Question

A survey shows that 10% of students are victimized by bullies in school each year. A random sample of 527 students shows a victimization rate of 14%. Are students more likely to be bullied in school? Assume significance level at 0.01.

Answer 1

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

$H_0:p\leq 0.1$

$H_1: p>0.1$

Based on the information provided, the significance level is $\alpha = 0.01,$ and the critical value for a right-tailed test is $z_c = 2.3263.$

The rejection region for this right-tailed test is $R = \{z: z > 2.3263\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}=\dfrac{0.14-0.1}{\sqrt{\dfrac{0.1(1-0.1)}{527}}}$$

$$\approx3.060864$$

Since it is observed that $z = 3.060864 >2.3263= z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is $p=P(Z>3.060864)\approx0.0011035,$ and since $p=0.0011035<0.01=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population proportion $p$ is greater than $0.1,$ at the $\alpha = 0.01$ significance level.