Answer to Question #347862 in Statistics and Probability for Justine Jose

Question

3) A social worker reports that 30% of workers in a factory are below 15 years of age. Of the 120 employees surveyed, 38 said they were below 15 years old. Using ? = 0.05, interpret the p-value.

Answer 1

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

$H_0:p=0.30$

$H_a:p\not=0.30$

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:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{38/120-0.3}{\sqrt{\dfrac{0.3(1-0.3)}{120}}}=0.3984$$

Using the P-value approach:

The p-value is $p=2P(Z>0.3984)= 0.690335,$ and since $p=0.690335>0.05=\alpha,$ 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 0.30, at the $\alpha = 0.05$ significance level.