Question

In a random sample of 100 units from an assembly line, 15 were defective. Does this constitute sufficient evidence at the 10% significance to conclude that the defective rate among all units exceeds 10%?

Answer 1

$H_0 : p = 0.1 \\ H_1 : p > 0.1$

If H₀ is correct, is correct, $np = 100 \times 0.1=10$ and $nq = 100 \times 0.9=90$ are both above 5.

The test statistic is The test statistic is Z.

The significance level is 10%, so α = 0.10

This is a right-tail test and the critical value is z_α= z_0.10 = 1.282

Reject H₀ if the observed test statistic is greater than 1.282.

$σ_{\hat{p}}= \sqrt{ \frac{p_0q_0}{n} } = \sqrt{ \frac{0.1 \times 0.9}{100} }= 0.03 \\ z = \frac{\hat{p}-p_0}{σ_{\hat{p}}} = \frac{0.15-0.1}{0.03}=1.667$

z = 1.667 > 1.282 so we can reject H₀. At α = 0.10 there is sufficient evidence to conclude that the defective rate is above 10%.