Hypothesis Testing Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Hypothesis Testing.

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 systematic method to decide whether sample data provides enough evidence to reject a claim (null hypothesis) about a population parameter.

Think of a courtroom trial: the null hypothesis ( $H_0$ ) is 'innocent until proven guilty.' You look at the evidence (data) and ask: 'Is this evidence so strong that it would be very unlikely if the defendant were truly innocent?' If yes, you reject the null hypothesis. If not, you don't have enough evidence to convict—but that doesn't prove innocence.

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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: Hypothesis testing checks whether sample data is surprising enough to reject a default claim about a population.

Common stuck point: The procedure for hypothesis testing is the easy part; the trap is treating 'fail to reject $H_0$ ' as 'prove $H_0$ true'. Asking "Am I deciding whether sample data is surprising enough to reject a specific stated claim about a population?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I deciding whether sample data is surprising enough to reject a specific stated claim about a population?

Worked Examples

Example 1

medium

A school claims its students average 75 on standardized tests. A sample of

n=36

gives

\bar{x}=78

with

\sigma=12

. Test

H_0: \mu=75

H_a: \mu>75

\alpha=0.05

Answer

z=1.5

p=0.067 > 0.05

. Fail to reject

H_0

. Evidence is inconclusive.

First step

Calculate test statistic:

z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} = \frac{78 - 75}{12/\sqrt{36}} = \frac{3}{2} = 1.5

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Example 2

hard

A medication is claimed to reduce blood pressure by 10 mmHg on average. A clinical trial with

n=49

patients shows

\bar{x}=8.2

mmHg reduction,

s=7

mmHg. Test

H_0: \mu=10

H_a: \mu \neq 10

\alpha=0.05

Example 3

medium

A factory claims defect rate

5\%

. From

n = 400

\hat p = 0.08

. Test

H_0: p = 0.05

H_a: p > 0.05

\alpha = 0.05

Practice Problems

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

Example 1

easy

State null and alternative hypotheses for each scenario: (a) testing if a coin is fair, (b) testing if a new drug reduces fever faster than the standard drug.

Example 2

hard

A teacher claims students average 80 points. A skeptic samples

n=25

students:

\bar{x}=76

s=10

. Using

z = \frac{\bar{x}-\mu_0}{s/\sqrt{n}}

, test

H_0: \mu=80

H_a: \mu < 80

\alpha=0.01

Example 3

easy

In a hypothesis test, what is the null hypothesis

H_0

usually a statement of?

Example 4

easy

A test gives

p = 0.02

with

\alpha = 0.05

. Do you reject or fail to reject

H_0

Example 5

easy

A test gives

p = 0.08

with

\alpha = 0.05

. State the conclusion.

Example 6

easy

True or false: failing to reject

H_0

proves

H_0

is true.

Example 7

easy

When must the significance level

\alpha

be chosen?

Example 8

easy

A claim is 'the mean equals

100

.' Write the null hypothesis symbolically.

Example 9

easy

A researcher wants to show the mean is greater than

50

. What is the alternative hypothesis

H_a

Example 10

easy

Does a statistically significant result automatically mean the effect is large and important?

Example 11

medium

A test of

H_0: \mu = 100

H_a: \mu \ne 100

has

\bar{x} = 104

\sigma = 10

n = 25

. Compute the z test statistic.

Example 12

medium

A two-sided test has test statistic

z = 2

. Using

P(Z > 2) \approx 0.0228

, find the p-value.

Example 13

medium

With test statistic

z = 2

, two-sided p-value

0.0456

, and

\alpha = 0.05

, state the conclusion.

Example 14

medium

A one-sided test

H_a: \mu > 50

has

\bar{x} = 53

\sigma = 12

n = 36

. Find

z

Example 15

medium

For a right-tailed test,

z = 1.5

and

P(Z > 1.5) \approx 0.0668

. With

\alpha = 0.05

, conclude.

Example 16

medium

A test rejects

H_0

\alpha = 0.05

but a

95\%

CI for

\mu

(48, 52)

and

H_0: \mu = 50

. Is this consistent?

Example 17

medium

A test of

H_0: \mu = 200

H_a: \mu \ne 200

has

\bar{x} = 195

\sigma = 20

n = 16

. Find

z

Example 18

challenge

Explain why we say 'fail to reject

H_0

' rather than 'accept

H_0

', using the courtroom analogy.

Example 19

challenge

With

n

very large, a trivial effect (mean differs from

100

0.1

) becomes statistically significant. Explain the mechanism and the lesson.

Example 20

challenge

A researcher runs

20

independent tests at

\alpha = 0.05

on data where every

H_0

is true. What is the expected number of false rejections, and what is this problem called?

Example 21

medium

A two-sided z-test has

z = -1.8

. Using

P(Z < -1.8) \approx 0.0359

, find the p-value and conclude at

\alpha = 0.05

Example 22

medium

A test of

H_0: \mu = 25

H_a: \mu > 25

has

\bar{x} = 28

\sigma = 6

n = 9

. Find

z

Example 23

easy

A test has

p = 0.03

and

\alpha = 0.05

. State the decision and what it means.

Example 24

easy

State the null and alternative hypotheses for testing if a coin's heads probability differs from

0.5

Example 25

easy

A z-test gives test statistic

z = 0.5

and

P(Z > 0.5) \approx 0.3085

. Find the two-sided p-value.

Example 26

easy

For

H_0: \mu = 50

H_a: \mu < 50

with

z = -2.5

and

P(Z < -2.5) \approx 0.0062

, state the conclusion at

\alpha = 0.01

Example 27

medium

For

H_0: \mu = 100

H_a: \mu \ne 100

with

\bar x = 106

\sigma = 10

n = 25

, compute the z statistic.

Example 28

medium

Continuing: with

z = 3

in a two-sided test,

P(Z > 3) \approx 0.00135

. Find the p-value.

Example 29

medium

For

H_0: p = 0.5

H_a: p > 0.5

\hat p = 0.60

n = 100

, compute the z statistic.

Example 30

medium

With

z = 2

in a right-tailed proportion test and

P(Z > 2) \approx 0.0228

, conclude at

\alpha = 0.05

Example 31

medium

Compute the z statistic for

H_0: \mu = 70

\bar x = 67

\sigma = 8

n = 64

Example 32

medium

With

z = -3

in a two-sided test,

P(Z < -3) \approx 0.00135

. Find p-value and conclude at

\alpha = 0.01

Example 33

medium

State which error type is committed: a court convicts an innocent defendant.

Example 34

medium

State which error type is committed: a court fails to convict a guilty defendant.

Example 35

medium

A two-sided test produces

p = 0.04

. What is the smallest significance level at which we reject

H_0

Example 36

medium

Match:

\alpha = 0.05

. The 95% confidence interval for

\mu

(48, 52)

and we test

H_0: \mu = 50

. Decision?

Example 37

hard

A researcher runs

50

independent hypothesis tests at

\alpha = 0.05

under all true nulls. What is the expected number of false rejections and the probability of at least one?

Example 38

hard

With

n = 100

\bar x = 102

\sigma = 10

, test

H_0: \mu = 100

H_a: \mu \ne 100

. Compute z and the p-value (use

P(Z > 2) \approx 0.0228

Example 39

hard

Why does increasing

n

(with fixed

\sigma

and effect size) tend to make any nonzero deviation statistically significant?

Example 40

hard

For

H_0: \mu = 0

H_a: \mu > 0

\sigma = 2

, the rejection region is

\bar x > 1.0

. With true

\mu = 1.5

and

n = 16

, find the power.

Example 41

hard

Explain why p-hacking inflates the Type I error rate.

Example 42

challenge

For a one-sided test

H_a: \mu > \mu_0

with

\sigma

known, derive the sample size

n

needed to achieve power

1 - \beta

at significance

\alpha

for a specified effect

\delta = \mu_1 - \mu_0

Example 43

challenge

Using the Bonferroni correction for

m

independent tests, what individual

\alpha

should each test use to maintain family-wise error rate

\le 0.05

with

m = 10

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Related Concepts

Sampling Distribution Normal Distribution Probability P-Value Type I and Type II Errors

Background Knowledge

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

sampling distributionnormal distributionprobability