Have a language expert improve your writing
Run a free plagiarism check in 10 minutes, generate accurate citations for free.
- Knowledge Base
Hypothesis Testing | A Step-by-Step Guide with Easy Examples
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:
- State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a or H 1 ).
- Collect data in a way designed to test the hypothesis.
- Perform an appropriate statistical test .
- Decide whether to reject or fail to reject your null hypothesis.
- Present the findings in your results and discussion section.
Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.
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.
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.
- H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.
Prevent plagiarism. Run a free check.
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 .
- an estimate of the difference in average height between the two groups.
- a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.
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 ).
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.
- Normal distribution
- Descriptive statistics
- Measures of central tendency
- Correlation coefficient
Methodology
- Cluster sampling
- Stratified sampling
- Types of interviews
- Cohort study
- Thematic analysis
Research bias
- Implicit bias
- Cognitive bias
- Survivorship bias
- Availability heuristic
- Nonresponse bias
- Regression to the mean
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.
Cite this Scribbr article
If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.
Bevans, R. (2023, June 22). Hypothesis Testing | A Step-by-Step Guide with Easy Examples. Scribbr. Retrieved September 9, 2024, from https://www.scribbr.com/statistics/hypothesis-testing/
Is this article helpful?
Rebecca Bevans
Other students also liked, choosing the right statistical test | types & examples, understanding p values | definition and examples, what is your plagiarism score.
User Preferences
Content preview.
Arcu felis bibendum ut tristique et egestas quis:
- Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris
- Duis aute irure dolor in reprehenderit in voluptate
- Excepteur sint occaecat cupidatat non proident
Keyboard Shortcuts
7.4.1 - hypothesis testing, five step hypothesis testing procedure section .
In the remaining lessons, we will use the following five step hypothesis testing procedure. This is slightly different from the five step procedure that we used when conducting randomization tests.
- Check assumptions and write hypotheses. The assumptions will vary depending on the test. In this lesson we'll be confirming that the sampling distribution is approximately normal by visually examining the randomization distribution. In later lessons you'll learn more objective assumptions. The null and alternative hypotheses will always be written in terms of population parameters; the null hypothesis will always contain the equality (i.e., \(=\)).
- Calculate the test statistic. Here, we'll be using the formula below for the general form of the test statistic.
- Determine the p-value. The p-value is the area under the standard normal distribution that is more extreme than the test statistic in the direction of the alternative hypothesis.
- Make a decision. If \(p \leq \alpha\) reject the null hypothesis. If \(p>\alpha\) fail to reject the null hypothesis.
- State a "real world" conclusion. Based on your decision in step 4, write a conclusion in terms of the original research question.
General Form of a Test Statistic Section
When using a standard normal distribution (i.e., z distribution), the test statistic is the standardized value that is the boundary of the p-value. Recall the formula for a z score: \(z=\frac{x-\overline x}{s}\). The formula for a test statistic will be similar. When conducting a hypothesis test the sampling distribution will be centered on the null parameter and the standard deviation is known as the standard error.
This formula puts our observed sample statistic on a standard scale (e.g., z distribution). A z score tells us where a score lies on a normal distribution in standard deviation units. The test statistic tells us where our sample statistic falls on the sampling distribution in standard error units.
Hypothesis Testing Cheat Sheet
Easy to understand info about the main types of hypothesis tests.
This FREE PDF cheat sheet will show you the differences between all of the main types of hypothesis testing. Including examples on when to use the, the equations used, and how to easily implement them in Excel!
You're Almost there! Please complete this form and click the button below to gain instant access.
We hate SPAM and promise to keep your email address safe. We will send occasional updates from FairlyNerdy.com. If you do not wish to receive these updates you may close this box, or unsubscribe at anytime
What You'll Get from This Free, Downloadable PDF:
- What is a "Z" test and the main types of "T" tests.
- What are examples of each of the tests in use
- What are the equations behind each of the hypothesis tests
- How you can quickly & easily implement each of them in Excel
Download Your FREE Hypothesis Testing Cheat Sheet Today!
Copyright © 2015 Your Site Here. Legal Information
- Skip to secondary menu
- Skip to main content
- Skip to primary sidebar
Statistics By Jim
Making statistics intuitive
Test Statistic: Definition, Types & Formulas
By Jim Frost 10 Comments
What is a Test Statistic?
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.
How to Find Test Statistics
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.
Formulas for Test Statistics
| 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. |
Understanding the Null Values and the Test Statistic Formulas
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.
T-Tests, Null = 0
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
F-tests including ANOVA, Null = 1
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
Chi-squared Tests, Null = 0
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.
Interpreting Test Statistics
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!
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:
- Assume the null hypothesis is true in the population.
- Repeat your study many times by drawing many random samples of the same size from this population.
- Perform the same hypothesis test on all these samples and save the test statistics.
- Plot the distribution of the test statistics.
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
Example of a Test Statistic in a Sampling Distribution
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
Test Statistics and Critical Values
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:
- T-distribution table
- Chi-square table
Related post : Understanding Significance Levels
Using Test Statistics to Find P-values
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
Share this:
Reader Interactions
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
Comments and Questions Cancel reply
Hypothesis Testing
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.
| 1. | |
| 2. | |
| 3. | |
| 4. | |
| 5. | |
| 6. | |
| 7. | |
| 8. |
What is Hypothesis Testing in Statistics?
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 Definition
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.
Null Hypothesis
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.
Alternative Hypothesis
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.
Hypothesis Testing P Value
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%.
Hypothesis Testing Critical region
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.
Hypothesis Testing Formula
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:
- z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation and n is the size of the sample.
- t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\). s is the sample standard deviation.
- \(\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}\). \(O_{i}\) is the observed value and \(E_{i}\) is the expected value.
We will learn more about these test statistics in the upcoming section.
Types of Hypothesis Testing
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.
Hypothesis Testing Z Test
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:
- One sample: z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
- Two samples: z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).
Hypothesis Testing t Test
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.
- One sample: t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\).
- Two samples: t = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\).
Hypothesis Testing Chi Square
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
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.
Two Tailed Hypothesis Testing
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 Steps
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:
- Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
- Step 2: Set up the alternative hypothesis.
- Step 3: Choose the correct significance level, \(\alpha\), and find the critical value.
- Step 4: Calculate the correct test statistic (z, t or \(\chi\)) and p-value.
- Step 5: Compare the test statistic with the critical value or compare the p-value with \(\alpha\) to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.
Hypothesis Testing Example
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.
Hypothesis Testing and Confidence Intervals
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.
Related Articles:
- Probability and Statistics
- Data Handling
Important Notes on Hypothesis Testing
- Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant.
- It involves the setting up of a null hypothesis and an alternate hypothesis.
- There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
- Hypothesis testing can be classified as right tail, left tail, and two tail tests.
Examples on Hypothesis Testing
- Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) > 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 5, s = 18. \(\alpha\) = 0.05 Using the t-distribution table, the critical value is 2.132 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
- Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. \(H_{0}\): \(\mu\) = 80, \(H_{1}\): \(\mu\) ≠ 80 \(\overline{x}\) = 88, \(\mu\) = 80, n = 36, \(\sigma\) = 10. \(\alpha\) = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) z = \(\frac{88-80}{\frac{10}{\sqrt{36}}}\) = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
- Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) < 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = \(\frac{82-90}{\frac{18}{\sqrt{6}}}\) t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.
go to slide go to slide go to slide
Book a Free Trial Class
FAQs on Hypothesis Testing
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.
What is the z Test in Hypothesis Testing?
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.
What is the t Test in Hypothesis Testing?
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.
What is the formula for z test in Hypothesis Testing?
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 = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).
What is the p Value in Hypothesis Testing?
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.
What is One Tail Hypothesis Testing?
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.
What is the Alpha Level in Two Tail 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 Formula
Hypothesis Testing Formula (Table of Contents)
What is the hypothesis testing formula.
Before deep diving into hypothesis testing, we need to understand the hypothesis in the first place. In simple language, an idea is an educated and informed guess about anything around you, which can be tested by experiment or observation.
For example, A new mobile variant will be accepted by people; new medicine might work or not, etc. So a hypothesis test is a statistical tool for testing the hypothesis we will make and whether that statement is full or not. We select a sample from the data set and test a hypothesis statement by determining the likelihood that a sample statistics. So If your results from that test are not significant, it means that the hypothesis is not valid.
Formula For Hypothesis Testing:
Start Your Free Investment Banking Course
Download Corporate Valuation, Investment Banking, Accounting, CFA Calculator & others
The z-test gives hypothesis testing. The formula for Z – Test is given as follows:
- X – Sample Mean
- U – Population Mean
- SD – Standard Deviation
- n – Sample size
But this is not as simple as it seems. To correctly perform the hypothesis test, you need to follow specific steps:
Step 1: First and foremost, to perform a hypothesis test, we must define the null and alternative hypotheses. An example of the null and alternate hypothesis is given by:
- H0 (null hypothesis): Mean value > 0
- For this, Alternate Hypothesis (Ha): Mean < 0
Step 2: Next thing we have to do is that we need to find out the level of significance. Generally, its value is 0.05 or 0.01
Step 3: Find the z-test value, also called test statistic, as stated in the above formula.
Step 4: Find the z score from the z table given the significance level and mean .
Step 5: Compare these two values, and if the test statistic is greater than the z score, reject the null hypothesis. You cannot reject the null hypothesis if the test statistic is less than the z score.
Examples of Hypothesis Testing Formula (With Excel Template)
Let’s take an example to understand the calculation of the Hypothesis Testing formula in a better manner.
Hypothesis Testing Formula – Example #1
Suppose you have been given the following parameters, and you have to find the Z value and state if you accept the null hypothesis or not:
Null hypothesis H0: Population Mean = 30
Alternate hypothesis Ha: Population Mean ≠ 30
Z – Test is calculated using the formula given below
Z = (X – U) / (SD / √n)
- Z – Test = ( 27 – 30 ) / (20 / SQRT(10))
- Z – Test = -0.474
Level of significance = 0.05
This is a Two tail test, so the probability lies on both sides of the distribution. So 0.025 on each side, and we will look at this value on the z table.
Source: https://www.z-table.com/
Since the significance level is 0.025 on each side, we need to find 0.025 in the z table. Once we see that value from the table, we must extract the z value.
If you see here, on the left side, the values of z are given, and in the top row, decimal places are given. So from that, we can say that 0.025 will give a z value of -1.96
So Z – Score = -1.96
We can reject the null hypothesis since the Z Test > Z Score.
Hypothesis Testing Formula – Example #2
Let’s say you are a school principal; you are claiming that the students in your school are above average intelligence. An analyst wants to double-check your claim and use hypothesis testing. He measures the IQ of all the students in the school and then takes a sample of 20 students. The following are the data points:
- Z – Test = (112 – 110)/ (15 / SQRT(20))
- Z – Test = 3.58
Null Hypothesis: Since population mean = 100,
- H0 : Mean = 100
- Ha: Mean > 100
Level of Significance = 0.05
Since the significance level is 0.05, we must find 1 – 0.05 = 0.95 in the z table. Once we find that value from the table, we must extract the z value.
Z – Table:
If you see here, on the left side, the values of z are given, and in the top row, decimal places are given. So from that, we can say that 0.95 lies between 1.64 to 1.65, mid-point of 1.645.
So Z Score = 1.645
Since the Z Test > Z Score, we can reject the null hypothesis and say students’ intelligence is above average.
Explanation
Everyone should remember that No hypothesis test is 100% correct, and there is always a chance of making an error. There is 2 type of errors that can arise in hypothesis testing: type I and type II.
Type 1: When the null hypothesis is true but rejected in the model. The level of significance gives the probability of this. So if the significance level is 0.05, there is a 5% chance that you will reject the true null.
Type 2: When the null hypothesis is not true but not rejected in the model. The probability of this is given the power of the test. Large sample size can help reduce the probability of this type of error, providing greater confidence in the model.
Relevance and Uses of Hypothesis Testing Formula
As discussed above, the hypothesis test helps the analyst test the statistical sample and, in the end, will either accept or reject the null hypothesis. The test assists in determining the accuracy of the formed hypothesis. If unexpected results occur, it may necessitate the formulation of a new hypothesis, which can then be tested. There are steps for any hypothesis test. The first step is to state the hypothesis, both the null and alternate hypothesis.
The next step is determining all the relevant parameters like mean, standard deviation , level of significance, etc., which helps determine the z-test value . The third step determines the z score from the z table, and for this step, we need to see if it is a two-tail or single-tail test and accordingly extract the z score. The fourth and final step is to compare the results and then, based on that, either accept or reject the null hypothesis.
Hypothesis Testing Formula Calculator
You can use the following Hypothesis Testing Calculator
| X | |
| U | |
| SD | |
| √n | |
| Z | |
| Z = |
Recommended ArticlesThis has been a guide to Hypothesis Testing Formula. Here we discuss how to calculate Hypothesis Testing along with practical examples. We also provide a Hypothesis Testing calculator with a downloadable exceExcelplate. You may also look at the following articles to learn more –
 *Please provide your correct email id. Login details for this Free course will be emailed to you By signing up, you agree to our Terms of Use and Privacy Policy . Download Hypothesis Testing Formula Excel Template Corporate Valuation, Investment Banking, Accounting, CFA Calculator & others Forgot Password? डाउनलोड Hypothesis Testing Formula Excel Template This website or its third-party tools use cookies, which are necessary to its functioning and required to achieve the purposes illustrated in the cookie policy. By closing this banner, scrolling this page, clicking a link or continuing to browse otherwise, you agree to our Privacy Policy  Explore 1000+ varieties of Mock tests View more Submit Next Question Early-Bird Offer: ENROLL NOW | ||||||||||||||||||||||||||||||
IMAGES
VIDEO
COMMENTS
Hypothesis Testing Cheat Sheet 23 June 2022 Hypothesis Terms Definitions Significance Level (𝜶) Defines the strength of evidence in probabilistic terms. Specifically, alpha represents the probability that tests will produce statistically significant results when the null hypothesis is correct. In most fields, α = ì. ì5 is used most often.
Hypothesis Testing Equation Sheet Chapter 9, 10 & 14 Steps in Hypothesis testing 1. Statement of hypothesis 2. Identification of the test statistic and its distribution 3. Specification of the significance level 4. Statement of the decision rule 5. Collection of the data and performance of the calculations 6. Making the statistical decision 7.
a. one b. two c. more than two. 1a. Hypothesis test about mean (one mean value) - the test is called hypothesis test about a population mean - we're interested if the population mean is equal to a specific value which is known (a constant) - notation (H0): µ=µ0. o if the population parameters are known ( µ,σ2,σ) we use the formula (1) to ...
0. 1. Left-tailed Test. H0 : μ = k H1 : μ < k P-value = P (z < zø) x This is the probability of getting a test statistic as low as or lower than zø x. If P-value ↵, we reject H0 and say the data are statistically significant at the level ↵. If P-value > ↵, we do not reject H0.
Hypothesis Testing | A Step-by-Step Guide with Easy ...
Type I Error: Reject the null hypothesis when the null hypothesis is true Type II Error: Do not reject the null hypothesis when the al-ternative hypothesis is true Test Statistics (t): A single number that summarizes the sam-ple data used to conduct the test hypothesis Standard Error: How far sample statistics (e.g., mean) devia-
The alternative hypothesis may be classified as two-tailed or one-tailed. Two-tailed test. is a two-sided alternative. we do the test with no preconceived notion that the true value of μ is either above or below the hypothesised value of μ 0. the alternative hypothesis is written: H1: μ =/= μo. One-tailed test. one-sided alternative.
Here, we'll be using the formula below for the general form of the test statistic. Determine the p-value. The p-value is the area under the standard normal distribution that is more extreme than the test statistic in the direction of the alternative hypothesis. Make a decision. If \(p \leq \alpha\) reject the null hypothesis.
Page 14/ Hypothesis Testing Formulas Z-test for a single score: Z = (X - µ) / σ"" " Z$test"with"means"of"a"sample:" Z = (M - µM) / σ M"" where:"" µ M = µ ...
n = count of one set of the pairs (don't double count) =TTEST(Array1,Array2,*,1) -> 1 for 1 tailed, 2 for 2 tailed. 2 Sample T-Test. Equal Variance. Test if the difference between the averages of two independent populations is equal to a target value. Do cats eat more of type A food than type B food. n1, n2 = count of sample 1, 2.
Hypothesis Testing Cheat Sheet. Easy To Understand Info About The Main Types Of Hypothesis Tests. This FREE PDF cheat sheet will show you the differences between all of the main types of hypothesis testing. Including examples on when to use the, the equations used, and how to easily implement them in Excel!
steps of formulating a hypothesis. state the two hypothesis: Null hypothesis and Alternative hypothesis. set the significance levels usually α = 0.05. carrying out the hypothesis testing and calculate the test statistics and corresponding P-value. compare P-value with significance levels and then decide to accept or reject null hypothesis.
Steps to Follow. Define the null and alternative hypothesis. Conduct the test. Using data from the test: Calculate the test statistic (i.e. F) and the critical value (i.e. F crit). Calculate a p value and compare it to a significance level (α) or confidence level (1-α). For example, if the significance level = 5%, then the confidence level = 95%.
The treatment group's mean is 58.70, compared to the control group's mean of 48.12. The mean difference is 10.67 points. Use the test's p-value and significance level to determine whether this difference is likely a product of random fluctuation in the sample or a genuine population effect.. Because the p-value (0.000) is less than the standard significance level of 0.05, the results are ...
9 Hypothesis Tests
Hypothesis Testing Steps. Step 1: Identify the claim and express in symbolic form. Step 2: Write the null and alternative hypothesis. Step 3: Calculate the test statistic. Step 4: Find the P-value or critical value. Step 5: Decide to reject or not to reject.
Test Statistic: Definition, Types & Formulas
Hypothesis Testing - Definition, Examples, Formula, Types
Unit 12: Significance tests (hypothesis testing)
H0 (null hypothesis): Mean value > 0. For this, Alternate Hypothesis (Ha): Mean < 0. Step 2: Next thing we have to do is that we need to find out the level of significance. Generally, its value is 0.05 or 0.01. Step 3: Find the z-test value, also called test statistic, as stated in the above formula.
17 Statistical Hypothesis Tests in Python (Cheat Sheet)
The Complete Guide: Hypothesis Testing in Excel