Answer to Question #208099 in Statistics and Probability for pallabi

Question

To evaluate the following hypotheses

H₀ :p = 0.3

H_A :p ≠ 0.3

Answer 1

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

$H_0:p=0.3$

$H_1:p\not=0.3$

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:

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

$$=\dfrac{0.36-0.3}{\sqrt{\dfrac{0.3(1-0.3)}{50}}}\approx0.92582$$

Since it is observed that $|z|=0.92582<1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population proportion $p$ is different than $0.3,$ at the $\alpha=0.05$ significance level.

Using the P-value approach: The p-value is $p=2P(Z<-0.92582)=0.35454,$ and since $p=0.35454>0.05=\alpha,$ 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 $0.3,$ at the $\alpha=0.05$ significance level.