1st cycle match with the 2nd one or not. This will help us know how the model responds to the sample training data. As we saw above, hypothesis testing is always needed when we are planning to contrast more than 2 groups. While checking on the results, it is important to check on the flexibility of the results for the sample and the population. Later, we can judge on the disagreement of the results are appropriate or vague. This is all we can do using hypothesis testing.
Hypothesis testing can be seen in several types. In total, we have 5 types of hypothesis testing. They are described below:
The alternative hypothesis explains and defines the relationship between two variables. It simply indicates a positive relationship between two variables which means they do have a statistical bond. It indicates that the sample observed is going to influence or affect the outcome. An alternative hypothesis is described using H a or H 1 . Ha indicates an alternative hypothesis and H 1 explains the possibility of influenced outcome which is 1. For example, children who study from the beginning of the class have fewer chances to fail. An alternate hypothesis will be accepted once the statistical predictions become significant. The alternative hypothesis can be further divided into 3 parts.
The null hypothesis simply states that there is no relation between statistical variables. If the facts presented at the start do not match with the outcomes, then we can say, the testing is null hypothesis testing. The null hypothesis is represented as H 0 . For example, children who study from the beginning of the class have no fewer chances to fail. There are types of Null Hypothesis described below:
Simple Hypothesis: It helps in denoting and indicating the distribution of the population.
Composite Hypothesis: It does not denote the population distribution
Exact Hypothesis: In the exact hypothesis, the value of the hypothesis is the same as the sample distribution. Example- μ= 10
Inexact Hypothesis: Here, the hypothesis values are not equal to the sample. It will denote a particular range of values.
The non-directional hypothesis is a tow-tailed hypothesis that indicates the true value does not equal the predicted value. In simpler terms, there is no direction between the 2 variables. For an example of a non-directional hypothesis, girls and boys have different methodologies to solve a problem. Here the example explains that the thinking methodologies of a girl and a boy is different, they don’t think alike.
In the Directional hypothesis, there is a direct relationship between two variables. Here any of the variables influence the other.
Statistical hypothesis helps in understanding the nature and character of the population. It is a great method to decide whether the values and the data we have with us satisfy the given hypothesis or not. It helps us in making different probabilistic and certain statements to predict the outcome of the population... We have several types of tests which are the T-test, Z-test, and Anova tests.
1. frequentist hypothesis testing.
Frequentist hypotheses mostly work with the approach of making predictions and assumptions based on the current data which is real-time data. All the facts are based on current data. The most famous kind of frequentist approach is null hypothesis testing.
Bayesian testing is a modern and latest way of hypothesis testing. It is known to be the test that works with past data to predict the future possibilities of the hypothesis. In Bayesian, it refers to the prior distribution or prior probability samples for the observed data. In the medical Industry, we observe that Doctors deal with patients’ diseases using past historical records. So, with this kind of record, it is helpful for them to understand and predict the current and upcoming health conditions of the patient.
Most of the time, people assume that data science is all about applying machine learning algorithms and getting results, that is true but in addition to the fact that to work in the data science field, one needs to be well versed with statistics as most of the background work in Data science is done through statistics. When we deal with data for pre-processing, manipulating, and analyzing, statistics play. Specifically speaking Hypothesis testing helps in making confident decisions, predicting the correct outcomes, and finding insightful conclusions regarding the population. Hypothesis testing helps us resolve tough things easily. To get more familiar with Hypothesis testing and other prediction models attend the superb useful KnowledgeHut Data Science full course which will give you more domain knowledge and will assist you in working with industry-related projects.
1. null and alternative hypothesis.
After we have done our initial research about the predictions that we want to find out if true, it is important to mention whether the hypothesis done is a null hypothesis(H0) or an alternative hypothesis (Ha). Once we understand the type of hypothesis, it will be easy for us to do mathematical research on it. A null hypothesis will usually indicate the no-relationship between the variables whereas an alternative hypothesis describes the relationship between 2 variables.
To prove our statistical test validity, it is essential and critical to check the data and proceed with sampling them to get the correct hypothesis results. If the target data is not prepared and ready, it will become difficult to make the predictions or the statistical inference on the population that we are planning to make. It is important to prepare efficient data, so that hypothesis findings can be easy to predict.
To perform various analyses on the data, we need to choose a statistical test. There are various types of statistical tests available. Based on the wide spread of the data that is variance within the group or how different the data category is from one another that is variance without a group, we can proceed with our further research study.
Once we get the result and outcome of the statistical test, we have to then proceed further to decide whether the reject or accept the null hypothesis. The significance level is indicated by alpha (α). It describes the probability of rejecting or accepting the null hypothesis. Example- Suppose the value of the significance level which is alpha is 0.05. Now, this value indicates the difference from the null hypothesis.
P value is simply the probability value and expected determined outcome which is at least as extreme and close as observed results of a hypothetical test. It helps in evaluating and verifying hypotheses against the sample data. This happens while assuming the null hypothesis is true. The lower the value of P, the higher and better will be the results of the significant value which is alpha (α). For example, if the P-value is 0.05 or even less than this, then it will be considered statistically significant. The main thing is these values are predicted based on the calculations done by deviating the values between the observed one and referenced one. The greater the difference between values, the lower the p-value will be.
After knowing the P-value and statistical significance, we can determine our results and take the appropriate decision of whether to accept or reject the null hypothesis based on the facts and statistics presented to us.
Hypothesis testing can be done using various statistical tests. One is Z-test. The formula for Z-test is given below:
Z = ( x̅ – μ 0 ) / (σ /√n)
In the above equation, x̅ is the sample mean
Now depending on the Z-test result, the examination will be processed further. The result is either going to be a null hypothesis or it is going to be an alternative hypothesis. That can be measured through below formula-
In this way, we calculate the hypothesis testing and can apply it to real-world scenarios.
Hypothesis testing has a wide variety of use cases that proves to be beneficial for various industries.
In the healthcare industry, all the research and experiments which are done to predict the success of any medicine or drug are done successfully with the help of Hypothesis testing.
Hypothesis testing assists in experimenting with different teaching techniques to deal with the understanding capability of different students.
Hypothesis testing helps in indicating the factors that may cause some serious mental health issues.
Testing whether the new change in the process of manufacturing helped in the improvement of the process as well as in the quantity or not. In the same way, there are many other use cases that we get to see in different sectors for hypothesis testing.
1. type-i error.
Type I error occurs during the process of hypothesis testing when the null hypothesis is rejected even though it is accurate. This kind of error is also known as False positive because even though the statement is positive or correct but results are given as false. For example, an innocent person still goes to jail because he is considered to be guilty.
Type II error occurs during the process of hypothesis testing when the null hypothesis is not rejected even though it is inaccurate. This Kind of error is also called a False-negative which means even though the statements are false and inaccurate, it still says it is correct and doesn’t reject it. For example, a person is guilty, but in court, he has been proven innocent where he is guilty, so this is a Type II error.
The level of significance is majorly used to measure the confidence with which a null hypothesis can be rejected. It is the value with which one can reject the null hypothesis which is H0. The level of significance gauges whether the hypothesis testing is significant or not.
P-value stands for probability value, which tells us the probability or likelihood to find the set of observations when the null hypothesis is true using statistical tests. The main purpose is to check the significance of the statistical statement.
A higher P-value indicates that the testing is not statistically significant. For example, a P value greater than 0.05 is considered to be having higher P value. A higher P-value also means that our evidence and proofs are not strong enough to influence the population.
In hypothesis testing, each step is responsible for getting the outcomes and the results, whether it is the selection of statistical tests or working on data, each step contributes towards the better consequences of the hypothesis testing. It is always a recommendable step when planning for predicting the outcomes and trying to experiment with the sample; hypothesis testing is a useful concept to apply.
We can test a hypothesis by selecting a correct hypothetical test and, based on those getting results.
Many statistical tests are used for hypothetical testing which includes Z-test, T-test, etc.
Hypothesis helps us in doing various experiments and working on a specific research topic to predict the results.
The null and alternative hypothesis, data collection, selecting a statistical test, selecting significance value, calculating p-value, check your findings.
In simple words, parametric tests are purely based on assumptions whereas non-parametric tests are based on data that is collected and acquired from a sample.
Gauri Guglani works as a Data Analyst at Deloitte Consulting. She has done her major in Information Technology and holds great interest in the field of data science. She owns her technical skills as well as managerial skills and also is great at communicating. Since her undergraduate, Gauri has developed a profound interest in writing content and sharing her knowledge through the manual means of blog/article writing. She loves writing on topics affiliated with Statistics, Python Libraries, Machine Learning, Natural Language processes, and many more.
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Statistics By Jim
Making statistics intuitive
By Jim Frost 10 Comments
A test statistic assesses how consistent your sample data are with the null hypothesis in a hypothesis test. Test statistic calculations take your sample data and boil them down to a single number that quantifies how much your sample diverges from the null hypothesis. As a test statistic value becomes more extreme, it indicates larger differences between your sample data and the null hypothesis.
When your test statistic indicates a sufficiently large incompatibility with the null hypothesis, you can reject the null and state that your results are statistically significant—your data support the notion that the sample effect exists in the population . To use a test statistic to evaluate statistical significance, you either compare it to a critical value or use it to calculate the p-value .
Statisticians named the hypothesis tests after the test statistics because they’re the quantity that the tests actually evaluate. For example, t-tests assess t-values, F-tests evaluate F-values, and chi-square tests use, you guessed it, chi-square values.
In this post, learn about test statistics, how to calculate them, interpret them, and evaluate statistical significance using the critical value and p-value methods.
Each test statistic has its own formula. I present several common test statistics examples below. To see worked examples for each one, click the links to my more detailed articles.
T-value for 1-sample t-test | Take the sample mean, subtract the hypothesized mean, and divide by the . | |
T-value for 2-sample t-test | Take one sample mean, subtract the other, and divide by the pooled standard deviation. | |
F-value for F-tests and ANOVA | Calculate the ratio of two . | |
Chi-squared value (χ ) for a Chi-squared test | Sum the squared differences between observed and expected values divided by the expected values. |
In the formulas above, it’s helpful to understand the null condition and the test statistic value that occurs when your sample data match that condition exactly. Also, it’s worthwhile knowing what causes the test statistics to move further away from the null value, potentially becoming significant. Test statistics are statistically significant when they exceed a critical value.
All these test statistics are ratios, which helps you understand their null values.
When a t-value equals 0, it indicates that your sample data match the null hypothesis exactly.
For a 1-sample t-test, when the sample mean equals the hypothesized mean, the numerator is zero, which causes the entire t-value ratio to equal zero. As the sample mean moves away from the hypothesized mean in either the positive or negative direction, the test statistic moves away from zero in the same direction.
A similar case exists for 2-sample t-tests. When the two sample means are equal, the numerator is zero, and the entire test statistic ratio is zero. As the two sample means become increasingly different, the absolute value of the numerator increases, and the t-value becomes more positive or negative.
Related post : How T-tests Work
When an F-value equals 1, it indicates that the two variances in the numerator and denominator are equal, matching the null hypothesis.
As the numerator and denominator become less and less similar, the F-value moves away from one in either direction.
Related post : The F-test in ANOVA
When a chi-squared value equals 0, it indicates that the observed values always match the expected values. This condition causes the numerator to equal zero, making the chi-squared value equal zero.
As the observed values progressively fail to match the expected values, the numerator increases, causing the test statistic to rise from zero.
Related post : How a Chi-Squared Test Works
You’ll never see a test statistic that equals the null value precisely in practice. However, trivial differences been sample values and the null value are not uncommon.
Test statistics are unitless. This fact can make them difficult to interpret on their own. You know they evaluate how well your data agree with the null hypothesis. If your test statistic is extreme enough, your data are so incompatible with the null hypothesis that you can reject it and conclude that your results are statistically significant. But how does that translate to specific values of your test statistic? Where do you draw the line?
For instance, t-values of zero match the null value. But how far from zero should your t-value be to be statistically significant? Is 1 enough? 2? 3? If your t-value is 2, what does it mean anyway? In this case, we know that the sample mean doesn’t equal the null value, but how exceptional is it? To complicate matters, the dividing line changes depending on your sample size and other study design issues.
Similar types of questions apply to the other test statistics too.
To interpret individual values of a test statistic, we need to place them in a larger context. Towards this end, let me introduce you to sampling distributions for test statistics!
Performing a hypothesis test on a sample produces a single test statistic. Now, imagine you carry out the following process:
This process produces the distribution of test statistic values that occurs when the effect does not exist in the population (i.e., the null hypothesis is true). Statisticians refer to this type of distribution as a sampling distribution, a kind of probability distribution.
Why would we need this type of distribution?
It provides the larger context required for interpreting a test statistic. More specifically, it allows us to compare our study’s single test statistic to values likely to occur when the null is true. We can quantify our sample statistic’s rareness while assuming the effect does not exist in the population. Now that’s helpful!
Fortunately, we don’t need to collect many random samples to create this distribution! Statisticians have developed formulas allowing us to estimate sampling distributions for test statistics using the sample data.
To evaluate your data’s compatibility with the null hypothesis, place your study’s test statistic in the distribution.
Related post : Understanding Probability Distributions
Suppose our t-test produces a t-value of two. That’s our test statistic. Let’s see where it fits in.
The sampling distribution below shows a t-distribution with 20 degrees of freedom, equating to a 1-sample t-test with a sample size of 21. The distribution centers on zero because it assumes the null hypothesis is correct. When the null is true, your analysis is most likely to obtain a t-value near zero and less likely to produce t-values further from zero in either direction.
The sampling distribution indicates that our test statistic is somewhat rare when we assume the null hypothesis is correct. However, the chances of observing t-values from -2 to +2 are not totally inconceivable. We need a way to quantify the likelihood.
From this point, we need to use the sampling distributions’ ability to calculate probabilities for test statistics.
Related post : Sampling Distributions Explained
The significance level uses critical values to define how far the test statistic must be from the null value to reject the null hypothesis. When the test statistic exceeds a critical value, the results are statistically significant.
The percentage of the area beneath the sampling distribution curve that is shaded represents the probability that the test statistic will fall in those regions when the null is true. Consequently, to depict a significance level of 0.05, I’ll shade 5% of the sampling distribution furthest away from the null value.
The two shaded areas are equidistant from the null value in the center. Each region has a likelihood of 0.025, which sums to our significance level of 0.05. These shaded areas are the critical regions for a two-tailed hypothesis test. Let’s return to our example t-value of 2.
Related post : What are Critical Values?
In this example, the critical values are -2.086 and +2.086. Our test statistic of 2 is not statistically significant because it does not exceed the critical value.
Other hypothesis tests have their own test statistics and sampling distributions, but their processes for critical values are generally similar.
Learn how to find critical values for test statistics using tables:
Related post : Understanding Significance Levels
P-values are the probability of observing an effect at least as extreme as your sample’s effect if you assume no effect exists in the population.
Test statistics represent effect sizes in hypothesis tests because they denote the difference between your sample effect and no effect —the null hypothesis. Consequently, you use the test statistic to calculate the p-value for your hypothesis test.
The above p-value definition is a bit tortuous. Fortunately, it’s much easier to understand how test statistics and p-values work together using a sampling distribution graph.
Let’s use our hypothetical test statistic t-value of 2 for this example. However, because I’m displaying the results of a two-tailed test, I need to use t-values of +2 and -2 to cover both tails.
Related post : One-tailed vs. Two-Tailed Hypothesis Tests
The graph below displays the probability of t-values less than -2 and greater than +2 using the area under the curve. This graph is specific to our t-test design (1-sample t-test with N = 21).
The sampling distribution indicates that each of the two shaded regions has a probability of 0.02963—for a total of 0.05926. That’s the p-value! The graph shows that the test statistic falls within these areas almost 6% of the time when the null hypothesis is true in the population.
While this likelihood seems small, it’s not low enough to justify rejecting the null under the standard significance level of 0.05. P-value results are always consistent with the critical value method. Learn more about using test statistics to find p values .
While test statistics are a crucial part of hypothesis testing, you’ll probably let your statistical software calculate the p-value for the test. However, understanding test statistics will boost your comprehension of what a hypothesis test actually assesses.
Related post : Interpreting P-values
July 5, 2024 at 8:21 am
“As the observed values progressively fail to match the observed values, the numerator increases, causing the test statistic to rise from zero”.
Sir, this sentence is written in the Chi-squared Test heading. There the observed value is written twice. I think the second one to be replaced with ‘expected values’.
July 5, 2024 at 4:10 pm
Thanks so much, Dr. Raj. You’re correct about the typo and I’ve made the correction.
May 9, 2024 at 1:40 am
Thank you very much (great page on one and two-tailed tests)!
May 6, 2024 at 12:17 pm
I would like to ask a question. If only positive numbers are the possible values in a sample (e.g. absolute values without 0), is it meaningful to test if the sample is significantly different from zero (using for example a one sample t-test or a Wilcoxon signed-rank test) or can I assume that if given a large enough sample, the result will by definition be significant (even if a small or very variable sample results in a non-significant hypothesis test).
Thank you very much,
May 6, 2024 at 4:35 pm
If you’re talking about the raw values you’re assessing using a one-sample t-test, it doesn’t make sense to compare them to zero given your description of the data. You know that the mean can’t possibly equal zero. The mean must be some positive value. Yes, in this scenario, if you have a large enough sample size, you should get statistically significant results. So, that t-test isn’t tell you anything that you don’t already know!
However, you should be aware of several things. The 1-sample test can compare your sample mean to values other than zero. Typically, you’ll need to specify the value of the null hypothesis for your software. This value is the comparison value. The test determines whether your sample data provide enough evidence to conclude that the population mean does not equal the null hypothesis value you specify. You’ll need to specify the value because there is no obvious default value to use. Every 1-sample t-test has its subject-area context with a value that makes sense for its null hypothesis value and it is frequently not zero.
I suspect that you’re getting tripped up with the fact that t-tests use a t-value of zero for its null hypothesis value. That doesn’t mean your 1-sample t-test is comparing your sample mean to zero. The test converts your data to a single t-value and compares the t-value to zero. But your actual null hypothesis value can be something else. It’s just converting your sample to a standardized value to use for testing. So, while the t-test compares your sample’s t-value to zero, you can actually compare your sample mean to any value you specify. You need to use a value that makes sense for your subject area.
I hope that makes sense!
May 8, 2024 at 8:37 am
Thank you very much Jim, this helps a lot! Actually, the value I would like to compare my sample to is zero, but I just couldn’t find the right way to test it apparently (it’s about EEG data). The original data was a sample of numbers between -1 and +1, with the question if they are significantly different from zero in either direction (in which case a one sample t-test makes sense I guess, since the sample mean can in fact be zero). However, since a sample mean of 0 can also occur if half of the sample differs in the negative, and the other half in the positive direction, I also wanted to test if there is a divergence from 0 in ‘absolute’ terms – that’s how the absolute valued numbers came about (I know that absolute values can also be zero, but in this specific case, they were all positive numbers) And a special thanks for the last paragraph – I will definitely keep in mind, it is a potential point of confusion.
May 8, 2024 at 8:33 pm
You can use a 1-sample t test for both cases but you’ll need to set them up slightly different. To detect a positive or negative difference from zero, use a 2-tailed test. For the case with absolute values, use a one-tailed test with a critical region in the positive end. To learn more, read about One- and Two-Tailed Tests Explained . Use zero for the comparison value in both cases.
February 12, 2024 at 1:00 am
Very helpful and well articulated! Thanks Jim 🙂
September 18, 2023 at 10:01 am
Thank you for brief explanation.
July 25, 2022 at 8:32 am
the content was helpful to me. thank you
Formal hypothesis testing is perhaps the most prominent and widely-employed form of statistical analysis. It is sometimes seen as the most rigorous and definitive part of a statistical analysis, but it is also the source of many statistical controversies. The currently-prevalent approach to hypothesis testing dates to developments that took place between 1925 and 1940, especially the work of Ronald Fisher , Jerzy Neyman , and Egon Pearson .
In recent years, many prominent statisticians have argued that less emphasis should be placed on the formal hypothesis testing approaches developed in the early twentieth century, with a correspondingly greater emphasis on other forms of uncertainty analysis. Our goal here is to give an overview of some of the well-established and widely-used approaches for hypothesis testing. We will also provide some perspectives on how these tools can be effectively used, and discuss their limitations. We will also discuss some new approaches to hypothesis testing that may eventually come to be as prominent as these classical approaches.
A falsifiable hypothesis is a statement, or hypothesis, that can be contradicted with evidence. In empirical (data-driven) research, this evidence will always be obtained through the data. In statistical hypothesis testing, the hypothesis that we formally test is called the null hypothesis . The alternative hypothesis is a second hypothesis that is our proposed explanation for what happens if the null hypothesis is wrong.
The key element of a statistical hypothesis test is the test statistic , which (like any statistic) is a function of the data. A test statistic takes our entire dataset, and reduces it to one number. This one number ideally should contain all the information in the data that is relevant for assessing the two hypotheses of interest, and exclude any aspects of the data that are irrelevant for assessing the two hypotheses. The test statistic measures evidence against the null hypothesis. Most test statistics are constructed so that a value of zero represents the lowest possible level of evidence against the null hypothesis. Test statistic values that deviate from zero represent greater levels of evidence against the null hypothesis. The larger the magnitude of the test statistic, the stronger the evidence against the null hypothesis.
A major theme of statistical research is to devise effective ways to construct test statistics. Many useful ways to do this have been devised, and there is no single approach that is always the best. In this introductory course, we will focus on tests that starting with an estimate of a quantity that is relevant for assessing the hypotheses, then proceed by standardizing this estimate by dividing it by its standard error. This approach is sometimes referred to as “Wald testing”, after Abraham Wald .
As a basic example, let’s consider risk perception related to COVID-19. As you will see below, hypothesis testing can appear at first to be a fairly elaborate exercise. Using this example, we describe each aspect of this exercise in detail below.
The data shown below are simulated but are designed to reflect actual surveys conducted in the United States in March of 2020. Partipants were asked whether they perceive that they have a substantial risk of dying if they are infected with the novel coronavirus. The number of people stating each response, stratified on age, are shown below (only two age groups are shown):
High risk | Not high risk | |
---|---|---|
Age < 30 | 25 | 202 |
Age 60-69 | 30 | 124 |
Each subject’s response is binary – they either perceive themselves to be high risk, or not to be at high risk. When working with this type of data, we are usually interested in the proportion of people who provide each response within each stratum (age group). These are conditional proportions, conditioning on the age group. The numerical values of the conditional proportions are given below:
High risk | Not high risk | |
---|---|---|
Age < 30 | 0.110 | 0.890 |
Age 60-69 | 0.195 | 0.805 |
There are four conditional proportions in the table above – the proportion of younger people who perceive themselves to be at higher risk, 0.110=25/(25+202); the proportion of younger people who do not perceive themselves to be at high risk, 0.890=202/(25+202); the proportion of older people who perceive themselves to be at high risk 0.195=30/(30+124); and the proportion of older people who do not perceive themselves to be at high risk, 0.805=124/(30+124).
The trend in the data is that younger people perceive themselves to be at lower risk of dying than older people, by a difference of 0.195-0.110=0.085 (in terms of proportions). But is this trend only present in this sample, or is it generalizable to a broader population (say the entire US population)? That is the goal of conducting a statistical hypothesis test in this setting.
Corresponding to our data above is the unobserved population structure, which we can denote as follows
High risk | Not high risk | |
---|---|---|
Age < 30 | \(p\) | \(1-p\) |
Age 60-69 | \(q\) | \(1-q\) |
The symbols \(p\) and \(q\) in the table above are population parameters . These are quantitites that we do not know, and wish to assess using the data. In this case, our null hypothesis can be expressed as the statement \(p = q\) . We can estimate \(p\) using the sample proportion \(\hat{p} = 0.110\) , and similarly estimate \(q\) using \(\hat{q} = 0.195\) . However these estimates do not immediately provide us with a way of expressing the evidence relating to the hypothesis that \(p=q\) . This is provided by the test statistic.
As noted above, a test statistic is a reduction of the data to one number that captures all of the relevant information for assessing the hypotheses. A natural first choice for a test statistic here would be the difference in sample proportions between the two age groups, which is 0.195 - 0.110 = 0.085. There is a difference of 0.085 between the perceived risks of death in the younger and older age groups.
The difference in rates (0.085) does not on its own make a good test statistic, although it is a good start toward obtaining one. The reason for this is that the evidence underlying this difference in rates depends also on the absolute rates (0.110 and 0.195), and on the sample sizes (227 and 154). If we only know that the difference in rates is 0.085, this is not sufficient to evaluate the hypothesis in a statistical manner. A given difference in rates is much stronger evidence if it is obtained from a larger sample. If we have a difference of 0.085 with a very large sample, say one million people, then we should be almost certain that the true rates differ (i.e. the data are highly incompatiable with the hypothesis that \(p=q\) ). If we have the same difference in rates of 0.085, but with a small sample, say 50 people per age group, then there would be almost no evidence for a true difference in the rates (i.e. the data are compatiable with the hypothesis \(p=q\) ).
To address this issue, we need to consider the uncertainty in the estimated rate difference, which is 0.085. Recall that the estimated rate difference is obtained from the sample and therefore is almost certain to deviate somewhat from the true rate difference in the population (which is unknown). Recall from our study of standard errors that the standard error for an estimated proportion is \(\sqrt{p(1-p)/n}\) , where \(p\) is the outcome probability (here the outcome is that a person perceives a high risk of dying), and \(n\) is the sample size.
In the present analysis, we are comparing two proportions, so we have two standard errors. The estimated standard error for the younger people is \(\sqrt{0.11\cdot 0.89/227} \approx 0.021\) . The estimated standard error for the older people is \(\sqrt{0.195\cdot 0.805/154} \approx 0.032\) . Note that both standard errors are estimated, rather than exact, because we are plugging in estimates of the rates (0.11 and 0.195). Also note that the standard error for the rate among older people is greater than that for younger people. This is because the sample size for older people is smaller, and also because the estimated rate for older people is closer to 1/2.
In our previous discussion of standard errors, we saw how standard errors for independent quantities \(A\) and \(B\) can be used to obtain the standard error for the difference \(A-B\) . Applying that result here, we see that the standard error for the estimated difference in rates 0.195-0.11=0.085 is \(\sqrt{0.021^2 + 0.032^2} \approx 0.038\) .
The final step in constructing our test statistic is to construct a Z-score from the estimated difference in rates. As with all Z-scores, we proceed by taking the estimated difference in rates, and then divide it by its standard error. Thus, we get a test statistic value of \(0.085 / 0.038 \approx 2.24\) .
A test statistic value of 2.24 is not very close to zero, so there is some evidence against the null hypothesis. But the strength of this evidence remains unclear. Thus, we must consider how to calibrate this evidence in a way that makes it more interpretable.
By the central limit theorem (CLT), a Z-score approximately follows a normal distribution. When the null hypothesis holds, the Z-score approximately follows the standard normal distribution (recall that a standard normal distribution is a normal distribution with expected value equal to 0 and variance equal to 1). If the null hypothesis does not hold, then the test statistic continues to approximately follow a normal distribution, but it is not the standard normal distribution.
A test statistic of zero represents the least possible evidence against the null hypothesis. Here, we will obtain a test statistic of zero when the two proportions being compared are identical, i.e. exactly the same proportions of younger and older people perceive a substantial risk of dying from a disease. Even if the test statistic is exactly zero, this does not guarantee that the null hypothesis is true. However it is the least amount of evidence that the data can present against the null hypothesis.
In a hypothesis testing setting using normally-distrbuted Z-scores, as is the case here (due to the CLT), the standard normal distribution is the reference distribution for our test statistic. If the Z-score falls in the center of the reference distribution, there is no evidence against the null hypothesis. If the Z-score falls into either tail of the reference distribution, then there is evidence against the null distribution, and the further into the tails of the reference distribution the Z-score falls, the greater the evidence.
The most conventional way to quantify the evidence in our test statistic is through a probability called the p-value . The p-value has a somewhat complex definition that many people find difficult to grasp. It is the probability of observing as much or more evidence against the null hypothesis as we actually observe, calculated when the null hypothesis is assumed to be true. We will discuss some ways to think about this more intuitively below.
For our purposes, “evidence against the null hypothesis” is reflected in how far into the tails of the reference distribution the Z-score (test statistic) falls. We observed a test statistic of 2.24 in our COVID risk perception analysis. Recall that due to the “empirical rule”, 95% of the time, a draw from a standard normal distribution falls between -2 and 2. Thus, the p-value must be less than 0.05, since 2.24 falls outside this interval. The p-value can be calculated using a computer, in this case it happens to be approximately 0.025.
As stated above, the p-value tells us how likely it would be for us to obtain as much evidence against the the null hypothesis as we observed in our actual data analysis, if we were certain that the null hypothesis were true. When the null hypothesis holds, any evidence against the null hypothesis is spurious. Thus, we will want to see stronger evidence against the null from our actual analysis than we would see if we know that the null hypothesis were true. A smaller p-value therefore reflects more evidence against the null hypothesis than a larger p-value.
By convention, p-values of 0.05 or smaller are considered to represent sufficiently strong evidence against the null hypothesis to make a finding “statistically significant”. This threshold of 0.05 was chosen arbitrarily 100 years ago, and there is no objective reason for it. In recent years, people have argued that either a lesser or a greater p-value threshold should be used. But largely due to convention, the practice of deeming p-values smaller than 0.05 to be statistically significant continues.
Here is a restatement of the above discussion, using slightly different language. In our analysis of COVID risk perceptions, we found a difference in proportions of 0.085 between younger and older subjects, with younger people perceiving a lower risk of dying. This is a difference based on the sample of data that we observed, but what we really want to know is whether there is a difference in COVID risk perception in the population (say, all US adults).
Suppose that in fact there is no difference in risk perception between younger and older people. For instance, suppose that in the population, 15% of people believe that they have a substantial risk of dying should they become infected with the novel coronavirus, regardless of their age. Even though the rates are equal in this imaginary population (both being 15%), the rates in our sample would typically not be equal. Around 3% of the time (0.024=2.4% to be exact), if the rates are actually equal in the population, we would see a test statistic that is 2.4 or larger. Since 3% represents a fairly rare event, we can conclude that our observed data are not compatible with the null hypothesis. We can also say that there is statistically significant evidence against the null hypothesis, and that we have “rejected” the null hypothesis at the 3% level.
In this data analysis, as in any data analysis, we cannot confirm definitively that the alternative hypothesis is true. But based on our data and the analysis performed above, we can claim that there is substantial evidence against the null hypothesis, using standard criteria for what is considered to be “substantial evidence”.
A very common setting where hypothesis testing is used arises when we wish to compare the means of a quantitative measurement obtained for two populations. Imagine, for example, that we have two ways of manufacturing a battery, and we wish to assess which approach yields batteries that are longer-lasting in actual use. To do this, suppose we obtain data that tells us the number of charge cycles that were completed in 200 batteries of type A, and in 300 batteries of type B. For the test developed below to be meaningful, the data must be independent and identically distributed samples.
The raw data for this study consists of 500 numbers, but it turns out that the most relevant information from the data is contained in the sample means and sample standard deviations computed within each battery type. Note that this is a huge reduction in complexity, since we started with 500 measurements and are able to summarize this down to just four numbers.
Suppose the summary statistics are as follows, where \(\bar{x}\) , \(\hat{\sigma}_x\) , and \(n\) denote the sample mean, sample standard deviation, and sample size, respectively.
Type | \(\bar{x}\) | \(\hat{\sigma}_x\) | \(n\) |
---|---|---|---|
420 | 70 | 200 | |
403 | 90 | 300 |
The simplest measure comparing the two manufacturing approaches is the difference 420 - 403 = 17. That is, batteries of type A tend to have 17 more charge cycles compared to batteries of type B. This difference is present in our sample, but is it also true that the entire population of type A batteries has more charge cycles than the entire population of type B batteries? That is the goal of conducting a hypothesis test.
The next step in the present analysis is to divide the mean difference, which is 17, by its standard error. As we have seen, the standard error of the mean, or SEM, is \(\sigma/n\) , where \(\sigma\) is the standard deviation and \(n\) is the sample size. Since \(\sigma\) is almost never known, we plug in its estimate \(\hat{\sigma}\) . For the type A batteries, the estimated SEM is thus \(70/\sqrt{200} \approx 4.95\) , and for the type B batteries the estimated SEM is \(90/\sqrt{300} \approx 5.2\) .
Since we are comparing two estimated means that are obtained from independent samples, we can pool the standard deviations to obtain an overall standard deviation of \(\sqrt{4.95^2 + 5.2^2} \approx 7.18\) . We can now obtain our test statistic \(17/7.18 \approx 2.37\) .
The test statistic can be calibrated against a standard normal reference distribution. The probability of observing a standard normal value that is greater in magnitude than 2.37 is 0.018 (this can be obtained from a computer). This is the p-value, and since it is smaller than the conventional threshold of 0.05, we can claim that there is a statistically significant difference between the average number of charge cycles for the two types of batteries, with the A batteries having more charge cycles on average.
The analysis illustrated here is called a two independent samples Z-test , or just a two sample Z-test . It may be the most commonly employed of all statistical tests. It is also common to see the very similar two sample t-test , which is different only in that it uses the Student t distribution rather than the normal (Gaussian) distribution to calculate the p-values. In fact, there are quite a few minor variations on this testing framework, including “one sided” and “two sided” tests, and tests based on different ways of pooling the variance. Due to the CLT, if the sample size is modestly large (which is the case here), the results of all of these tests will be almost identical. For simplicity, we only cover the Z-test in this course.
The tests for comparing proportions and means presented above are quite similar in many ways. To provide one more example of a hypothesis test that is somewhat different, we consider a test for a correlation coefficient.
Recall that the sample correlation coefficient \(\hat{r}\) is used to assess the relationship, or association, between two quantities X and Y that are measured on the same units. For example, we may ask whether two biomarkers, serum creatinine and D-dimer, are correlated with each other. These biomarkers are both commonly used in medical settings and are obtained using blood tests. D-dimer is used to assess whether a person has blood clots, and serum creatinine is used to measure kidney performance.
Suppose we are interested in whether there is a correlation in the population between D-dimer and serum creatinine. The population correlation coefficient between these two quantitites can be denoted \(r\) . Our null hypothesis is \(r=0\) . Suppose that we observe a sample correlation coefficient of \(\hat{r}=0.15\) , using an independent and identically distributed sample of pairs \((x, y)\) , where \(x\) is a D-dimer measurement and \(y\) is a serum creatinine measurement. Are these data consistent with the null hypothesis?
As above, we proceed by constructing a test statistic by taking the estimated statistic and dividing it by its standard error. The approximate standard error for \(\hat{r}\) is \(1/\sqrt{n}\) , where \(n\) is the sample size. The test statistic is therefore \(\sqrt{n}\cdot \hat{r} \approx 1.48\) .
We now calibrate this test statistic by comparing it to a standard normal reference distribution. Recall from the empirical rule that 5% of the time, a standard normal value falls outside the interval (-2, 2). Therefore, if the test statistic is smaller than 2 in magnitude, as is the case here, its p-value is greater than 0.05. Thus, in this case we know that the p-value will exceed 0.05 without calculating it, and therefore there is no basis for claiming that D-dimer and serum creatinine levels are correlated in this population.
A p-value is the most common way of calibrating evidence. Smaller p-values indicate stronger evidence against a null hypothesis. By convention, if the p-value is smaller than some threshold, usually 0.05, we reject the null hypothesis and declare a finding to be “statistically significant”. How can we understand more deeply what this means? One major concern should be obtaining a small p-value when the null hypothesis is true. If the null hypothesis is true, then it is incorrect to reject it. If we reject the null hypothesis, we are making a false claim. This can never be prevented with complete certainty, but we would like to have a very clear understanding of how likely it is to reject the null hypothesis when the null hypothesis is in fact true.
P-values have a special property that when the null distribution is true, the probability of observing a p-value smaller than 0.05 is 0.05 (5%). In fact, the probability of observing a p-value smaller than \(t\) is equal to \(t\) , for any threshold \(t\) . For example, the probability of observing a p-value smaller than 0.1, when the null hypothesis is true, is 10%.
This fact gives a more concrete understanding of how strong the evidence is for a particular p-value. If we always reject the null hypothesis when the p-value is 0.1 or smaller, then over the long run we will reject the null hypothesis 10% of the time when the null hypothesis is true. If we always reject the null hypothesis when the p-value is 0.05 or smaller, then over the long run we will reject the null hypothesis 5% of the time when the null hypothesis is true.
The approach to hypothesis testing discussed above largely follows the framework developed by RA Fisher around 1925. Note that although we mentioned the alternative hypothesis above, we never actually used it. A more elaborate approach to hypothesis testing was developed somewhat later by Egon Pearson and Jerzy Neyman. The “Neyman-Pearson” approach to hypothesis testing is even more formal than Fisher’s approach, and is most suited to highly planned research efforts in which the study is carefully designed, then executed. While ideally all research projects should be carried out this way, in reality we often conduct research using data that are already available, rather than using data that are specifically collected to address the research question.
Neyman-Pearson hypothesis testing involves specifying an alternative hypothesis that we anticipate encountering. Usually this alternative hypothesis represents a realistic guess about what we might find once the data are collected. In each of the three examples above, imagine that the data are not yet collected, and we are asked to specify an alternative hypothesis. We may arrive at the following:
In comparing risk perceptions for COVID, we may anticipate that older people will perceive a 30% risk of dying, and younger people will anticipate a 5% risk of dying.
In comparing the number of charge cycles for two types of batteries, we may anticipate that batter type A will have on average 500 charge cycles, and battery type B will have on average 400 charge cycles.
In assessing the correlation between D-dimer and serum creatinine levels, we may anticipate a correlation of 0.3.
Note that none of the numbers stated here are data-driven – they are specified before any data are collected, so they do not match the results from the data, which were collected only later. These alternative hypotheses are all essentially speculations, based perhaps on related data or theoretical considerations.
There are several benefits of specifying an explicit alternative hypothesis, as done here, even though it is not strictly necessary and can be avoided entirely by adopting Fisher’s approach to hypothesis testing. One benefit of specifying an alternative hypothesis is that we can use it to assess the power of our planned study, which can in turn inform the design of the study, in particular the sample size. The power is the probability of rejecting the null hypothesis when the alternative hypothesis is true. That is, it is the probability of discovering something real. The power should be contrasted with the level of a hypothesis test, which is the probability of rejecting the null hypothesis when the null hypothesis is true. That is, the level is the probability of “discovering” something that is not real.
To calculate the power, recall that for many of the test statistics that we are considering here, the test statistic has the form \(\hat{\theta}/{\rm SE}(\hat{\theta})\) , where \(\hat{\theta}\) is an estimate. For example, \(\hat{\theta}\) ) may be the correlation coefficient between D-dimer and serum creatinine levels. As stated above, the power is the probability of rejecting the null hypothesis when the alternative hypothesis is true. Suppose we decide to reject the null hypothesis when the test statistic is greater than 2, which is approximately equivalent to rejecting the null hypothesis when the p-value is less than 0.05. The following calculation tells us how to obtain the power in this setting:
Under the alternative hypothesis, \(\sqrt{n}(\hat{r} - r)\) approximately follows a standard normal distribution. Therefore, if \(r\) and \(n\) are given, we can easily use the computer to obtain the probability of observing a value greater than \(2 - \sqrt{n}r\) . This gives us the power of the test. For example, if we anticipate \(r=0.3\) and plan to collect data for \(n=100\) observations, the power is 0.84. This is generally considered to be good power – if the true value of \(r\) is in fact 0.3, we would reject the null hypothesis 84% of the time.
A study usually has poor power because it has too small of a sample size. Poorly powered studies can be very misleading, but since large sample sizes are expensive to collect, a lot of research is conducted using sample sizes that yield moderate or even low power. If a study has low power, it is unlikely to reject the null hypothesis even when the alternative hypothesis is true, but it remains possible to reject the null hypothesis when the null hypothesis is true (usually this probability is 5%). Therefore the most likely outcome of a poorly powered study may be an incorrectly rejected null hypothesis.
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We introduce a projection-based test for assessing logistic regression models using the empirical residual marked empirical process and suggest a model-based bootstrap procedure to calculate critical values. We comprehensively compare this test and Stute and Zhu's test with several commonly used goodness-of-fit (GoF) tests: the Hosmer-Lemeshow test, modified Hosmer-Lemeshow test, Osius ...
Statistical tests are used in hypothesis testing. They can be used to: determine whether a predictor variable has a statistically significant relationship with an outcome variable. estimate the difference between two or more groups. Statistical tests assume a null hypothesis of no relationship or no difference between groups. Then they ...
Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables. Let's discuss few examples of statistical hypothesis from real-life - ...
Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test. Step 4: Decide whether to reject or fail to reject your null hypothesis. Step 5: Present your findings. Other interesting articles. Frequently asked questions about hypothesis testing.
A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis. The Two Types of Statistical Hypotheses. To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data. There are two types of ...
Statistical hypothesis testing is a key technique of both frequentist inference and Bayesian inference, although the two types of inference have notable differences. Statistical hypothesis tests define a procedure that controls (fixes) the probability of incorrectly deciding that a default position (null hypothesis) is incorrect. The procedure ...
Make a Decision: Based on the statistical test results, decide whether to reject the null hypothesis or fail to reject it. Report the Results: Summarize and present the outcomes in your report's results and discussion sections. While the specifics of these steps can vary depending on the research context and the data type, the fundamental ...
Hypothesis testing is a crucial procedure to perform when you want to make inferences about a population using a random sample. These inferences include estimating population properties such as the mean, differences between means, proportions, and the relationships between variables. This post provides an overview of statistical hypothesis testing.
In data science and statistics, hypothesis testing is an important step as it involves the verification of an assumption that could help develop a statistical parameter. For instance, a researcher establishes a hypothesis assuming that the average of all odd numbers is an even number. ... All in all, there are 2 most common types of hypothesis ...
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted \(H_0\) while the alternative hypothesis is usually denoted \(H_1\). An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...
Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid. A null hypothesis and an alternative ...
Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. ... There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater ...
HYPOTHESIS TESTING. A clinical trial begins with an assumption or belief, and then proceeds to either prove or disprove this assumption. In statistical terms, this belief or assumption is known as a hypothesis. Counterintuitively, what the researcher believes in (or is trying to prove) is called the "alternate" hypothesis, and the opposite ...
Here's a closer look at the three fundamental types of hypothesis tests: 1. Z-Test: The z-test is a statistical method primarily employed when comparing means from two datasets, particularly when the population standard deviation is known. Its main objective is to ascertain if the means are statistically equivalent.
S.3 Hypothesis Testing. In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail. The general idea of hypothesis testing involves: Making an initial assumption. Collecting evidence (data).
A statistical hypothesis test may return a value called p or the p-value. This is a quantity that we can use to interpret or quantify the result of the test and either reject or fail to reject the null hypothesis. This is done by comparing the p-value to a threshold value chosen beforehand called the significance level.
To perform various analyses on the data, we need to choose a statistical test. There are various types of statistical tests available. Based on the wide spread of the data that is variance within the group or how different the data category is from one another that is variance without a group, we can proceed with our further research study. 4.
A test statistic assesses how consistent your sample data are with the null hypothesis in a hypothesis test. Test statistic calculations take your sample data and boil them down to a single number that quantifies how much your sample diverges from the null hypothesis. As a test statistic value becomes more extreme, it indicates larger ...
Outcomes and the Type I and Type II Errors. When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H0 and the decision to reject or not. The outcomes are summarized in the following table: ACTION. H0 is Actually True.
Hypothesis tests # Formal hypothesis testing is perhaps the most prominent and widely-employed form of statistical analysis. It is sometimes seen as the most rigorous and definitive part of a statistical analysis, but it is also the source of many statistical controversies. The currently-prevalent approach to hypothesis testing dates to developments that took place between 1925 and 1940 ...
Abstract. Statistical hypothesis testing is common in research, but a conventional understanding sometimes leads to mistaken application and misinterpretation. The logic of hypothesis testing presented in this article provides for a clearer understanding, application, and interpretation. Key conclusions are that (a) the magnitude of an estimate ...
Hypothesis testing is a statistical procedure in which a choice is made between a null hypothesis and an alternative hypothesis based on information in a sample. The end result of a hypotheses testing procedure is a choice of one of the following two possible conclusions: Reject H0. H 0. (and therefore accept Ha.
Components of a Formal Hypothesis Test. The null hypothesis is a statement about the value of a population parameter, such as the population mean (µ) or the population proportion (p).It contains the condition of equality and is denoted as H 0 (H-naught).. H 0: µ = 157 or H0 : p = 0.37. The alternative hypothesis is the claim to be tested, the opposite of the null hypothesis.
148. 4. Photo by Anna Nekrashevich from Pexels. Hypothesis testing is a common statistical tool used in research and data science to support the certainty of findings. The aim of testing is to answer how probable an apparent effect is detected by chance given a random data sample. This article provides a detailed explanation of the key concepts ...