Hypothesis Testing: Complete AP Statistics Guide

AP Statistics Topic Guide

Hypothesis Testing: Complete AP Statistics Guide

Build a significance test from the statistical question outward: define the parameter, write hypotheses, verify the procedure, measure evidence, and make a contextual conclusion without overstating what the data prove.

StatusCurrent AP Statistics inference framework

Main keywordhypothesis testing

Worked analysis24 cases

Focused practice20 MCQs + 8 FRQs

Study progress0 completed

Hypothesis Testing: Complete AP Statistics Guide: direct answer

In a hypothesis testing problem, begin by identifying the population target and the inferential role of the data. Build a significance test from the statistical question outward: define the parameter, write hypotheses, verify the procedure, measure evidence, and make a contextual conclusion without overstating what the data prove.

This page uses hypothesis testing as its single primary search focus. The lesson, numerical cases, and retained questions are restricted to that intent so the page does not function as a generic inference question bank.

Quick reference for Hypothesis Testing: Complete AP Statistics Guide

Hypothesis Testing: Complete AP Statistics Guide quick reference
Step 1	Define the parameter and research claim
Step 2	Write H₀ and Hₐ
Step 3	Check design and procedure conditions
Step 4	Compute the test statistic
Step 5	Convert to a p-value using the correct tail
Step 6	Compare with α and conclude in context

Concept mastery: hypothesis testing

Start with the population question

Hypothesis testing begins with a population parameter and a claim, not with a calculator menu. Decide whether the target is a proportion, mean, difference in proportions, difference in means, or another parameter before selecting a procedure.

The null hypothesis supplies a benchmark

H0 states a specific benchmark or equality that the test treats as the reference model. The alternative hypothesis describes the direction of departure that would count as evidence for the research question.

Choose direction before looking at the result

A one-sided or two-sided alternative should follow from the research question and design, not from which direction the observed sample happened to move. Changing the alternative after seeing the statistic corrupts the stated error rate.

Verify conditions for the selected procedure

Different tests have different approximation conditions, but all require attention to the design and the model. Check the sampling or assignment mechanism, independence, and any success-failure or shape requirements before trusting a reference distribution.

Compute a statistic that measures null discrepancy

The test statistic expresses how far the observed estimate lies from the null benchmark in standard-error units. A large magnitude indicates that the data are difficult to reconcile with H0 under the chosen model.

Translate the statistic into a p-value

The p-value is a tail probability computed assuming H0 is true. Its tail or tails must match Ha. It is not the probability that H0 is true and not the probability that the result occurred only by chance.

Compare evidence with alpha

If p<=alpha, reject H0; otherwise, fail to reject H0. Alpha is a decision threshold chosen before the data analysis. It is not a measure of effect size and does not tell you how important a detected difference is.

Write the conclusion in evidence language

Use phrases such as “there is convincing evidence” or “there is not convincing evidence” about the alternative claim. Avoid saying that H0 has been proved true or that Ha has been proved with certainty.

Separate statistical significance from practical importance

A tiny effect can be statistically significant with a large sample, while an important effect can miss significance when a study is underpowered. Report or examine effect size and interval estimates alongside the test decision when possible.

Respect the scope of inference

A significance test does not fix confounding, selection bias, or lack of random assignment. Generalization and causation depend on design. The p-value only measures evidence relative to the model and hypotheses that were specified.

Worked analysis for hypothesis testing

Testing workflow case 1: Community College

The research question concerns students who completed the placement module and calls for a one-proportion z test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 2: County Library

The research question concerns visitors who attended a weekly program and calls for a one-sample t test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 3: Public High School

The research question concerns seniors who submitted the financial-aid form and calls for a two-proportion z test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Suppose the correctly computed p-value is 0.041 and α=0.10. Because 0.041 ≤ 0.10, the formal decision is to reject H₀. In context, the data provide convincing evidence for the prespecified alternative. This wording does not say that H₀ has probability 0.041, does not equate statistical significance with practical importance, and does not expand the scope of inference beyond what the data-collection design permits.

Testing workflow case 4: Municipal Water Office

The research question concerns households reporting no service interruption and calls for a paired t test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Suppose the correctly computed p-value is 0.074 and α=0.01. Because 0.074 > 0.01, the formal decision is to fail to reject H₀. In context, the data provide insufficient evidence for the prespecified alternative at this threshold. This wording does not say that H₀ has probability 0.074, does not equate statistical significance with practical importance, and does not expand the scope of inference beyond what the data-collection design permits.

Testing workflow case 5: State Park

The research question concerns visitors who used the marked trail system and calls for a two-sample t test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Suppose the correctly computed p-value is 0.126 and α=0.05. Because 0.126 > 0.05, the formal decision is to fail to reject H₀. In context, the data provide insufficient evidence for the prespecified alternative at this threshold. This wording does not say that H₀ has probability 0.126, does not equate statistical significance with practical importance, and does not expand the scope of inference beyond what the data-collection design permits.

Testing workflow case 6: Food Cooperative

The research question concerns members who renewed before the deadline and calls for a chi-square test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Suppose the correctly computed p-value is 0.287 and α=0.10. Because 0.287 > 0.10, the formal decision is to fail to reject H₀. In context, the data provide insufficient evidence for the prespecified alternative at this threshold. This wording does not say that H₀ has probability 0.287, does not equate statistical significance with practical importance, and does not expand the scope of inference beyond what the data-collection design permits.

Testing workflow case 7: University Advising Center

The research question concerns appointments starting within ten minutes and calls for a one-proportion z test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Suppose the correctly computed p-value is 0.003 and α=0.01. Because 0.003 ≤ 0.01, the formal decision is to reject H₀. In context, the data provide convincing evidence for the prespecified alternative. This wording does not say that H₀ has probability 0.003, does not equate statistical significance with practical importance, and does not expand the scope of inference beyond what the data-collection design permits. In the Testing workflow case 7: University Advising Center context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Testing workflow case 8: Regional Manufacturer

The research question concerns parts meeting the diameter specification and calls for a one-sample t test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Suppose the correctly computed p-value is 0.018 and α=0.05. Because 0.018 ≤ 0.05, the formal decision is to reject H₀. In context, the data provide convincing evidence for the prespecified alternative. This wording does not say that H₀ has probability 0.018, does not equate statistical significance with practical importance, and does not expand the scope of inference beyond what the data-collection design permits. In the Testing workflow case 8: Regional Manufacturer context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Testing workflow case 9: Public Health Clinic

The research question concerns clients returning for the scheduled checkup and calls for a two-proportion z test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 10: Urban Recreation Program

The research question concerns participants completing the eight-week session and calls for a paired t test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 11: School District

The research question concerns families responding to the annual survey and calls for a two-sample t test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 12: Local Election Office

The research question concerns mailed ballots returned before election day and calls for a chi-square test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 13: Energy-Efficiency Pilot

The research question concerns homes meeting the target reduction and calls for a one-proportion z test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Suppose the correctly computed p-value is 0.003 and α=0.01. Because 0.003 ≤ 0.01, the formal decision is to reject H₀. In context, the data provide convincing evidence for the prespecified alternative. This wording does not say that H₀ has probability 0.003, does not equate statistical significance with practical importance, and does not expand the scope of inference beyond what the data-collection design permits. In the Testing workflow case 13: Energy-Efficiency Pilot context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Testing workflow case 14: Community Broadband Project

The research question concerns households achieving the advertised speed and calls for a one-sample t test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Suppose the correctly computed p-value is 0.018 and α=0.05. Because 0.018 ≤ 0.05, the formal decision is to reject H₀. In context, the data provide convincing evidence for the prespecified alternative. This wording does not say that H₀ has probability 0.018, does not equate statistical significance with practical importance, and does not expand the scope of inference beyond what the data-collection design permits. In the Testing workflow case 14: Community Broadband Project context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Testing workflow case 15: Regional Bus Network

The research question concerns trips arriving within the on-time window and calls for a two-proportion z test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 16: Campus Dining Service

The research question concerns transactions using reusable containers and calls for a paired t test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 17: Workforce Training Program

The research question concerns participants earning the credential and calls for a two-sample t test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 18: County Recycling Audit

The research question concerns sampled loads meeting contamination limits and calls for a chi-square test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 19: University Residence Halls

The research question concerns rooms passing the first safety inspection and calls for a one-proportion z test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Suppose the correctly computed p-value is 0.003 and α=0.01. Because 0.003 ≤ 0.01, the formal decision is to reject H₀. In context, the data provide convincing evidence for the prespecified alternative. This wording does not say that H₀ has probability 0.003, does not equate statistical significance with practical importance, and does not expand the scope of inference beyond what the data-collection design permits. In the Testing workflow case 19: University Residence Halls context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Testing workflow case 20: Telehealth Pilot

The research question concerns appointments completed without rescheduling and calls for a one-sample t test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Suppose the correctly computed p-value is 0.018 and α=0.05. Because 0.018 ≤ 0.05, the formal decision is to reject H₀. In context, the data provide convincing evidence for the prespecified alternative. This wording does not say that H₀ has probability 0.018, does not equate statistical significance with practical importance, and does not expand the scope of inference beyond what the data-collection design permits. In the Testing workflow case 20: Telehealth Pilot context, this checkpoint must be tied to the stated population and data structure before the numerical conclusion is accepted.

Testing workflow case 21: Public Museum

The research question concerns visitors using the audio guide and calls for a two-proportion z test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 22: Youth Sports League

The research question concerns players completing concussion training and calls for a paired t test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 23: Rural Pharmacy Network

The research question concerns prescriptions filled within the service target and calls for a two-sample t test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Testing workflow case 24: City Tree Program

The research question concerns new plantings surviving the first year and calls for a chi-square test. Before calculation, define the population parameter and write H₀ and Hₐ so the alternative matches the substantive claim. The design must justify the selected inferential model, and the procedure-specific approximation conditions must be checked rather than replaced by a generic statement that “the sample is large.”

Hypothesis Testing multiple-choice practice

Question 1. Hypothesis-testing workflow

Community Health Network studies follow-up completion rate. The parameter and data structure call for one-proportion z test. After verifying the relevant conditions, the analysis gives p=0.004 with α=0.01. Which statement correctly connects the procedure, decision, and conclusion?

A. Use the one-proportion z test; reject H₀ and state convincing evidence for the prespecified alternative in the population.
B. Accept H₀ because the p-value is the probability that H₀ is true.
C. Change the alternative direction to match the observed sample result, then recompute the p-value.
D. Skip the design and condition checks because a p-value has already been produced.

Answer: A

The research question determines the parameter and the one-proportion z test before the result is inspected. Since p=0.004 ≤ α=0.01, the formal decision is to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 2. Hypothesis-testing workflow

Regional College studies mean weekly study time. The parameter and data structure call for one-sample t test. After verifying the relevant conditions, the analysis gives p=0.023 with α=0.05. Which statement correctly connects the procedure, decision, and conclusion?

A. Accept H₀ because the p-value is the probability that H₀ is true.
B. Change the alternative direction to match the observed sample result, then recompute the p-value.
C. Skip the design and condition checks because a p-value has already been produced.
D. Use the one-sample t test; reject H₀ and state convincing evidence for the prespecified alternative in the population.

Answer: D

The research question determines the parameter and the one-sample t test before the result is inspected. Since p=0.023 ≤ α=0.05, the formal decision is to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 3. Hypothesis-testing workflow

City Transit Agency studies on-time arrival proportion. The parameter and data structure call for two-proportion z test. After verifying the relevant conditions, the analysis gives p=0.047 with α=0.10. Which statement correctly connects the procedure, decision, and conclusion?

A. Change the alternative direction to match the observed sample result, then recompute the p-value.
B. Skip the design and condition checks because a p-value has already been produced.
C. Use the two-proportion z test; reject H₀ and state convincing evidence for the prespecified alternative in the population.
D. Accept H₀ because the p-value is the probability that H₀ is true.

Answer: C

The research question determines the parameter and the two-proportion z test before the result is inspected. Since p=0.047 ≤ α=0.10, the formal decision is to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 4. Hypothesis-testing workflow

County Library System studies program participation rate. The parameter and data structure call for two-sample t test. After verifying the relevant conditions, the analysis gives p=0.068 with α=0.01. Which statement correctly connects the procedure, decision, and conclusion?

A. Skip the design and condition checks because a p-value has already been produced.
B. Use the two-sample t test; fail to reject H₀ and state insufficient evidence for the prespecified alternative in the population.
C. Accept H₀ because the p-value is the probability that H₀ is true.
D. Change the alternative direction to match the observed sample result, then recompute the p-value.

Answer: B

The research question determines the parameter and the two-sample t test before the result is inspected. Since p=0.068 > α=0.01, the formal decision is to fail to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 5. Hypothesis-testing workflow

Manufacturing Plant studies mean component strength. The parameter and data structure call for one-proportion z test. After verifying the relevant conditions, the analysis gives p=0.119 with α=0.05. Which statement correctly connects the procedure, decision, and conclusion?

A. Use the one-proportion z test; fail to reject H₀ and state insufficient evidence for the prespecified alternative in the population.
B. Accept H₀ because the p-value is the probability that H₀ is true.
C. Change the alternative direction to match the observed sample result, then recompute the p-value.
D. Skip the design and condition checks because a p-value has already been produced.

Answer: A

The research question determines the parameter and the one-proportion z test before the result is inspected. Since p=0.119 > α=0.05, the formal decision is to fail to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 6. Hypothesis-testing workflow

School District studies graduation-plan completion proportion. The parameter and data structure call for one-sample t test. After verifying the relevant conditions, the analysis gives p=0.244 with α=0.10. Which statement correctly connects the procedure, decision, and conclusion?

A. Accept H₀ because the p-value is the probability that H₀ is true.
B. Change the alternative direction to match the observed sample result, then recompute the p-value.
C. Skip the design and condition checks because a p-value has already been produced.
D. Use the one-sample t test; fail to reject H₀ and state insufficient evidence for the prespecified alternative in the population.

Answer: D

The research question determines the parameter and the one-sample t test before the result is inspected. Since p=0.244 > α=0.10, the formal decision is to fail to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 7. Hypothesis-testing workflow

Public Parks Department studies visitor satisfaction proportion. The parameter and data structure call for two-proportion z test. After verifying the relevant conditions, the analysis gives p=0.032 with α=0.01. Which statement correctly connects the procedure, decision, and conclusion?

A. Change the alternative direction to match the observed sample result, then recompute the p-value.
B. Skip the design and condition checks because a p-value has already been produced.
C. Use the two-proportion z test; fail to reject H₀ and state insufficient evidence for the prespecified alternative in the population.
D. Accept H₀ because the p-value is the probability that H₀ is true.

Answer: C

The research question determines the parameter and the two-proportion z test before the result is inspected. Since p=0.032 > α=0.01, the formal decision is to fail to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 8. Hypothesis-testing workflow

Energy Pilot studies mean household savings. The parameter and data structure call for two-sample t test. After verifying the relevant conditions, the analysis gives p=0.081 with α=0.05. Which statement correctly connects the procedure, decision, and conclusion?

A. Skip the design and condition checks because a p-value has already been produced.
B. Use the two-sample t test; fail to reject H₀ and state insufficient evidence for the prespecified alternative in the population.
C. Accept H₀ because the p-value is the probability that H₀ is true.
D. Change the alternative direction to match the observed sample result, then recompute the p-value.

Answer: B

The research question determines the parameter and the two-sample t test before the result is inspected. Since p=0.081 > α=0.05, the formal decision is to fail to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 9. Hypothesis-testing workflow

Hospital System studies 30-day readmission proportion. The parameter and data structure call for one-proportion z test. After verifying the relevant conditions, the analysis gives p=0.004 with α=0.10. Which statement correctly connects the procedure, decision, and conclusion?

A. Use the one-proportion z test; reject H₀ and state convincing evidence for the prespecified alternative in the population.
B. Accept H₀ because the p-value is the probability that H₀ is true.
C. Change the alternative direction to match the observed sample result, then recompute the p-value.
D. Skip the design and condition checks because a p-value has already been produced.

Answer: A

The research question determines the parameter and the one-proportion z test before the result is inspected. Since p=0.004 ≤ α=0.10, the formal decision is to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 10. Hypothesis-testing workflow

Workforce Program studies credential completion rate. The parameter and data structure call for one-sample t test. After verifying the relevant conditions, the analysis gives p=0.023 with α=0.01. Which statement correctly connects the procedure, decision, and conclusion?

A. Accept H₀ because the p-value is the probability that H₀ is true.
B. Change the alternative direction to match the observed sample result, then recompute the p-value.
C. Skip the design and condition checks because a p-value has already been produced.
D. Use the one-sample t test; fail to reject H₀ and state insufficient evidence for the prespecified alternative in the population.

Answer: D

The research question determines the parameter and the one-sample t test before the result is inspected. Since p=0.023 > α=0.01, the formal decision is to fail to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 11. Hypothesis-testing workflow

Municipal Call Center studies mean resolution time. The parameter and data structure call for two-proportion z test. After verifying the relevant conditions, the analysis gives p=0.047 with α=0.05. Which statement correctly connects the procedure, decision, and conclusion?

A. Change the alternative direction to match the observed sample result, then recompute the p-value.
B. Skip the design and condition checks because a p-value has already been produced.
C. Use the two-proportion z test; reject H₀ and state convincing evidence for the prespecified alternative in the population.
D. Accept H₀ because the p-value is the probability that H₀ is true.

Answer: C

The research question determines the parameter and the two-proportion z test before the result is inspected. Since p=0.047 ≤ α=0.05, the formal decision is to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 12. Hypothesis-testing workflow

Broadband Project studies household reliability proportion. The parameter and data structure call for two-sample t test. After verifying the relevant conditions, the analysis gives p=0.068 with α=0.10. Which statement correctly connects the procedure, decision, and conclusion?

A. Skip the design and condition checks because a p-value has already been produced.
B. Use the two-sample t test; reject H₀ and state convincing evidence for the prespecified alternative in the population.
C. Accept H₀ because the p-value is the probability that H₀ is true.
D. Change the alternative direction to match the observed sample result, then recompute the p-value.

Answer: B

The research question determines the parameter and the two-sample t test before the result is inspected. Since p=0.068 ≤ α=0.10, the formal decision is to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 13. Hypothesis-testing workflow

University Housing studies mean repair turnaround time. The parameter and data structure call for one-proportion z test. After verifying the relevant conditions, the analysis gives p=0.119 with α=0.01. Which statement correctly connects the procedure, decision, and conclusion?

A. Use the one-proportion z test; fail to reject H₀ and state insufficient evidence for the prespecified alternative in the population.
B. Accept H₀ because the p-value is the probability that H₀ is true.
C. Change the alternative direction to match the observed sample result, then recompute the p-value.
D. Skip the design and condition checks because a p-value has already been produced.

Answer: A

The research question determines the parameter and the one-proportion z test before the result is inspected. Since p=0.119 > α=0.01, the formal decision is to fail to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 14. Hypothesis-testing workflow

Food Cooperative studies member renewal proportion. The parameter and data structure call for one-sample t test. After verifying the relevant conditions, the analysis gives p=0.244 with α=0.05. Which statement correctly connects the procedure, decision, and conclusion?

A. Accept H₀ because the p-value is the probability that H₀ is true.
B. Change the alternative direction to match the observed sample result, then recompute the p-value.
C. Skip the design and condition checks because a p-value has already been produced.
D. Use the one-sample t test; fail to reject H₀ and state insufficient evidence for the prespecified alternative in the population.

Answer: D

The research question determines the parameter and the one-sample t test before the result is inspected. Since p=0.244 > α=0.05, the formal decision is to fail to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 15. Hypothesis-testing workflow

Recreation Department studies mean weekly attendance. The parameter and data structure call for two-proportion z test. After verifying the relevant conditions, the analysis gives p=0.032 with α=0.10. Which statement correctly connects the procedure, decision, and conclusion?

A. Change the alternative direction to match the observed sample result, then recompute the p-value.
B. Skip the design and condition checks because a p-value has already been produced.
C. Use the two-proportion z test; reject H₀ and state convincing evidence for the prespecified alternative in the population.
D. Accept H₀ because the p-value is the probability that H₀ is true.

Answer: C

The research question determines the parameter and the two-proportion z test before the result is inspected. Since p=0.032 ≤ α=0.10, the formal decision is to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Question 16. Hypothesis-testing workflow

Water Authority studies compliance proportion. The parameter and data structure call for two-sample t test. After verifying the relevant conditions, the analysis gives p=0.081 with α=0.01. Which statement correctly connects the procedure, decision, and conclusion?

A. Skip the design and condition checks because a p-value has already been produced.
B. Use the two-sample t test; fail to reject H₀ and state insufficient evidence for the prespecified alternative in the population.
C. Accept H₀ because the p-value is the probability that H₀ is true.
D. Change the alternative direction to match the observed sample result, then recompute the p-value.

Answer: B

The research question determines the parameter and the two-sample t test before the result is inspected. Since p=0.081 > α=0.01, the formal decision is to fail to reject H₀. The conclusion is evidence language about the alternative claim; the p-value is not the probability that H₀ is true, and neither a calculator output nor statistical significance repairs a weak design.

Hypothesis Testing free-response practice

FRQ 1. Build the full testing chain

City Transit Agency investigates on-time arrival proportion. Describe a complete hypothesis-testing plan: define an appropriate population parameter, state how H₀ and Hₐ should be constructed, name the type of condition checks that belong before calculation, explain what a test statistic measures, interpret a reported p-value of 0.018, and give the decision at α=0.05 without claiming certainty.

Model response

A strong plan begins with the population parameter that matches on-time arrival proportion; H₀ supplies a specific equality benchmark and Hₐ records the prespecified research direction. The sampling or assignment design and the procedure-specific approximation conditions must be checked before using the reference distribution. The test statistic measures the observed estimate’s discrepancy from H₀ in standard-error units, and p=0.018 is the probability, assuming H₀ and the model, of data at least as extreme in the Hₐ direction. Because 0.018 ≤ 0.05, reject H₀. Conclude that the data provide convincing evidence for Hₐ in the population, while keeping causal or generalization claims within the study design.

FRQ 2. Build the full testing chain

Manufacturing Plant investigates mean component strength. Describe a complete hypothesis-testing plan: define an appropriate population parameter, state how H₀ and Hₐ should be constructed, name the type of condition checks that belong before calculation, explain what a test statistic measures, interpret a reported p-value of 0.072, and give the decision at α=0.01 without claiming certainty.

Model response

A strong plan begins with the population parameter that matches mean component strength; H₀ supplies a specific equality benchmark and Hₐ records the prespecified research direction. The sampling or assignment design and the procedure-specific approximation conditions must be checked before using the reference distribution. The test statistic measures the observed estimate’s discrepancy from H₀ in standard-error units, and p=0.072 is the probability, assuming H₀ and the model, of data at least as extreme in the Hₐ direction. Because 0.072 > 0.01, fail to reject H₀. Conclude that the data do not provide convincing evidence for Hₐ in the population, while keeping causal or generalization claims within the study design.

FRQ 3. Build the full testing chain

Public Parks Department investigates visitor satisfaction proportion. Describe a complete hypothesis-testing plan: define an appropriate population parameter, state how H₀ and Hₐ should be constructed, name the type of condition checks that belong before calculation, explain what a test statistic measures, interpret a reported p-value of 0.006, and give the decision at α=0.10 without claiming certainty.

Model response

A strong plan begins with the population parameter that matches visitor satisfaction proportion; H₀ supplies a specific equality benchmark and Hₐ records the prespecified research direction. The sampling or assignment design and the procedure-specific approximation conditions must be checked before using the reference distribution. The test statistic measures the observed estimate’s discrepancy from H₀ in standard-error units, and p=0.006 is the probability, assuming H₀ and the model, of data at least as extreme in the Hₐ direction. Because 0.006 ≤ 0.10, reject H₀. Conclude that the data provide convincing evidence for Hₐ in the population, while keeping causal or generalization claims within the study design.

FRQ 4. Build the full testing chain

Hospital System investigates 30-day readmission proportion. Describe a complete hypothesis-testing plan: define an appropriate population parameter, state how H₀ and Hₐ should be constructed, name the type of condition checks that belong before calculation, explain what a test statistic measures, interpret a reported p-value of 0.134, and give the decision at α=0.05 without claiming certainty.

Model response

A strong plan begins with the population parameter that matches 30-day readmission proportion; H₀ supplies a specific equality benchmark and Hₐ records the prespecified research direction. The sampling or assignment design and the procedure-specific approximation conditions must be checked before using the reference distribution. The test statistic measures the observed estimate’s discrepancy from H₀ in standard-error units, and p=0.134 is the probability, assuming H₀ and the model, of data at least as extreme in the Hₐ direction. Because 0.134 > 0.05, fail to reject H₀. Conclude that the data do not provide convincing evidence for Hₐ in the population, while keeping causal or generalization claims within the study design.

FRQ 5. Build the full testing chain

Municipal Call Center investigates mean resolution time. Describe a complete hypothesis-testing plan: define an appropriate population parameter, state how H₀ and Hₐ should be constructed, name the type of condition checks that belong before calculation, explain what a test statistic measures, interpret a reported p-value of 0.043, and give the decision at α=0.01 without claiming certainty.

Model response

A strong plan begins with the population parameter that matches mean resolution time; H₀ supplies a specific equality benchmark and Hₐ records the prespecified research direction. The sampling or assignment design and the procedure-specific approximation conditions must be checked before using the reference distribution. The test statistic measures the observed estimate’s discrepancy from H₀ in standard-error units, and p=0.043 is the probability, assuming H₀ and the model, of data at least as extreme in the Hₐ direction. Because 0.043 > 0.01, fail to reject H₀. Conclude that the data do not provide convincing evidence for Hₐ in the population, while keeping causal or generalization claims within the study design.

FRQ 6. Build the full testing chain

University Housing investigates mean repair turnaround time. Describe a complete hypothesis-testing plan: define an appropriate population parameter, state how H₀ and Hₐ should be constructed, name the type of condition checks that belong before calculation, explain what a test statistic measures, interpret a reported p-value of 0.201, and give the decision at α=0.10 without claiming certainty.

Model response

A strong plan begins with the population parameter that matches mean repair turnaround time; H₀ supplies a specific equality benchmark and Hₐ records the prespecified research direction. The sampling or assignment design and the procedure-specific approximation conditions must be checked before using the reference distribution. The test statistic measures the observed estimate’s discrepancy from H₀ in standard-error units, and p=0.201 is the probability, assuming H₀ and the model, of data at least as extreme in the Hₐ direction. Because 0.201 > 0.10, fail to reject H₀. Conclude that the data do not provide convincing evidence for Hₐ in the population, while keeping causal or generalization claims within the study design.

Next steps after mastering hypothesis testing

Use this general framework as a method-selection checkpoint, then move to a specific test page. Given a new prompt, identify the parameter, data structure, null benchmark, alternative direction, and design before naming the procedure. If those five items are correct, the remaining calculation is usually much less error-prone.

For a complete review, take one previously solved test and rewrite only the conclusion three ways: an invalid proof statement, an invalid probability-of-H₀ statement, and a valid evidence statement. Explaining why the first two are wrong is a stronger diagnostic than merely memorizing “reject” or “fail to reject.”