Question

Calculate the 𝑝-value of the test of the following hypothesis given that the sample proportion 𝑝 =

0.63, n = 800. The null hypothesis and alternative hypothesis are: H0: 𝑝 = 0.60 vs H1: 𝑝 > 0.60.

Will the null hypothesis be rejected at a 5% level of significance?

Answer 1

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

$H_0:p=0.60$

$H_1:p>0.60$

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

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{0.63-0.6}{\sqrt{\dfrac{0.6(1-0.6)}{800}}}$$

$$\approx1.7320508$$

The p-value is $p=P(Z>1.7650508)=0.041632.$

Since $p = 0.041632 < 0.05=\alpha,$ it is concluded that the null hypothesis is rejected.