In order to compare the means of two populations, independent random samples of $420$ observations are selected from each population, with the following results:

Sample 1	Sample 2
$\overline{x}_1 = 5286$	$\overline{x}_2 = 5093$
$s_1 = 180$	$s_2 = 200$

(a) $\ $ Use a $98$ % confidence interval to estimate the difference between the population means $(\mu_1 - \mu_2)$.

$\leq (\mu_1 - \mu_2) \leq$
(b) $\ $ Test the null hypothesis: $H_0:(\mu_1 - \mu_2) = 0$ versus the alternative hypothesis: $H_a:(\mu_1 - \mu_2) \ne 0$. Using $\alpha = 0.02$, give the following:

(i) $\ $ the test statistic $z =$

(ii) $\ $ the positive critical $z$ score $\ $

(iii) $\ $ the negative critical $z$ score $\ $

The final conclustion is

A. There is not sufficient evidence to reject the null hypothesis that $(\mu_1 - \mu_2) = 0$.
B. We can reject the null hypothesis that $(\mu_1 - \mu_2) = 0$ and accept that $(\mu_1 - \mu_2) \ne 0$.

(c) $\ $ Test the null hypothesis: $H_0:(\mu_1 - \mu_2) = 25$ versus the alternative hypothesis: $H_a:(\mu_1 - \mu_2) \ne 25$. Using $\alpha = 0.02$, give the following:

(i) $\ $ the test statistic $z =$

(ii) $\ $ the positive critical $z$ score $\ $

(iii) $\ $ the negative critical $z$ score $\ $

The final conclustion is

A. There is not sufficient evidence to reject the null hypothesis that $(\mu_1 - \mu_2) = 25$.
B. We can reject the null hypothesis that $(\mu_1 - \mu_2) = 25$ and accept that $(\mu_1 - \mu_2) \ne 25$.