Question

Directions: Draw a conclusion for the given situation using the five-step hypothesis testing

procedure for population proportion in both methods (critical value method and

the p-value method)

1. A nationwide poll claims that the country’s president has a less than 64% approval rating.

In a random sample of 120 people, 69 of them gave the president a positive approval

rating. Test the claim at 0.05 level of significance.

Answer 1

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

$H_0:p\ge0.64$

$H_a:p<0.64$

This corresponds to a left-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 left-tailed test is $z_c = -1.6449.$

The rejection region for this left-tailed test is $R = \{z: z < -1.6449\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{69/120-0.64}{\sqrt{\dfrac{0.64(1-0.64)}{120}}}=-1.4834$$

Since it is observed that $z = -1.4834 \ge-1.6449= z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is $p=P(Z<-1.4834)=0.068984,$ and since $p=0.068984>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population proportion $p$ is less than 0.64, at the $\alpha = 0.05$ significance level.