Using the z-distribution, as we are working with a proportion, it is found that the correct p-value for the test is given by 0.1608.
What are the hypothesis tested?
At the null hypothesis, we test if the proportions are the same, that is:
[tex]H_0: p_1 - p_2 = 0[/tex]
At the alternative hypothesis, we test if they are different, that is:
[tex]H_1: p_1 - p_2 \neq 0[/tex].
What is the mean and the standard error of the distribution of differences?
For each sample, they are given by:
[tex]p_1 = \frac{27}{50} = 0.54, s_1 = \sqrt{\frac{0.54(0.46)}{50}} = 0.0705[/tex]
[tex]p_2 = \frac{20}{50} = 0.4, s_2 = \sqrt{\frac{0.4(0.6)}{50}} = 0.0693[/tex]
Hence, for the distribution of differences, they are given by:
[tex]\overline{p} = p_1 - p_2 = 0.54 - 0.4 = 0.14[/tex]
[tex]s = \sqrt{s_1^2 + s_2^2} = \sqrt{0.0705^2 + 0.0693^2} = 0.0989[/tex]
What are the test statistic and the p-value?
The test statistic is given by:
[tex]z = \frac{\overline{p} - p}{s}[/tex]
In which p = 0 is the value tested at the null hypothesis.
Hence:
[tex]z = \frac{\overline{p} - p}{s}[/tex]
[tex]z = \frac{0.14 - 0}{0.0989}[/tex]
[tex]z = 1.42[/tex]
The p-value of the test is found using a z-distribution calculator, with a two-tailed test, as we are testing if the proportion is different of a value, with z = 1.42, and it is of 0.1608.
More can be learned about the z-distribution at https://brainly.com/question/26454209
