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

A formal procedure for using sample data to decide between two competing claims (hypotheses) about a population parameter.

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.

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: Hypothesis testing evaluates whether observed data provides enough evidence to reject the null hypothesis in favor of an alternative, using a predetermined significance threshold.

Common stuck point: Failing to reject the null hypothesis does NOT prove it is true — it only means the data did not provide strong enough evidence to reject it.

Worked Examples

Example 1

hard
A company claims its light bulbs last an average of 1000 hours. A sample of 50 bulbs has \bar{x} = 985 hours with \sigma = 40. Set up the null and alternative hypotheses for a two-tailed test.

Solution

  1. 1
    Step 1: Null hypothesis H_0: \mu = 1000 (the claim is true).
  2. 2
    Step 2: Alternative hypothesis H_a: \mu \neq 1000 (the mean differs from the claim).
  3. 3
    Step 3: This is two-tailed because we are testing whether the mean is different (in either direction) from 1000.

Answer

H_0: \mu = 1000, H_a: \mu \neq 1000 (two-tailed test).
Hypothesis testing starts by stating the null hypothesis (status quo or claim) and the alternative hypothesis (what we suspect). The test then determines if the data provides enough evidence to reject H_0.

Example 2

hard
Continuing the light bulb example (H_0: \mu = 1000, n = 50, \bar{x} = 985, \sigma = 40), calculate the test statistic.

Practice Problems

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

Example 1

hard
A school claims students sleep an average of 8 hours. A sample of 36 students has \bar{x} = 7.5 with \sigma = 1.2. State the hypotheses for a one-tailed test (testing if students sleep less) and compute the z-statistic.

Example 2

hard
In a two-tailed z-test at \alpha = 0.05, the test statistic is z = 2.10. Using critical values \pm 1.96, should H_0 be rejected?
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Background Knowledge

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

sampling distributionstandard errorprobability basic