P-Value Examples in Math

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

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

Concept Recap

The probability of observing a test statistic at least as extreme as the one computed from the sample data, assuming the null hypothesis $H_0$ is true.

The p-value answers: 'If nothing special is going on ( $H_0$ is true), how surprising is my data?' A tiny p-value means the data would be very rare under $H_0$ , so maybe $H_0$ is wrong. Think of it like this: you flip a coin 100 times and get 92 heads. If the coin is fair, the chance of that happening is astronomically small (tiny p-value)—so you'd conclude the coin is probably not fair.

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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: The p-value is the probability of getting data at least this extreme assuming the null hypothesis is true.

Common stuck point: The procedure for p-value is the easy part; the trap is reading the p-value as the probability the null hypothesis is true. Asking "Am I computing the probability of data this extreme assuming the null is true (not the probability the null is true)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I computing the probability of data this extreme assuming the null is true (not the probability the null is true)?

Worked Examples

Example 1

medium

A hypothesis test produces

z=2.3

for a two-tailed test. Calculate the p-value and interpret it at both

\alpha=0.05

and

\alpha=0.01

Answer

p=0.0214

. Significant at

\alpha=0.05

but not at

\alpha=0.01

First step

Two-tailed p-value:

p = 2 \times P(Z > 2.3) = 2 \times (1 - 0.9893) = 2 \times 0.0107 = 0.0214

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Example 2

hard

Correct the following misconceptions about p-values: (a) 'p=0.03 means there's a 3% chance H₀ is true.' (b) 'p=0.03 means the effect is large.'

Example 3

easy

A right-tailed test produces

z = 0.5

. Use

P(Z > 0.5) = 0.3085

to find the p-value and decide at

\alpha = 0.05

Example 4

medium

A one-sample z-test has

\bar{x} = 52

\mu_0 = 50

\sigma = 8

n = 64

, with

H_a: \mu > 50

. Compute

z

and the p-value using

P(Z > 2) = 0.0228

Example 5

medium

A coin is flipped 100 times and lands heads 60 times. Test

H_0: p = 0.5

H_a: p \ne 0.5

using a z-test. Use

P(Z > 2) = 0.0228

Example 6

medium

A two-tailed test produces

z = 2.58

with

P(Z > 2.58) = 0.0049

. Find the p-value and decide at

\alpha = 0.01

Example 7

hard

A t-test with

df = 9

produces

t = 2.262

, the critical value for

\alpha = 0.05

two-sided. Find the p-value.

Example 8

hard

Show that the probability of getting at least one

p < 0.05

k = 10

independent tests, all under

H_0

, is

1 - 0.95^{10}

. Compute the value.

Example 9

hard

Explain the difference between

P(\text{data} \mid H_0)

(the p-value) and

P(H_0 \mid \text{data})

(the posterior on

H_0

). Why are they not equal?

Example 10

hard

A test gives

z = 1.0

with

n = 25

. If

n

were quadrupled to

100

with the same

\bar{x} - \mu_0

and

\sigma

, what would

z

become, and how does the p-value change?

Example 11

challenge

Prove: if the test statistic

T

has a continuous distribution under

H_0

, then the p-value

P = P(T \ge T_{\text{obs}} \mid H_0)

is uniformly distributed on

(0,1)

when

H_0

holds.

Practice Problems

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

Example 1

easy

A one-tailed test has

z=1.7

. Find the p-value and determine if we reject

H_0

\alpha=0.05

Example 2

hard

Study A:

n=50

p=0.04

, effect size

d=0.15

(tiny). Study B:

n=10

p=0.06

, effect size

d=0.8

(large). Discuss which study's result is more practically meaningful and why we shouldn't rely solely on p-values.

Example 3

easy

Define in words what a p-value measures.

Example 4

easy

A test gives

p = 0.03

with

\alpha = 0.05

. Reject or fail to reject

H_0

Example 5

easy

A test gives

p = 0.20

. Does this prove the null hypothesis is true?

Example 6

easy

True or false: the p-value is the probability that

H_0

is true.

Example 7

easy

Which p-value is stronger evidence against

H_0

0.001

0.04

Example 8

easy

For a right-tailed test with

z = 1.645

, the p-value equals which tail area? Use

P(Z > 1.645) = 0.05

Example 9

easy

\alpha = 0.05

a fundamental law of nature or a convention?

Example 10

easy

A two-sided test has one-tail area

0.015

. What is the p-value?

Example 11

medium

A right-tailed test has

z = 2.33

with

P(Z > 2.33) \approx 0.0099

. With

\alpha = 0.01

, conclude.

Example 12

medium

A two-sided test has

z = -2.5

with

P(Z < -2.5) \approx 0.0062

. Find the p-value.

Example 13

medium

A study with

\bar{x} = 105

\mu_0 = 100

\sigma = 15

n = 9

tests

H_a: \mu \ne 100

. Find

z

, then the p-value uses

P(Z > 1) = 0.1587

Example 14

medium

Two studies report

p = 0.049

and

p = 0.051

\alpha = 0.05

. How different is the underlying evidence?

Example 15

medium

A test gives

p = 0.30

. A student writes 'there is a

30\%

chance

H_0

is true.' Correct the interpretation.

Example 16

medium

A left-tailed test

H_a: \mu < 50

has

z = -1.28

with

P(Z < -1.28) \approx 0.10

. With

\alpha = 0.05

, conclude.

Example 17

medium

H_0

is true, what is the distribution of the p-value (for a continuous test statistic)?

Example 18

challenge

Show why, under

H_0

P(\text{p-value} \le \alpha) = \alpha

, and connect this to the Type I error rate.

Example 19

challenge

A journal publishes only

p < 0.05

results. Explain why published effect sizes tend to be overstated (the 'winner's curse').

Example 20

challenge

A test statistic has p-value

0.04

. The researcher then collects more data hoping to 'confirm' significance, stopping when

p < 0.05

again. Why does this invalidate the p-value?

Example 21

medium

A right-tailed test has

z = 1.96

with

P(Z > 1.96) = 0.025

. Conclude at

\alpha = 0.05

Example 22

medium

A two-sided test has

z = 3

with

P(Z > 3) \approx 0.0013

. Find the p-value.

Example 23

easy

A right-tailed z-test gives

z = 1.28

. Using

P(Z > 1.28) = 0.10

, state the p-value.

Example 24

easy

A two-sided test produces one-tail area

0.025

. Find the p-value.

Example 25

easy

With

\alpha = 0.10

and

p = 0.07

, do we reject

H_0

Example 26

easy

Order from strongest to weakest evidence against

H_0

p = 0.4

p = 0.001

p = 0.05

Example 27

medium

A two-sided test produces

z = -1.96

with

P(Z < -1.96) = 0.025

. Find the p-value.

Example 28

medium

A study reports

p = 0.045

. A reader claims 'there is a

4.5\%

chance the treatment has no effect.' Why is this wrong?

Example 29

medium

A two-sided test has

z = 1.5

with

P(Z > 1.5) = 0.0668

. Find the p-value.

Example 30

medium

Two studies test the same drug. Study A:

n = 1000

p = 0.001

, effect tiny. Study B:

n = 20

p = 0.20

, effect large. Which has more practical significance?

Example 31

medium

What value of

\alpha

corresponds to a

99\%

confidence level?

Example 32

hard

A researcher runs 20 independent tests, all with true

H_0

, at

\alpha = 0.05

each. Expected number of false positives?

Example 33

hard

Bonferroni correction for

k

tests sets the per-test threshold at

\alpha/k

. For

\alpha = 0.05

family-wise and

k = 10

tests, what is the per-test threshold?

Example 34

hard

A study reports 'no significant difference,

p = 0.32

.' Can we conclude there is no difference between groups?

Example 35

hard

Explain why p-hacking (running many tests until something is significant) inflates the actual Type I error far above the nominal

\alpha

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Related Concepts

Hypothesis Testing Probability Type I and Type II Errors

Background Knowledge

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

hypothesis testingprobability