Correct
Incorrect
(TYPE 1 Error - a)
Incorrect
(TYPE 2 Error - b)
Correct
The “Correct” cases are the ones where the decisions taken on the samples are truly applicable to the entire population. The cases of errors arise when one decides to retain (or reject) the null hypothesis based on the sample calculations, but that decision does not really apply for the entire population. These cases constitute Type 1 ( alpha ) and Type 2 ( beta ) errors, as indicated in the table above.
Selecting the correct critical value allows eliminating the type-1 alpha errors or limiting them to an acceptable range.
Alpha denotes the error on the level of significance and is determined by the researcher. To maintain the standard 5% significance or confidence level for probability calculations, this is retained at 5%.
According to the applicable decision-making benchmarks and definitions:
A few more examples will demonstrate this and other calculations.
A monthly income investment scheme exists that promises variable monthly returns. An investor will invest in it only if they are assured of an average $180 monthly income. The investor has a sample of 300 months’ returns which has a mean of $190 and a standard deviation of $75. Should they invest in this scheme?
Let’s set up the problem. The investor will invest in the scheme if they are assured of the investor's desired $180 average return.
H 0 : Null Hypothesis: mean = 180
H 1 : Alternative Hypothesis: mean > 180
Identify a critical value X L for the sample mean, which is large enough to reject the null hypothesis – i.e. reject the null hypothesis if the sample mean >= critical value X L
P (identify a Type I alpha error) = P (reject H 0 given that H 0 is true),
This would be achieved when the sample mean exceeds the critical limits.
= P (given that H 0 is true) = alpha
Graphically, it appears as follows:
Taking alpha = 0.05 (i.e. 5% significance level), Z 0.05 = 1.645 (from the Z-table or normal distribution table)
= > X L = 180 +1.645*(75/sqrt(300)) = 187.12
Since the sample mean (190) is greater than the critical value (187.12), the null hypothesis is rejected, and the conclusion is that the average monthly return is indeed greater than $180, so the investor can consider investing in this scheme.
One can also use standardized value z.
Test Statistic, Z = (sample mean – population mean) / (std-dev / sqrt (no. of samples).
Then, the rejection region becomes the following:
Z= (190 – 180) / (75 / sqrt (300)) = 2.309
Our rejection region at 5% significance level is Z> Z 0.05 = 1.645.
Since Z= 2.309 is greater than 1.645, the null hypothesis can be rejected with a similar conclusion mentioned above.
We aim to identify P (sample mean >= 190, when mean = 180).
= P (Z >= (190- 180) / (75 / sqrt (300))
= P (Z >= 2.309) = 0.0084 = 0.84%
The following table to infer p-value calculations concludes that there is confirmed evidence of average monthly returns being higher than 180:
p-value | Inference |
less than 1% | supporting alternative hypothesis |
between 1% and 5% | supporting alternative hypothesis |
between 5% and 10% | supporting alternative hypothesis |
greater than 10% | supporting alternative hypothesis |
A new stockbroker (XYZ) claims that their brokerage fees are lower than that of your current stock broker's (ABC). Data available from an independent research firm indicates that the mean and std-dev of all ABC broker clients are $18 and $6, respectively.
A sample of 100 clients of ABC is taken and brokerage charges are calculated with the new rates of XYZ broker. If the mean of the sample is $18.75 and std-dev is the same ($6), can any inference be made about the difference in the average brokerage bill between ABC and XYZ broker?
H 0 : Null Hypothesis: mean = 18
H 1 : Alternative Hypothesis: mean <> 18 (This is what we want to prove.)
Rejection region: Z <= - Z 2.5 and Z>=Z 2.5 (assuming 5% significance level, split 2.5 each on either side).
Z = (sample mean – mean) / (std-dev / sqrt (no. of samples))
= (18.75 – 18) / (6/(sqrt(100)) = 1.25
This calculated Z value falls between the two limits defined by:
- Z 2.5 = -1.96 and Z 2.5 = 1.96.
This concludes that there is insufficient evidence to infer that there is any difference between the rates of your existing broker and the new broker.
Alternatively, The p-value = P(Z< -1.25)+P(Z >1.25)
= 2 * 0.1056 = 0.2112 = 21.12% which is greater than 0.05 or 5%, leading to the same conclusion.
Graphically, it is represented by the following:
Criticism Points for the Hypothetical Testing Method:
Hypothesis testing allows a mathematical model to validate a claim or idea with a certain confidence level. However, like the majority of statistical tools and models, it is bound by a few limitations. The use of this model for making financial decisions should be considered with a critical eye, keeping all dependencies in mind. Alternate methods like Bayesian Inference are also worth exploring for similar analysis.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 4-5.
Rice University, OpenStax. " Introductory Statistics 2e: 7.1 The Central Limit Theorem for Sample Means (Averages) ."
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 5-6.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 13.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 6.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 6-7.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 10.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 11.
Gregory J. Privitera. " Chapter 8: Introduction to Hypothesis Testing ." Statistics for Behavioral Sciences, Part III: Probability and the Foundations of Inferential Statistics. Sage Publications , pp. 7, 10-11.
Numpy: integration with scipy exercise-7 with solution.
Write a NumPy program to generate random samples and perform a hypothesis test using SciPy's stats module.
Sample Solution:
Python Code:
Explanation:
Python-Numpy Code Editor:
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If the biologist set her significance level \(\alpha\) at 0.05 and used the critical value approach to conduct her hypothesis test, she would reject the null hypothesis if her test statistic t* were less than -1.6939 (determined using statistical software or a t-table):s-3-3. Since the biologist's test statistic, t* = -4.60, is less than -1.6939, the biologist rejects the 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.
In statistics, hypothesis tests are used to test whether or not some hypothesis about a population parameter is true. To perform a hypothesis test in the real world, researchers will obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:. Null Hypothesis (H 0): The sample data occurs purely from chance.
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 ...
Example 3: Public Opinion About President Step 1. Determine the null and alternative hypotheses. Null hypothesis: There is no clear winning opinion on this issue; the proportions who would answer yes or no are each 0.50. Alternative hypothesis: Fewer than 0.50, or 50%, of the population would answer yes to this question.
Registered nurses earned an average annual salary of $69,110. For that same year, a survey was conducted of 41 California registered nurses to determine if the annual salary is higher than $69,110 for California nurses. The sample average was $71,121 with a sample standard deviation of $7,489. Conduct a hypothesis test.
Unit 12: Significance tests (hypothesis testing) Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values ...
Full Hypothesis Test Examples. Example 8.6.4 8.6. 4. Statistics students believe that the mean score on the first statistics test is 65. A statistics instructor thinks the mean score is higher than 65. He samples ten statistics students and obtains the scores 65 65 70 67 66 63 63 68 72 71.
Test Statistic: z = x¯¯¯ −μo σ/ n−−√ z = x ¯ − μ o σ / n since it is calculated as part of the testing of the hypothesis. Definition 7.1.4 7.1. 4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true.
where μ μ denotes the mean distance between the holes. Step 2. The sample is small and the population standard deviation is unknown. Thus the test statistic is. T = x¯ −μ0 s/ n−−√ T = x ¯ − μ 0 s / n. and has the Student t t -distribution with n − 1 = 4 − 1 = 3 n − 1 = 4 − 1 = 3 degrees of freedom. Step 3.
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.
A teacher believes that 85% of students in the class will want to go on a field trip to the local zoo. The teacher performs a hypothesis test to determine if the percentage is the same or different from 85%. The teacher samples 50 students and 39 reply that they would want to go to the zoo. For the hypothesis test, use a 1% level of significance.
It tests the null hypothesis that the population variances are equal (called homogeneity of variance or homoscedasticity). Suppose the resulting p-value of Levene's test is less than the significance level (typically 0.05).In that case, the obtained differences in sample variances are unlikely to have occurred based on random sampling from a population with equal variances.
Full Hypothesis Test Examples Tests on Means. Example 6. Jeffrey, as an eight-year old, established a mean time of 16.43 seconds for swimming the 25-yard freestyle, with a standard deviation of 0.8 seconds. His dad, Frank, thought that Jeffrey could swim the 25-yard freestyle faster using goggles.
View Solution to Question 1. Question 2. A professor wants to know if her introductory statistics class has a good grasp of basic math. Six students are chosen at random from the class and given a math proficiency test. The professor wants the class to be able to score above 70 on the test. The six students get the following scores:62, 92, 75 ...
3. One-Sided vs. Two-Sided Testing. When it's time to test your hypothesis, it's important to leverage the correct testing method. The two most common hypothesis testing methods are one-sided and two-sided tests, or one-tailed and two-tailed tests, respectively. Typically, you'd leverage a one-sided test when you have a strong conviction ...
Developing a hypothesis (with example) Step 1. Ask a question. Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project. Example: Research question.
(A z-score and a t-score are examples of test statistics.) Compare the preconceived α with the p-value, make a decision (reject or do not reject H0), and write a clear conclusion. 10.7: Hypothesis Testing of a Single Mean and Single Proportion (Worksheet) A statistics Worksheet: The student will select the appropriate distributions to use in ...
The coach thought the different grip helped Marco throw farther than 40 yards. Conduct a hypothesis test using a preset \(\alpha = 0.05\). Assume the throw distances for footballs are normal. First, determine what type of test this is, set up the hypothesis test, find the p-value, sketch the graph, and state your conclusion.
A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process. Consider a study designed to examine the relationship between sleep deprivation and test ...
9.1: Prelude to Hypothesis Testing. A statistician will make a decision about claims via a process called "hypothesis testing." A hypothesis test involves collecting data from a sample and evaluating the data. Then, the statistician makes a decision as to whether or not there is sufficient evidence, based upon analysis of the data, to reject ...
Hypothesis or significance testing is a mathematical model for testing a claim, idea or hypothesis about a parameter of interest in a given population set, using data measured in a sample set.
A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the population proportions. 10.5: Matched or Paired Samples When using a hypothesis test for matched or paired samples, the following characteristics should be present: Simple random sampling is used. Sample sizes are often small.
Create two normally distributed samples with different means (loc) but the same standard deviation (scale). Perform an independent t-test using SciPy's stats module: Use ttest_ind to perform a t-test to compare the means of the two samples. Print the results of the hypothesis test: Display the t-statistic and p-value to determine if there is a ...
8.1: Introduction to Two-Sample Tests; 8.2: Comparing Two Independent Population Means; 8.3: Cohen's Standards for Small, Medium, and Large Effect Sizes; 8.4: Test for Differences in Means- Assuming Equal Population Variances; 8.5: Comparing Two Independent Population Proportions; 8.6: Two Population Means with Known Standard Deviations