Answer to Question #210268 in Statistics and Probability for Mariah

Question

every year, the assigned teachers determine the body mass index of students. in a certain public junior high school, a study finds that 10% of grade 7 students observe are underweight. A sample of 780 Grade 7 students was randomly chosen and it was found out that 125 of them are underweight. Is this claim different for their grade level age? Use 0.05 level of significance

Answer 1

$$\hat{p}=\dfrac{125}{780}=\dfrac{25}{156}$$

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

$H_0:p=0.1$

​$H_1:p\not=0.1$

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:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}$$

$$=\dfrac{\dfrac{25}{156}-0.1}{\sqrt{\dfrac{0.1(1-0.1)}{780}}}\approx5.61$$

Since it is observed that $|z|=5.61>1.96=z_c,$ it is then concluded that the null hypothesis is rejected. Therefore, there is enough evidence to claim that the population proportion $p$ is different than $0.1,$ at the $\alpha=0.05$ significance level.

Using the P-value approach: The p-value is $p=2P(z>5.61)=0,$ and since $p=0<0.05=\alpha,$

it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population proportion $p$ is different than $0.1,$ at the $\alpha=0.05$ significance level.