Practice Hypothesis Testing in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A systematic method to decide whether sample data provides enough evidence to reject a claim (null hypothesis) about a population parameter.

Think of a courtroom trial: the null hypothesis (H0H_0) is 'innocent until proven guilty.' You look at the evidence (data) and ask: 'Is this evidence so strong that it would be very unlikely if the defendant were truly innocent?' If yes, you reject the null hypothesis. If not, you don't have enough evidence to convict—but that doesn't prove innocence.

Showing a random 20 of 50 problems.

Example 1

easy
State null and alternative hypotheses for each scenario: (a) testing if a coin is fair, (b) testing if a new drug reduces fever faster than the standard drug.

Example 2

medium
A two-sided test produces p=0.04p = 0.04. What is the smallest significance level at which we reject H0H_0?

Example 3

challenge
Using the Bonferroni correction for mm independent tests, what individual α\alpha should each test use to maintain family-wise error rate ≤0.05\le 0.05 with m=10m = 10?

Example 4

challenge
With nn very large, a trivial effect (mean differs from 100100 by 0.10.1) becomes statistically significant. Explain the mechanism and the lesson.

Example 5

easy
A researcher wants to show the mean is greater than 5050. What is the alternative hypothesis HaH_a?

Example 6

hard
A teacher claims students average 80 points. A skeptic samples n=25n=25 students: xˉ=76\bar{x}=76, s=10s=10. Using z=xˉ−μ0s/nz = \frac{\bar{x}-\mu_0}{s/\sqrt{n}}, test H0:μ=80H_0: \mu=80 vs Ha:μ<80H_a: \mu < 80 at α=0.01\alpha=0.01.

Example 7

medium
With test statistic z=2z = 2, two-sided p-value 0.04560.0456, and α=0.05\alpha = 0.05, state the conclusion.

Example 8

easy
What is power of a test?

Example 9

challenge
A researcher runs 2020 independent tests at α=0.05\alpha = 0.05 on data where every H0H_0 is true. What is the expected number of false rejections, and what is this problem called?

Example 10

medium
For H0:μ=100H_0: \mu = 100 vs Ha:μ≠100H_a: \mu \ne 100 with xˉ=106\bar x = 106, σ=10\sigma = 10, n=25n = 25, compute the z statistic.

Example 11

medium
A test of H0:μ=100H_0: \mu = 100 vs Ha:μ≠100H_a: \mu \ne 100 has xˉ=104\bar{x} = 104, σ=10\sigma = 10, n=25n = 25. Compute the z test statistic.

Example 12

easy
A claim is 'the mean equals 100100.' Write the null hypothesis symbolically.

Example 13

easy
For H0:μ=50H_0: \mu = 50 vs Ha:μ<50H_a: \mu < 50 with z=−2.5z = -2.5 and P(Z<−2.5)≈0.0062P(Z < -2.5) \approx 0.0062, state the conclusion at α=0.01\alpha = 0.01.

Example 14

medium
State which error type is committed: a court fails to convict a guilty defendant.

Example 15

medium
Continuing: with z=3z = 3 in a two-sided test, P(Z>3)≈0.00135P(Z > 3) \approx 0.00135. Find the p-value.

Example 16

hard
A researcher runs 5050 independent hypothesis tests at α=0.05\alpha = 0.05 under all true nulls. What is the expected number of false rejections and the probability of at least one?

Example 17

medium
A factory claims defect rate 5%5\%. From n=400n = 400, p^=0.08\hat p = 0.08. Test H0:p=0.05H_0: p = 0.05 vs Ha:p>0.05H_a: p > 0.05 at α=0.05\alpha = 0.05.

Example 18

challenge
Explain why we say 'fail to reject H0H_0' rather than 'accept H0H_0', using the courtroom analogy.

Example 19

medium
With z=−3z = -3 in a two-sided test, P(Z<−3)≈0.00135P(Z < -3) \approx 0.00135. Find p-value and conclude at α=0.01\alpha = 0.01.

Example 20

medium
A test rejects H0H_0 at α=0.05\alpha = 0.05 but a 95%95\% CI for μ\mu is (48,52)(48, 52) and H0:μ=50H_0: \mu = 50. Is this consistent?
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