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 groupsβ€”say, students taught with Method A vs Method Bβ€”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\hat{p}_1 = 0.55 (n1=200n_1 = 200) and p^2=0.45\hat{p}_2 = 0.45 (n2=200n_2 = 200). Test H0:p1=p2H_0: p_1 = p_2 β€” compute zz.

Example 2

easy
The pooled proportion is p^=x1+x2n1+n2\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}. With x1=30,x2=50,n1=100,n2=100x_1 = 30, x_2 = 50, n_1 = 100, n_2 = 100, compute p^\hat{p}.

Example 3

easy
A two-sample test of means gives p-value 0.0120.012. At Ξ±=0.05\alpha = 0.05, what is the decision about H0H_0?

Example 4

medium
True or false: doubling each sample size n1,n2n_1, n_2 (variances unchanged) cuts the SE of xΛ‰1βˆ’xΛ‰2\bar{x}_1 - \bar{x}_2 by a factor of 2\sqrt{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 xΛ‰1βˆ’xΛ‰2\bar{x}_1 - \bar{x}_2 given s1=5,n1=25,s2=4,n2=16s_1 = 5, n_1 = 25, s_2 = 4, n_2 = 16.

Example 7

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

Example 8

medium
A 99% CI for ΞΌ1βˆ’ΞΌ2\mu_1 - \mu_2 is (2,10)(2, 10). Does the test reject H0:ΞΌ1=ΞΌ2H_0: \mu_1 = \mu_2 at Ξ±=0.01\alpha = 0.01?

Example 9

medium
Diet A produced xˉ1=4.5\bar{x}_1 = 4.5 lb loss (s1=2,n1=40s_1 = 2, n_1 = 40); Diet B produced xˉ2=3.0\bar{x}_2 = 3.0 lb (s2=2.5,n2=40s_2 = 2.5, n_2 = 40). Compute the Welch t-statistic.

Example 10

easy
Sample proportions: p^1=0.45\hat{p}_1 = 0.45 from n1=200n_1 = 200 and p^2=0.35\hat{p}_2 = 0.35 from n2=200n_2 = 200. Compute p^1βˆ’p^2\hat{p}_1 - \hat{p}_2.

Example 11

hard
A two-sample test rejects H0H_0 with p-value 0.0030.003 and observed difference xΛ‰1βˆ’xΛ‰2=1.2\bar{x}_1 - \bar{x}_2 = 1.2 on a 100-point exam. Is the difference necessarily practically important?

Example 12

medium
A 95% CI for p1βˆ’p2p_1 - p_2 is (βˆ’0.04,0.06)(-0.04, 0.06). Does the corresponding two-sided test reject H0:p1=p2H_0: p_1 = p_2?

Example 13

challenge
An investigator wants 80% power to detect a true difference of ΞΌ1βˆ’ΞΌ2=5\mu_1 - \mu_2 = 5 with common Οƒ=10\sigma = 10. Using the rule-of-thumb nβ‰ˆ2(Οƒ)2(zΞ±/2+zΞ²)2Ξ΄2n \approx \frac{2(\sigma)^2(z_{\alpha/2} + z_{\beta})^2}{\delta^2} with z0.025=1.96z_{0.025} = 1.96 and z0.20=0.84z_{0.20} = 0.84, estimate nn per group.

Example 14

medium
Two independent samples of means: xΛ‰1βˆ’xΛ‰2=6\bar{x}_1 - \bar{x}_2 = 6, standard error of the difference =2= 2. Compute the two-sample t-statistic.

Example 15

medium
Pooled SE for a two-proportion z-test: p^=0.4\hat{p} = 0.4, n1=n2=100n_1 = n_2 = 100. Compute the pooled SE.

Example 16

easy
A two-sample test of proportions gives p-value 0.210.21 at Ξ±=0.05\alpha = 0.05. State the decision.

Example 17

medium
Test whether two teaching methods differ in effectiveness. Method A (nA=30n_A=30, xˉA=75\bar{x}_A=75, sA=8s_A=8) vs. Method B (nB=30n_B=30, xˉB=80\bar{x}_B=80, sB=10s_B=10). Use a two-sample z-test at α=0.05\alpha=0.05.

Example 18

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

Example 19

easy
Group 1 has xΛ‰1=72\bar{x}_1 = 72 and Group 2 has xΛ‰2=68\bar{x}_2 = 68. Compute xΛ‰1βˆ’xΛ‰2\bar{x}_1 - \bar{x}_2.

Example 20

challenge
Two independent samples: xΛ‰1=50,xΛ‰2=47,s1=s2=6,n1=n2=36\bar{x}_1 = 50, \bar{x}_2 = 47, s_1 = s_2 = 6, n_1 = n_2 = 36. Construct a 95% CI for ΞΌ1βˆ’ΞΌ2\mu_1 - \mu_2 using tβˆ—β‰ˆ1.99t^* \approx 1.99.
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