Practice Hypothesis Testing in Statistics

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

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Quick Recap

Hypothesis testing is a formal statistical procedure for using sample data to decide between two competing claims about a population parameter. You state a null hypothesis (no effect) and an alternative hypothesis, collect data, compute a test statistic, and determine whether the evidence is strong enough to reject the null.

Hypothesis testing is like a courtroom trial for data. You start by assuming innocence (null hypothesis: nothing special is happening). Then you look at the evidence (data). If the evidence is strong enough to be very unlikely under the assumption of innocence, you reject it and conclude something real is happening.

Showing a random 20 of 76 problems.

Example 1

easy
What is a Type I error?

Example 2

easy
True or false: failing to reject H0H_0 proves the null is true.

Example 3

medium
A test gives z=1.5z = 1.5 (two-sided). The two-tailed p-value is roughly 0.134. At α=0.05\alpha=0.05, what is the decision?

Example 4

easy
Fill in: hypothesis testing decides between two competing claims about a population ____.

Example 5

challenge
A study tests 20 independent hypotheses, each at α=0.05\alpha=0.05, with all nulls actually true. About how many false 'significant' results are expected?

Example 6

medium
A new drug shows a tiny, clinically meaningless improvement, but the test rejects H0H_0 with n=100,000n=100{,}000. What does this illustrate?

Example 7

medium
Why must hypotheses be stated before collecting or examining the data?

Example 8

easy
A teacher believes a new method increases mean test scores above the historical mean of 7272. State HaH_a.

Example 9

hard
A paired-sample test of n=25n=25 differences has dˉ=3\bar{d}=3, sd=10s_d=10. Test H0:μd=0H_0: \mu_d = 0. Compute tt.

Example 10

medium
Power of a test equals 1−β1 - \beta. If β=0.20\beta = 0.20, what is the power?

Example 11

easy
Which hypothesis carries the burden of proof?

Example 12

medium
A sample of n=36n=36 has mean xˉ=52\bar{x}=52 and known σ=6\sigma=6. Test H0:μ=50H_0: \mu = 50 vs. Ha:μ≠50H_a: \mu \ne 50. Compute the z-test statistic.

Example 13

medium
True or false: a statistically significant result with p=0.04p=0.04 always means the practical effect is meaningful.

Example 14

medium
A sample of n=49n=49 gives xˉ=53\bar{x}=53, the null claims μ0=50\mu_0=50, and σ=14\sigma=14. Compute the test statistic zz.

Example 15

easy
At significance level α=0.05\alpha = 0.05, a p-value of 0.030.03 leads to what decision about H0H_0?

Example 16

hard
Continuing the light bulb example (H0:μ=1000H_0: \mu = 1000, n=50n = 50, xˉ=985\bar{x} = 985, σ=40\sigma = 40), calculate the test statistic.

Example 17

challenge
If 2020 independent tests are run at α=0.05\alpha = 0.05 when all nulls are true, what is the expected number of false rejections?

Example 18

medium
We test H0:p=0.3H_0: p=0.3 vs Ha:p>0.3H_a: p>0.3 with p^=0.35\hat{p}=0.35, n=100n=100, SE under H0H_0 =0.3⋅0.7/100≈0.0458= \sqrt{0.3\cdot 0.7/100} \approx 0.0458. Compute zz.

Example 19

easy
At α=0.01\alpha = 0.01, a p-value of 0.040.04 leads to what decision about H0H_0?

Example 20

medium
A drug company tests whether its pill lowers blood pressure (Ha:μ<130H_a: \mu < 130). State the direction of the rejection region for a z-test.
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