Comparing the marks of students from different universities The standard deviation of first year statistics exam marks is known to be $14$. A sample of $50$
first year statistics students from University A had a mean exam mark of $75$, while a sample
of $36$ University B students had a sample mean of $80$. Test at the $10$% level of significance
whether the marks for University B are significantly better than A. 
I want to know if a one-sided ($\mu_{A}<\mu_{B}$) hypothesis test is appropriate for this question and if the critical value $z$ is $-1.28$ (very important!) and the test statistic $Z$ is $-1.634$.
 A: A one-sided test is appropriate if we were expecting
or hoping the University B tend to have significantly
higher scores. That seems to be the case because of
the wording of the question: "Test at the 10% level of significance whether the marks for University B are significantly better than A."
Then, once we see that the sample means from the two universities, the specific question becomes whether the difference between $\bar X_a = 75$ and $\bar X_b = 80$ is large enough to be called 'statistically significant' at the 10% level (or whether a difference of that size
might have happened because of the random sampling of the the students).
The test statistic for the one-sided test you describe is
$$Z = \frac{\bar X_a - \bar X_b}{\sigma\sqrt{\frac{1}{n_a}+\frac{1}{n_b}}} = 
\frac{75- 80}{14\sqrt{\frac{1}{50}+\frac{1}{36}}} = -1.634,$$
obtained using R as a calculator:
se = 14*sqrt(1/50 + 1/36)
z = (75 - 80)/se;  z
[1] -1.633913

Because the critical value at the 10% level for this left-test is $t_c =-1.28$ and $Z = -1.634 \le t_c = 1.28.$
The critical value is $t_c - 1.28$ from printed normal
tables or from software, because that value cuts probability $0.1$ from the lower tail of the standard
normal distribution.
= qnorm(0.1)
[1] -1.281552

Note: If the issue is whether the two universities differ (in either direction) then you'd want to test
$H_0: \mu_a = \mu_b$ against $H_a: \mu_a \ne \mu_b.$
For that two-sided test, we would reject $H_0$ if $|Z| > t_c^\prime = 1.645.$ The critical value is chosen
because values $\pm 1.645$ cut probabilities 5% from upper and lower tail, respectively, of the standard
normal distribution.  Because $|Z| = 1.634 < 1.645,$
we would not reject $H_0$ against the two-sided alternative.
qnorm(0.95)
[1] 1.644854

Notice that the one-tailed test rejects and and two-tailed test does not. Proper statistical practice is
to decide before seeing the data whether to do a one or two-sided test.
