Practice Two-Sample Tests in Math

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

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Quick Recap

Hypothesis tests and confidence intervals for comparing parameters (means or proportions) of two independent populations. The two-sample t-test compares means; the two-proportion z-test compares proportions.

You have two separate groups—say, students taught with Method A vs Method B—and want to know if there's a real difference. Unlike paired tests where the same subjects appear in both groups, here the groups are completely independent. You compare the two sample statistics and ask: 'Is the gap between these groups larger than what random variation alone would produce?'

Showing a random 20 of 50 problems.

Example 1

hard
Two-proportion z-test: p^1=0.55 (n1=200) and p^2=0.45 (n2=200). Test H0:p1=p2 — compute z.

Example 2

easy
The pooled proportion is p^=x1+x2n1+n2. With x1=30,x2=50,n1=100,n2=100, compute p^.

Example 3

easy
A two-sample test of means gives p-value 0.012. At α=0.05, what is the decision about H0?

Example 4

medium
True or false: doubling each sample size n1,n2 (variances unchanged) cuts the SE of xˉ1−xˉ2 by a factor of 2.

Example 5

medium
Why must you NOT use a paired t-test on two independent random samples of different people?

Example 6

medium
Compute the standard error of xˉ1−xˉ2 given s1=5,n1=25,s2=4,n2=16.

Example 7

hard
Two independent samples: Group 1 (n=20,xˉ=100,s=15), Group 2 (n=20,xˉ=95,s=20). Calculate the t-statistic and df for a Welch's t-test (unequal variances).

Example 8

medium
A 99% CI for μ1−μ2 is (2,10). Does the test reject H0:μ1=μ2 at α=0.01?

Example 9

medium
Diet A produced xˉ1=4.5 lb loss (s1=2,n1=40); Diet B produced xˉ2=3.0 lb (s2=2.5,n2=40). Compute the Welch t-statistic.

Example 10

easy
Sample proportions: p^1=0.45 from n1=200 and p^2=0.35 from n2=200. Compute p^1−p^2.

Example 11

hard
A two-sample test rejects H0 with p-value 0.003 and observed difference xˉ1−xˉ2=1.2 on a 100-point exam. Is the difference necessarily practically important?

Example 12

medium
A 95% CI for p1−p2 is (−0.04,0.06). Does the corresponding two-sided test reject H0:p1=p2?

Example 13

challenge
An investigator wants 80% power to detect a true difference of μ1−μ2=5 with common σ=10. Using the rule-of-thumb n≈2(σ)2(zα/2+zβ)2δ2 with z0.025=1.96 and z0.20=0.84, estimate n per group.

Example 14

medium
Two independent samples of means: xˉ1−xˉ2=6, standard error of the difference =2. Compute the two-sample t-statistic.

Example 15

medium
Pooled SE for a two-proportion z-test: p^=0.4, n1=n2=100. Compute the pooled SE.

Example 16

easy
A two-sample test of proportions gives p-value 0.21 at α=0.05. State the decision.

Example 17

medium
Test whether two teaching methods differ in effectiveness. Method A (nA=30, xˉA=75, sA=8) vs. Method B (nB=30, xˉB=80, sB=10). Use a two-sample z-test at α=0.05.

Example 18

medium
A two-sample t-test gives t=1.0, p-value =0.33, at α=0.05. Conclude about the difference in means.

Example 19

easy
Group 1 has xˉ1=72 and Group 2 has xˉ2=68. Compute xˉ1−xˉ2.

Example 20

challenge
Two independent samples: xˉ1=50,xˉ2=47,s1=s2=6,n1=n2=36. Construct a 95% CI for μ1−μ2 using t∗≈1.99.
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