Hypothesis Testing Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
A teacher claims students average 80 points. A skeptic samples n=25n=25 students: xˉ=76\bar{x}=76, s=10s=10. Using z=xˉ−μ0s/nz = \frac{\bar{x}-\mu_0}{s/\sqrt{n}}, test H0:μ=80H_0: \mu=80 vs Ha:μ<80H_a: \mu < 80 at α=0.01\alpha=0.01.

Solution

  1. 1
    z=76−8010/25=−42=−2.0z = \frac{76-80}{10/\sqrt{25}} = \frac{-4}{2} = -2.0
  2. 2
    One-tailed p-value: P(Z<−2.0)=0.0228P(Z < -2.0) = 0.0228
  3. 3
    Compare to α=0.01\alpha = 0.01: p=0.0228>0.01p = 0.0228 > 0.01
  4. 4
    Decision: Fail to reject H0H_0 at α=0.01\alpha=0.01. Evidence is insufficient to conclude the mean is below 80 at the 1% level.

Answer

z=−2.0z=-2.0, p=0.023>0.01p=0.023 > 0.01. Fail to reject H0H_0 at α=0.01\alpha=0.01. (Would reject at α=0.05\alpha=0.05.)
The significance level α changes the conclusion. At α=0.05 we would reject H₀; at the stricter α=0.01 we don't. Choosing α before the test is critical — adjusting it after seeing data is p-hacking (data dredging). Stricter α reduces Type I errors but increases Type II errors.

About Hypothesis Testing

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

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More Hypothesis Testing Examples

Example 1 medium

A school claims its students average 75 on standardized tests. A sample of [formula] gives [formula]

Example 2 hard

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

Example 3 easy

State null and alternative hypotheses for each scenario: (a) testing if a coin is fair, (b) testing

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