Answer to Question #338362 in Statistics and Probability for ethan

Question

In a public senior high school, a survey conducted last year showed that 25%

of the students’ parents are farmers. This year, a new survey was conducted

randomly on 150 students from the same school. It was found that 63 of them

have parents who are farmers. Test if the claim was higher at α = 0.01 level.

Answer 1

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

$H_0:p\le0.25$

$H_a:p>0.25$

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

Based on the information provided, the significance level is $\alpha = 0.01,$ 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:

$$z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}=\dfrac{\dfrac{63}{150}-0.25}{\sqrt{\dfrac{0.25(1-0.25)}{150}}}=4.8083$$

Since it is observed that $z = 4.8083 > 2.3263=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is $p=P(Z>4.8083)=0.000001,$ and since $p = 0.000001 < 0.01=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population proportion $p$

is greater than 0.25, at the $\alpha = 0.01$ significance level.