Question

A researcher claims that 13% of all motorcycle helmets have manufacturing flaws that could potentially cause injury to the wearer. A sample of 150 of these helmets revealed that 18 contained such defects.

Answer 1

The following null and alternative hypotheses for the population proportion needs to be tested:

$H_0:p=0.13$

$H_a:p\not=0.13$

This corresponds to a two-tailed test, for which a z-test for one population proportion will be used.

Based on the information provided, the significance level is $\alpha = 0.05,$ and the critical value for a two-tailed test is $z_c =1.96.$

The rejection region for this two-tailed test is $R = \{z: |z| > 1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{\dfrac{18}{150}-0.13}{\sqrt{\dfrac{0.13(1-0.13)}{150}}}\approx-0.3642$$

Since it is observed that $|z|=0.3642< 1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is $p=2P(Z<-0.3642)=0.715709,$ and since $p=0.715709>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population proportion $p$ is different than $0.13,$ at the $\alpha = 0.05$ significance level.