A random sample of $100$ observations from a population with standard deviation $9.97364640094697$ yielded a sample mean of $91.7$.
(a) $\ $ Given that the null hypothesis is $\mu = 90$ and the alternative hypothesis is $\mu > 90$ using $\alpha = .05$, find the following:

(i) $\ $ critical $z$ score $\ $

(ii) $\ $ test statistic $=$
(b) $\ $ Given that the null hypothesis is $\mu = 90$ and the alternative hypothesis is $\mu \ne 90$ using $\alpha = .05$, find the following:

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

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

(iii) $\ $ test statistic $=$

The conclusion from part (a) is:

A. Reject the null hypothesis
B. There is insufficient evidence to reject the null hypothesis
C. None of the above

The conclusion from part (b) is:

A. Reject the null hypothesis
B. There is insufficient evidence to reject the null hypothesis
C. None of the above