Two-Sample Tests Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Two-Sample Tests.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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?'

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Two-sample tests compare a mean or proportion between two INDEPENDENT groups to see if they truly differ.

Common stuck point: The procedure for two-sample tests is the easy part; the trap is running a two-sample test on paired data. Asking "Are the two groups made of different, unrelated subjects with no natural pairing between them?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the two groups made of different, unrelated subjects with no natural pairing between them?

Worked Examples

Example 1

medium

Test whether two teaching methods differ in effectiveness. Method A (

n_A=30

\bar{x}_A=75

s_A=8

) vs. Method B (

n_B=30

\bar{x}_B=80

s_B=10

). Use a two-sample z-test at

\alpha=0.05

Answer

z = -2.14

p \approx 0.032 < 0.05

. Reject

H_0

. Methods differ significantly.

First step

H_0: \mu_A = \mu_B

;

H_a: \mu_A \neq \mu_B

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Example 2

hard

Construct a 95% confidence interval for

\mu_A - \mu_B

given:

\bar{x}_A=50

\bar{x}_B=45

s_A=6

s_B=8

n_A=n_B=25

Example 3

medium

Compute the standard error of

\bar{x}_1 - \bar{x}_2

given

s_1 = 5, n_1 = 25, s_2 = 4, n_2 = 16

Example 4

medium

Diet A produced

\bar{x}_1 = 4.5

lb loss (

s_1 = 2, n_1 = 40

); Diet B produced

\bar{x}_2 = 3.0

lb (

s_2 = 2.5, n_2 = 40

). Compute the Welch t-statistic.

Example 5

hard

A two-sample Welch t-test gives

t = -1.10

with df

\approx 50

, two-sided p-value

\approx 0.28

. State the decision at

\alpha = 0.05

and interpret in plain English.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

When should a two-sample t-test be used instead of a z-test, and what is the key assumption about the two groups?

Example 2

hard

Two independent samples: Group 1 (

n=20, \bar{x}=100, s=15

), Group 2 (

n=20, \bar{x}=95, s=20

). Calculate the t-statistic and df for a Welch's t-test (unequal variances).

Example 3

easy

Two INDEPENDENT groups (Method A vs Method B students) are compared on mean test scores. Paired or two-sample test?

Example 4

easy

In a two-sample test of means, the usual null hypothesis says the two population means are what?

Example 5

easy

Group means are

\bar{x}_1 = 50

and

\bar{x}_2 = 44

. Compute the difference in sample means

\bar{x}_1 - \bar{x}_2

Example 6

easy

A two-proportion z-test compares two population what?

Example 7

easy

In a two-proportion z-test of

H_0: p_1 = p_2

, what special proportion is used to estimate the common value under the null?

Example 8

easy

The pooled proportion is

\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}

. With

x_1 = 30, x_2 = 50, n_1 = 100, n_2 = 100

, compute

\hat{p}

Example 9

easy

Two independent samples must satisfy which key relationship between them for a two-sample test?

Example 10

easy

A two-sample mean test gives a p-value of

0.30

\alpha = 0.05

. What is the conclusion?

Example 11

medium

Two independent samples of means:

\bar{x}_1 - \bar{x}_2 = 6

, standard error of the difference

= 2

. Compute the two-sample t-statistic.

Example 12

medium

The SE of a difference in means is

\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}

. With

s_1^2 = 16, n_1 = 4, s_2^2 = 9, n_2 = 9

, compute it.

Example 13

medium

Sample proportions

\hat{p}_1 = 0.6

(

n_1 = 50

) and

\hat{p}_2 = 0.4

(

n_2 = 50

). Compute the pooled proportion

\hat{p}

Example 14

medium

A two-sample t-test gives

t = 2.8

, p-value

= 0.008

, at

\alpha = 0.05

. Conclude about the two means.

Example 15

medium

A 95% CI for

\mu_1 - \mu_2

(-1, 5)

. What does it imply about a difference in means at the 5% level?

Example 16

medium

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

Example 17

medium

Build a CI for

\mu_1 - \mu_2

: difference

= 8

, SE

= 2.5

t^* = 2.0

Example 18

medium

For a two-proportion z-test, the pooled SE is

\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}

. With

\hat{p} = 0.5

n_1 = n_2 = 50

, compute it.

Example 19

medium

A two-sample t-test gives

t = 1.0

, p-value

= 0.33

, at

\alpha = 0.05

. Conclude about the difference in means.

Example 20

challenge

Two independent means:

\bar{x}_1 = 80, \bar{x}_2 = 74

s_1 = 10, n_1 = 25

s_2 = 8, n_2 = 16

. Compute the t-statistic.

Example 21

challenge

A two-proportion z-test of

H_0: p_1 = p_2

has successes

40/100

and

30/100

. Compute the pooled

\hat{p}

, the SE, and the z-statistic.

Example 22

challenge

Researchers compared a drug vs placebo in two independent random groups and found a significant difference in recovery rates. A colleague suggests a paired analysis instead. Explain why the paired test is inappropriate and what would justify one.

Example 23

easy

Group 1 has

\bar{x}_1 = 72

and Group 2 has

\bar{x}_2 = 68

. Compute

\bar{x}_1 - \bar{x}_2

Example 24

easy

A two-sample test of means gives p-value

0.012

. At

\alpha = 0.05

, what is the decision about

H_0

Example 25

easy

Sample proportions:

\hat{p}_1 = 0.45

from

n_1 = 200

and

\hat{p}_2 = 0.35

from

n_2 = 200

. Compute

\hat{p}_1 - \hat{p}_2

Example 26

easy

A two-sample test of proportions gives p-value

0.21

\alpha = 0.05

. State the decision.

Example 27

medium

Two-sample t-test:

\bar{x}_1 - \bar{x}_2 = 9

SE = 3

. Compute the t-statistic for

H_0: \mu_1 = \mu_2

Example 28

medium

Build a 95% CI for

\mu_1 - \mu_2

: difference

= 4

SE = 1.5

t^* = 2.0

Example 29

medium

Compute the pooled proportion

\hat{p}

for successes

36/120

and

24/80

Example 30

medium

Pooled SE for a two-proportion z-test:

\hat{p} = 0.4

n_1 = n_2 = 100

. Compute the pooled SE.

Example 31

medium

A 99% CI for

\mu_1 - \mu_2

(2, 10)

. Does the test reject

H_0: \mu_1 = \mu_2

\alpha = 0.01

Example 32

medium

Two-proportion z-test:

\hat{p}_1 - \hat{p}_2 = 0.08

, pooled

SE = 0.05

. Compute the z-statistic.

Example 33

medium

A 95% CI for

p_1 - p_2

(-0.04, 0.06)

. Does the corresponding two-sided test reject

H_0: p_1 = p_2

Example 34

medium

For independent samples with

s_1 = 6, n_1 = 36, s_2 = 8, n_2 = 16

, compute the SE of

\bar{x}_1 - \bar{x}_2

Example 35

hard

Welch t-test:

\bar{x}_1 = 102, \bar{x}_2 = 96

s_1 = 10, n_1 = 25, s_2 = 8, n_2 = 25

. Compute the t-statistic.

Example 36

hard

Two-proportion z-test:

\hat{p}_1 = 0.55

(

n_1 = 200

) and

\hat{p}_2 = 0.45

(

n_2 = 200

). Test

H_0: p_1 = p_2

— compute

z

Example 37

hard

Construct a 95% CI for

p_1 - p_2

given

\hat{p}_1 = 0.6, n_1 = 100

and

\hat{p}_2 = 0.5, n_2 = 100

. Use

z^* = 1.96

and the unpooled SE.

Example 38

hard

A study uses a paired t-test on two independent random samples of unrelated subjects. State why the inference may be invalid.

Example 39

hard

A two-sample test rejects

H_0

with p-value

0.003

and observed difference

\bar{x}_1 - \bar{x}_2 = 1.2

on a 100-point exam. Is the difference necessarily practically important?

Example 40

hard

Welch t-test gives

|t| = 2.50

with df

= 30

. Using the rough cutoff

t^* \approx 2.04

at the two-sided 5% level, decide whether to reject

H_0

Example 41

challenge

Two independent samples:

\bar{x}_1 = 50, \bar{x}_2 = 47, s_1 = s_2 = 6, n_1 = n_2 = 36

. Construct a 95% CI for

\mu_1 - \mu_2

using

t^* \approx 1.99

Example 42

challenge

An investigator wants 80% power to detect a true difference of

\mu_1 - \mu_2 = 5

with common

\sigma = 10

. Using the rule-of-thumb

n \approx \frac{2(\sigma)^2(z_{\alpha/2} + z_{\beta})^2}{\delta^2}

with

z_{0.025} = 1.96

and

z_{0.20} = 0.84

, estimate

n

per group.

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Related Concepts

Hypothesis Testing Confidence Interval Sampling Distribution Central Limit Theorem

Background Knowledge

These ideas may be useful before you work through the harder examples.

hypothesis testingconfidence intervalsampling distributioncentral limit theorem