College practice · Worked answers
Hypothesis testing practice problems
A hypothesis test asks whether your data are hard to explain under a stated population claim. For college statistics, the key is to connect the question, hypotheses, test, and conclusion. A small p-value is evidence against a model; it is not the probability that your conclusion is wrong.
Write the question before the formula
- Name the parameter. Is the question about a population mean, proportion, or difference?
- State the hypotheses. Put the equality in the null hypothesis. Choose a one-sided or two-sided alternative from the research question before seeing the result.
- Check the design and assumptions. Independence, sampling, and the shape or size conditions depend on the test.
- Calculate the statistic and p-value. The p-value uses the null model and the specified tail or tails.
- Compare with the chosen α. Under the usual p ≤ α convention, reject H₀ when p ≤ α; otherwise fail to reject it.
- Write a contextual conclusion. State what the evidence supports about the population parameter, including the direction if relevant.
Choose a test that matches the information
| Question | Typical test | Check first |
|---|---|---|
| One mean, population σ known | One-sample z test | Independent observations; suitable normal sampling model |
| One mean, population σ unknown | One-sample t test | Use sample s; check outliers and small-sample shape |
| One proportion | One-proportion z test | Independent trials; sufficient expected successes and failures under H₀ |
Course conventions for count thresholds differ. Check the threshold your instructor uses; do not switch to a normal approximation just because a calculator offers it.
Keep the conclusion honest
A p-value is calculated assuming the null hypothesis and model conditions. It measures how often a statistic at least as extreme as the observed one would occur under that setup. Failing to reject H₀ means insufficient evidence against it, not proof that it is true. Statistical significance also does not tell you whether an effect is large or useful.
Work through two examples
A mean with known σ
- A hypothetical independent random sample has n = 64, mean 52, and known population σ = 8. Assume the sampling model is appropriate. Test H₀: μ = 50 against Hₐ: μ ≠ 50 at α = 0.05.
- The standard error is 8/√64 = 1. The z statistic is (52 − 50)/1 = 2.
- A two-sided standard-normal tail area gives p ≈ 0.0455. Since p < 0.05, reject H₀. There is evidence that the population mean differs from 50 under this model.
Interpreting a result that misses the threshold
- A valid test of H₀: p = 0.40 against Hₐ: p > 0.40 gives p-value 0.12. The significance level was set at 0.05.
- Compare 0.12 with 0.05. The p-value is larger, so fail to reject H₀.
- Say there is insufficient evidence that the population proportion exceeds 0.40. Do not write that the proportion is definitely 0.40.
Try it yourself
Choose an answer, then check your reasoning. You can retry any question or open its solution.
Show solution 1
μ ≠ 18. “Differs” allows either direction, so the population alternative is μ ≠ 18. Hypotheses concern parameters, not the observed sample mean.
Why the other choices are wrong:
- μ > 18: That is one-sided. “Differs” allows either direction.
- The sample mean equals 18: Hypotheses are about the population mean μ, not the sample mean you already computed.
- μ = 18: μ = 18 is the null hypothesis. The alternative is what you’d conclude if you reject it.
Show solution 2
Reject H₀. 0.03 is below 0.05, so reject H₀ under the chosen rule. This is evidence, not mathematical proof of the alternative.
Why the other choices are wrong:
- Prove H₀: A small p-value is evidence against H₀, not for it, and a test never proves a hypothesis.
- Fail to reject H₀: That is the decision when p > α. Here 0.03 is less than 0.05.
- Prove that Hₐ is true: Right direction, too strong. Rejecting H₀ is evidence for Hₐ, not proof.
Show solution 3
There is insufficient evidence to reject H₀. The p-value exceeds α, so the data do not provide enough evidence to reject H₀. It does not assign a probability to H₀.
Why the other choices are wrong:
- H₀ has a 24% probability of being true: A p-value is calculated assuming H₀ is true. It isn’t the probability that H₀ is true.
- H₀ is proven: Failing to reject isn’t proof. The data just aren’t strong evidence against H₀.
- Reject H₀, because 0.24 is larger than 0.05: That flips the rule. You reject when p ≤ α; a bigger p-value means weaker evidence.
Show solution 4
1.5. The standard error is 12/√36 = 2. Then z = (43 − 40)/2 = 1.5.
Why the other choices are wrong:
- 0.25: That divides by σ = 12. A sample mean uses the standard error σ/√n = 12/6 = 2.
- 3: 3 is the raw difference 43 − 40. You still need to divide by the standard error, 2.
- 9: That divides σ by n instead of √n: 12/36 = 1/3, and 3 ÷ (1/3) = 9.
Show solution 5
Null-model probability of results at least as extreme as observed. A p-value is a tail probability under H₀ and the test assumptions. It is not the probability that H₀ is true or that sampling was random.
Why the other choices are wrong:
- Probability that H₀ is true: The p-value assumes H₀ is true, so it can’t also be the probability that H₀ is true.
- Probability the sample was random: Random sampling is a design condition you check. The p-value doesn’t measure it.
- Probability that the result happened by chance alone: This common phrasing really describes the chance that H₀ is true given the data. The p-value goes the other way: it assumes H₀ and asks how unusual the data are.
Show solution 6
One-sample t test. With σ unknown, the one-sample t test uses s and t-distribution critical values, provided its conditions hold.
Why the other choices are wrong:
- One-sample z test: A z test for a mean needs the population σ. With only s, use t.
- One-proportion z test: That test is for a yes/no proportion, not a numerical mean.
- Two-sample t test: There is only one sample here, compared with one hypothesized value.
Show solution 7
H₀: μ = 12, Hₐ: μ < 12. The null states equality with the old value, μ = 12. The research question asks about a decrease, so Hₐ: μ < 12.
Why the other choices are wrong:
- H₀: μ < 12, Hₐ: μ = 12: That is swapped. The equality goes in H₀, and the claim you want evidence for goes in Hₐ.
- H₀: μ = 12, Hₐ: μ ≠ 12: The question names a direction (“lowers”), so a one-sided alternative fits. Choose it before seeing the data.
- H₀: x̄ = 12, Hₐ: x̄ < 12: Hypotheses are about the population mean μ. You already know the sample mean x̄.
Show solution 8
About 0.0455. The area above z = 2 is about 0.02275. A two-sided test counts both tails: 2 × 0.02275 ≈ 0.0455.
Why the other choices are wrong:
- About 0.0228: That is one tail. A two-sided test counts both tails, so double it.
- About 0.9772: That is the area to the left of z = 2. The p-value is the tail area beyond the statistic.
- 0.05: 0.05 is a common α, not the p-value. Compute the tail area for z = 2.
Answer key
Open all answers
- μ ≠ 18. “Differs” allows either direction, so the population alternative is μ ≠ 18. Hypotheses concern parameters, not the observed sample mean.
- Reject H₀. 0.03 is below 0.05, so reject H₀ under the chosen rule. This is evidence, not mathematical proof of the alternative.
- There is insufficient evidence to reject H₀. The p-value exceeds α, so the data do not provide enough evidence to reject H₀. It does not assign a probability to H₀.
- 1.5. The standard error is 12/√36 = 2. Then z = (43 − 40)/2 = 1.5.
- Null-model probability of results at least as extreme as observed. A p-value is a tail probability under H₀ and the test assumptions. It is not the probability that H₀ is true or that sampling was random.
- One-sample t test. With σ unknown, the one-sample t test uses s and t-distribution critical values, provided its conditions hold.
- H₀: μ = 12, Hₐ: μ < 12. The null states equality with the old value, μ = 12. The research question asks about a decrease, so Hₐ: μ < 12.
- About 0.0455. The area above z = 2 is about 0.02275. A two-sided test counts both tails: 2 × 0.02275 ≈ 0.0455.
Common questions
Should I always use α = 0.05?
No. Use the significance level specified in the problem or set in advance for the study. Do not choose it after seeing the p-value.
Can a significant result have a small effect?
Yes. Report the estimated effect and uncertainty as well as the test result.
Is this useful for a college statistics final?
Yes, for practicing the logic of a test. You still need the conditions and calculations for each test covered in your course.
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Sources and fact checks
Definitions and methods checked against these textbook references. Practice scenarios and worked numbers were written for this guide.