Suppose you want to test the claim the the paired sample data given below come from a population for which the mean difference is $\mu_d = 0$. $$\begin{array}{c|ccccccc} x & 58 & 57 & 89 & 58 & 67 & 66 & 82 \cr \hline y & 78 & 87 & 76 & 84 & 79 & 84 & 78 \cr \end{array}$$ Use a $0.01$ significance level to find the following:
(a) $\ $ The mean value of the differnces $d$ for the paired sample data
$\overline{d} =$
(b) $\ $ The standard deviation of the differences $d$ for the paired sample data
$s_d =$
(c) $\ $ The t test statistic
$t =$
(d) $\ $ The positive critical value
$t =$
(e) $\ $ The negative critical value
$t =$
(f) $\ $ Does the test statistic fall in the critical region?

A. Yes
B. No

(g) $\ $ Construct a $99$% conficence interval for the population mean of all differences $x - y$.
$< \mu_d <$