Hypothesis Testing Examples in Statistics

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 Statistics.

Concept Recap

Hypothesis testing is a formal statistical procedure for using sample data to decide between two competing claims about a population parameter. You state a null hypothesis (no effect) and an alternative hypothesis, collect data, compute a test statistic, and determine whether the evidence is strong enough to reject the null.

Hypothesis testing is like a courtroom trial for data. You start by assuming innocence (null hypothesis: nothing special is happening). Then you look at the evidence (data). If the evidence is strong enough to be very unlikely under the assumption of innocence, you reject it and conclude something real is happening.

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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 uses a sample result and a variation model to make a careful population statement.

Common stuck point: Students often know a procedure related to hypothesis testing but skip the recognition step: Am I using sample-to-sample variation to make a population claim with uncertainty stated clearly? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I using sample-to-sample variation to make a population claim with uncertainty stated clearly?

Worked Examples

Example 1

medium

A sample of

n=36

has mean

\bar{x}=52

and known

\sigma=6

. Test

H_0: \mu = 50

vs.

H_a: \mu \ne 50

. Compute the z-test statistic.

Answer

z = 2

First step

Standard error:

\sigma/\sqrt{n} = 6/\sqrt{36} = 1

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Example 2

medium

Power of a test equals

1 - \beta

. If

\beta = 0.20

, what is the power?

Example 3

hard

A factory's mean weight is supposed to be

\mu_0 = 100

g. A sample of

n=25

has

\bar{x}=98

and

s = 5

. Compute the t-statistic for

H_0: \mu = 100

Example 4

hard

80\%

confidence interval for

\mu

(48.5, 51.5)

. Does a two-sided test of

H_0: \mu = 50

\alpha = 0.20

reject?

Example 5

medium

A coin is flipped

100

times and lands heads

60

times. Test

H_0: p = 0.5

vs.

H_a: p \ne 0.5

. Compute z.

Example 6

challenge

Two researchers test the same hypothesis at

\alpha = 0.05

. The chance both make a Type I error (when

H_0

is true and tests are independent) is approximately what?

Example 7

medium

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

H_0: p=0.5

H_a: p\ne 0.5

and SE

= \sqrt{0.5\cdot 0.5/100}=0.05

. Compute

z

Example 8

medium

A sample has

\bar{x}=52

s=8

n=64

. Test

H_0:\mu=50

H_a:\mu\ne 50

. Compute

t

Example 9

medium

We test

H_0: p=0.3

H_a: p>0.3

with

\hat{p}=0.35

n=100

, SE under

H_0

= \sqrt{0.3\cdot 0.7/100} \approx 0.0458

. Compute

z

Example 10

hard

Two independent samples have

\bar{x}_1=20, s_1=4, n_1=50

and

\bar{x}_2=18, s_2=5, n_2=50

. Test

H_0:\mu_1=\mu_2

H_a:\mu_1\ne\mu_2

. Compute

t

using

\text{SE}=\sqrt{s_1^2/n_1+s_2^2/n_2}

Example 11

hard

A paired-sample test of

n=25

differences has

\bar{d}=3

s_d=10

. Test

H_0: \mu_d = 0

. Compute

t

Example 12

challenge

Test

H_0: p=0.5

H_a: p>0.5

. Out of

n=400

, 220 successes. Compute

z

using SE under

H_0

Example 13

hard

A company claims its light bulbs last an average of 1000 hours. A sample of 50 bulbs has

\overset{ˉ}{x} = 985

hours with

σ = 40

. Set up the null and alternative hypotheses for a two-tailed test.

Example 14

hard

Continuing the light bulb example (

H_{0} : μ = 1000

n = 50

\overset{ˉ}{x} = 985

σ = 40

), calculate the test statistic.

Practice Problems

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

Example 1

easy

In hypothesis testing, what does the null hypothesis

H_0

typically state?

Example 2

easy

What is the alternative hypothesis

H_a

Example 3

easy

In a hypothesis test, which hypothesis do we assume true while computing evidence?

Example 4

easy

What do we do when the evidence against

H_0

is strong enough?

Example 5

easy

True or false: failing to reject

H_0

proves the null is true.

Example 6

easy

A test statistic measures how far the sample result is from

H_0

in units of what?

Example 7

easy

Fill in: hypothesis testing decides between two competing claims about a population ____.

Example 8

easy

Order the steps: (a) compute test statistic, (b) state hypotheses, (c) decide. What is the correct sequence?

Example 9

medium

A sample mean is 105, the null claims

\mu_0=100

, and SE

=2.5

. Compute the test statistic

z

Example 10

medium

State

H_0

and

H_a

for testing whether a coin is biased toward heads (

p

= probability of heads).

Example 11

medium

A test gives

z=2

for a two-sided alternative. Roughly what is the p-value (use that beyond

|z|=2

is about 5%)?

Example 12

medium

A two-sided test gives

z=2

\approx 0.05

). At

\alpha=0.05

, what is the decision?

Example 13

medium

A new drug shows a tiny, clinically meaningless improvement, but the test rejects

H_0

with

n=100{,}000

. What does this illustrate?

Example 14

medium

Why must hypotheses be stated before collecting or examining the data?

Example 15

medium

A one-sided test of

H_a: \mu>50

gives

z=1.7

. The upper-tail area beyond

z=1.7

is about 0.045. At

\alpha=0.05

, decide.

Example 16

medium

In the courtroom analogy, what plays the role of the null hypothesis?

Example 17

medium

A sample of

n=49

gives

\bar{x}=53

, the null claims

\mu_0=50

, and

\sigma=14

. Compute the test statistic

z

Example 18

challenge

A sample of

n=64

gives

\bar{x}=52

\sigma=8

, testing

H_0:\mu=50

H_a:\mu>50

. Compute

z

and decide at

\alpha=0.05

(one-sided

z^*=1.645

Example 19

challenge

Explain the difference between a Type I and a Type II error in hypothesis testing.

Example 20

challenge

A study tests 20 independent hypotheses, each at

\alpha=0.05

, with all nulls actually true. About how many false 'significant' results are expected?

Example 21

easy

A coffee company claims its bags contain

250

g. State

H_0

for testing whether the true mean weight differs from

250

Example 22

easy

A teacher believes a new method increases mean test scores above the historical mean of

72

. State

H_a

Example 23

easy

At significance level

\alpha = 0.05

, a p-value of

0.03

leads to what decision about

H_0

Example 24

easy

\alpha = 0.01

, a p-value of

0.04

leads to what decision about

H_0

Example 25

medium

For

H_a: \mu > 50

with

z = 2

, find the approximate p-value (use

P(Z>2) \approx 0.0228

Example 26

medium

For

H_a: \mu \ne 50

with

z = 2

, find the approximate two-sided p-value (use

P(Z>2) \approx 0.0228

Example 27

medium

A drug company tests whether its pill lowers blood pressure (

H_a: \mu < 130

). State the direction of the rejection region for a z-test.

Example 28

medium

A two-proportion z-test compares

\hat{p}_1 = 0.40

and

\hat{p}_2 = 0.50

with SE of the difference

=0.05

. Compute the z-statistic.

Example 29

medium

A claim says 'at least

60\%

approve.' A pollster wants to test against this claim. State

H_a

if she suspects approval is lower.

Example 30

hard

For a t-test with

n=25

, how many degrees of freedom does the t-distribution have?

Example 31

hard

A test has

\alpha = 0.05

and observed p-value

= 0.051

. State and justify the decision.

Example 32

hard

Doubling the sample size (other things equal) changes the standard error by what factor?

Example 33

hard

A chi-square goodness-of-fit test has

4

categories with no estimated parameters. How many degrees of freedom?

Example 34

challenge

20

independent tests are run at

\alpha = 0.05

when all nulls are true, what is the expected number of false rejections?

Example 35

easy

True or false: if

p > \alpha

, we reject

H_0

Example 36

easy

Which hypothesis carries the burden of proof?

Example 37

easy

If a two-sided test gives

z = -2.5

, is the result statistically significant at

\alpha=0.05

Example 38

medium

A drug trial wants to detect whether the drug raises the cure rate above 50%. Write

H_0

and

H_a

Example 39

medium

A test gives

z = 1.5

(two-sided). The two-tailed p-value is roughly 0.134. At

\alpha=0.05

, what is the decision?

Example 40

medium

Which is the better description of

\alpha

: (A) the probability

H_0

is true; (B) the probability we wrongly reject a true

H_0

Example 41

medium

Which decision is consistent with a 95% CI for

\mu - \mu_0

that contains 0?

Example 42

medium

If we shrink

\alpha

from 0.05 to 0.01, what happens to (i) Type I error rate, (ii) Type II error rate?

Example 43

hard

An advertiser tests 20 independent claims at

\alpha=0.05

each. If every

H_0

is true, about how many will be 'significant' by chance?

Example 44

hard

Why is 'fail to reject

H_0

' weaker than 'accept

H_0

Example 45

hard

Why might a researcher use a one-sided test rather than two-sided?

Example 46

medium

True or false: a statistically significant result with

p=0.04

always means the practical effect is meaningful.

Example 47

medium

A class average is claimed to be 70. We test

H_0:\mu=70

H_a:\mu<70

. A sample gives

\bar{x}=69

with

z=-0.8

. Are we 'getting evidence the average is lower'?

Example 48

hard

An experimenter runs the test, gets

p=0.07

, then collects more data and stops when

p

first drops below 0.05. Why is this wrong?

Example 49

hard

A school claims students sleep an average of 8 hours. A sample of 36 students has

\overset{ˉ}{x} = 7.5

with

σ = 1.2

. State the hypotheses for a one-tailed test (testing if students sleep less) and compute the z-statistic.

Example 50

hard

In a two-tailed z-test at

α = 0.05

, the test statistic is

z = 2.10

. Using critical values

\pm 1.96

, should

H_{0}

be rejected?

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Related Concepts

Sampling Distribution Standard Error Basic Probability P-Value Statistical Significance

Background Knowledge

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

sampling distributionstandard errorprobability basic