Respuesta :
Using the z-distribution, it is found that since the absolute value of the test statistic is less than the absolute value of the critical value for the two-tailed test, there is no reason to believe that the true percentage of those in the first group who suffer a second episode is different from the true percentage of those in the second group who suffer a second episode.
At the null hypothesis, it is tested if the proportions are not different, that is, their subtraction is 0, hence:
[tex]H_0: p_1 - p_2 = 0[/tex]
At the alternative hypothesis, it is tested if they are different, that is, their subtraction is not 0, hence:
[tex]H_1: p_1 - p_2 \neq 0[/tex]
For each sample, the proportion and the standard error are:
[tex]p_1 = 0.13, s_1 = \sqrt{\frac{0.13(0.87)}{163}} = 0.0263[/tex]
[tex]p_2 = 0.14, s_2 = \sqrt{\frac{0.14(0.86)}{160}} = 0.0274[/tex]
The mean and the standard error of the distribution of the difference is given by:
[tex]\overline{p} = p_2 - p_1 = 0.14 - 0.13 = 0.01[/tex]
[tex]s = \sqrt{s_1^2 + s_2^2} = \sqrt{0.0263^2 + 0.0274^2} = 0.038[/tex]
The test statistic is:
[tex]z = \frac{\overline{p} - p}{s}[/tex]
In which [tex]p = 0[/tex] is the value tested at the null hypothesis.
Hence:
[tex]z = \frac{0.01}{0.038}[/tex]
[tex]z = 0.26[/tex]
The critical value for a two-tailed test, as we are testing if the mean is different of a value, with a significance level of 0.1 is [tex]z^{\ast} = \pm 1.645[/tex]
Since the absolute value of the test statistic is less than the absolute value of the critical value for the two-tailed test, there is no reason to believe that the true percentage of those in the first group who suffer a second episode is different from the true percentage of those in the second group who suffer a second episode.
A similar problem is given at https://brainly.com/question/24166849
