Comparing the marks of students from different universities

Question

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 $-1.634$.

Answer 1

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.