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 convictβ€”but that doesn't prove innocence.

Read the full concept explanation β†’

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 follows a fixed procedure: (1) state H_0 and H_a, (2) choose significance level \alpha, (3) compute the test statistic, (4) find the p-value, (5) reject H_0 if p-value < \alpha, otherwise fail to reject.

Common stuck point: 'Fail to reject H_0' does NOT mean 'H_0 is true'β€”it means there's not enough evidence against it. Absence of evidence is not evidence of absence.

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 vs H_a: \mu>75 at \alpha=0.05.

Solution

  1. 1
    Calculate test statistic: z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} = \frac{78 - 75}{12/\sqrt{36}} = \frac{3}{2} = 1.5
  2. 2
    Find p-value (one-tailed): P(Z > 1.5) = 1 - 0.9332 = 0.0668
  3. 3
    Compare to \alpha = 0.05: p = 0.0668 > 0.05
  4. 4
    Decision: Fail to reject H_0. Conclusion: insufficient evidence that the true mean exceeds 75.

Answer

z=1.5, p=0.067 > 0.05. Fail to reject H_0. Evidence is inconclusive.
Hypothesis testing follows a structured procedure: state hypotheses, calculate test statistic, find p-value, compare to Ξ±, state conclusion. Failing to reject Hβ‚€ does not prove Hβ‚€ is true β€” it means the data is insufficient to reject it.

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 vs H_a: \mu \neq 10 at \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 vs H_a: \mu < 80 at \alpha=0.01.
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Background Knowledge

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

sampling distributionnormal distributionprobability