Paired t-Test Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Paired t-Test.

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

Concept Recap

A hypothesis test for the mean difference in a paired (matched) data design, where each subject provides two related measurements. The test analyzes the differences $d_i = x_{1i} - x_{2i}$ as a single sample.

You want to know if a tutoring program improves math scores. Instead of comparing two separate groups, you test the SAME students before and after tutoring. Each student is their own control. By looking at the difference (after $-$ before) for each student, you eliminate individual variation and focus purely on the change.

Read the full concept explanation →

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: The paired t-test reduces matched before/after pairs to a single column of differences and tests whether their mean is zero.

Common stuck point: The procedure for paired t-test is the easy part; the trap is treating paired data as two independent samples. Asking "Is every value in one group paired with exactly one specific value in the other (so I can form a difference per pair)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is every value in one group paired with exactly one specific value in the other (so I can form a difference per pair)?

Worked Examples

Example 1

medium

Students' scores before and after tutoring: Before:

\{70, 65, 80, 75, 60\}

, After:

\{75, 70, 85, 80, 70\}

. Conduct a paired t-test at

\alpha=0.05

to test if tutoring improved scores.

Answer

t \approx 6.51 > 2.132

. Reject

H_0

. Tutoring significantly improved scores.

First step

Differences

d_i = \text{After} - \text{Before}

5, 5, 5, 5, 10

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

hard

Explain when to use a paired t-test vs. a two-sample t-test. If shoe comfort is measured on the same subjects wearing Brand A and Brand B, which test is appropriate and why?

Example 3

easy

Five pairs give differences

\{0, 2, 0, 4, 4\}

. Compute

\bar{d}

and state the null hypothesis for a paired t-test.

Example 4

medium

Four pairs give differences

\{2, 4, 6, 8\}

. Find

\bar{d}

s_d

, and the paired t-statistic.

Example 5

medium

Six pairs of weights (before/after a diet, in kg) are: before

\{80, 85, 90, 75, 82, 88\}

, after

\{77, 82, 86, 73, 79, 84\}

. Compute the mean difference (after

-

before).

Example 6

medium

Build a

95\%

CI for

\mu_d

when

\bar{d} = 10

s_d = 4

n = 16

. Use

t^* = 2.131

Example 7

medium

Three twins' weights are recorded: twin A in group 1, twin B in group 2. The 10 pairs give differences with

\bar{d} = 1.5

kg,

s_d = 2

kg. Compute

t

and decide at

\alpha = 0.05

two-sided (

t^* = 2.262

Example 8

medium

Five pairs give differences

\{-1, 0, 2, 1, 3\}

. Find

\bar{d}

and

s_d

Example 9

hard

A paired study has

\bar{d} = 3

s_d = 6

n = 36

. Compute

t

, and find the two-sided p-value using

P(|T_{35}| > 3) \approx 0.005

Example 10

hard

Twelve patients' cholesterol before and after a drug have

\bar{d} = -8

mg/dL,

s_d = 12

. Construct a

90\%

CI for

\mu_d

using

t^* = 1.796

Example 11

hard

A study reports a paired test result as 'significant,

p < 0.05

', but doesn't report the effect size. Critique this reporting.

Example 12

hard

In a paired study, the correlation between before and after measurements is

\rho

. How does

\rho

affect the paired test's standard error relative to a two-sample test?

Example 13

challenge

Derive: if

X_{1i}, X_{2i}

are paired observations with

\text{Var}(X_{1i}) = \text{Var}(X_{2i}) = \sigma^2

and

\text{Cor}(X_{1i}, X_{2i}) = \rho

, show that

\text{Var}(d_i) = 2\sigma^2(1-\rho)

Practice Problems

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

Example 1

easy

Five paired differences are:

\{2, -1, 3, 0, 4\}

. Calculate

\bar{d}

and

s_d

, then set up the t-test statistic formula.

Example 2

hard

A paired t-test for blood pressure before and after medication:

\bar{d}=-8

mmHg,

s_d=5

mmHg,

n=16

. Construct a 95% CI for the true mean difference and interpret.

Example 3

easy

A paired design records each subject's before and after scores. What single quantity does the paired t-test analyze for each subject?

Example 4

easy

Subject scores before/after:

10 \to 13

. Compute the difference

d = \text{after} - \text{before}

Example 5

easy

For paired data, the null hypothesis is usually that the mean difference equals what value?

Example 6

easy

Differences are

2, 4, 6

. Compute the mean difference

\bar{d}

Example 7

easy

With

n = 16

pairs, how many degrees of freedom does the paired t-test use?

Example 8

easy

The paired t-statistic is

t = \frac{\bar{d}}{s_d / \sqrt{n}}

. If

\bar{d} = 6

s_d = 4

n = 16

, compute

t

Example 9

easy

Which condition must be checked for a paired t-test: normality of the differences, or normality of each original group?

Example 10

easy

Why does a paired design often have more power than using two independent groups for the same subjects?

Example 11

medium

Differences from a before/after study are

3, 5, 7, 5

. Compute

\bar{d}

Example 12

medium

A paired t-test gives

t = 3.2

df = 9

, p-value

= 0.011

, at

\alpha = 0.05

. Conclude about the mean difference.

Example 13

medium

A study measures the SAME 30 students before and after a course. Is a paired or two-sample t-test appropriate, and why?

Example 14

medium

Build a CI for the mean difference:

\bar{d} = 8

s_d/\sqrt{n} = 1.5

t^* = 2.0

Example 15

medium

A student computes the mean of the before group (

72

) and the after group (

78

) and tests the difference

6

with a two-sample test. Why is this wrong for paired data?

Example 16

medium

Differences

d

have

\bar{d} = 0

. Without computing further, what is the paired t-statistic and the likely conclusion?

Example 17

medium

Twins are matched and one of each pair gets a treatment. Each pair gives one difference. Is this a paired or two-sample design?

Example 18

medium

A one-sided paired test (

H_a: \mu_d > 0

) has

t = 2.1

df = 14

. The two-sided p-value would be

0.054

. What is the one-sided p-value (effect in predicted direction)?

Example 19

medium

Before/after pairs give differences

1, 2, 0, 1

. Compute the mean difference

\bar{d}

Example 20

challenge

Paired differences:

4, 6, 5, 5

. Compute

\bar{d}

s_d

, and the t-statistic.

Example 21

challenge

A researcher uses a two-sample t-test on paired data and gets p-value

0.08

; the correct paired test gives

0.02

. Explain why the paired test found significance when the two-sample test did not.

Example 22

challenge

Design A measures 20 people twice (before/after). Design B uses 20 different people in each of two groups (40 total). For detecting a within-person change, which is generally more efficient and why?

Example 23

easy

A subject scores

40

before and

45

after training. Compute the difference

d = \text{after} - \text{before}

Example 24

easy

Differences are

\{1, 2, 3, 4, 5\}

. Compute

\bar{d}

Example 25

easy

Why is matching subjects (e.g., before/after on the same person) called a paired design?

Example 26

easy

What is the standard error of

\bar{d}

when

s_d = 3

and

n = 9

Example 27

medium

A paired t-test with

\bar{d} = 1.2

s_d = 3

n = 36

produces what t-statistic?

Example 28

medium

A paired test gives

t = 2.5

df = 10

. The critical value for

\alpha = 0.05

two-sided is

t^* = 2.228

. Decide.

Example 29

medium

A study tests systolic BP before and after a drug. Why is a paired t-test better than a two-sample test here?

Example 30

medium

A paired t-test gives

t = 1.0

df = 9

, p-value

= 0.343

. At

\alpha = 0.05

, decide.

Example 31

hard

A researcher inadvertently uses a two-sample t-test on paired before/after data and gets

p = 0.20

. A paired test on the same data gives

p = 0.01

. Which p-value is correct, and why is the other wrong?

Example 32

hard

For a paired study with

\bar{d} = 0.5

s_d = 2

, what minimum sample size

n

achieves

|t| \ge 2

Example 33

hard

What happens to the paired t-test if you accidentally enter the data as

\text{before} - \text{after}

instead of

\text{after} - \text{before}

Example 34

hard

A paired t-test reveals

\bar{d} = 0.3

standard deviations (Cohen's

d_z = 0.3

) with

p = 0.001

from

n = 200

pairs. Is this practically large?

← Back to Paired t-Test Formula Explained Practice Mode

Related Concepts

Hypothesis Testing Confidence Interval Sampling Distribution Mean Standard Deviation Two-Sample Tests

Background Knowledge

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

hypothesis testingconfidence intervalsampling distributionmeanstandard deviation