The director of admissions at the University of Maryland, University College is concerned about the high cost of textbooks for the students each semester.
A sample of 25 students enrolled in the university indicates that X (bar) = $315.4 and s = $43.20.a. Using the 0.10 level of significance, is there evidence that the population mean is above $300?
b. What is your answer in (a) if s = $75 and the 0.05 level of significance is used?
c. What is your answer in (a) if X (bar) = $305.11 and s = $43.20?
d. Based on the information in part (a), what decision should the director make about the books used for the courses if the goal is to keep the cost below $300?

Let X be the random variable that represents the cost of textbooks. We have observed n = 25 values, [tex]\bar{x}[/tex] = 315.4 and s = 43.20. We suppose that X is normally distributed.

We have the following null and alternative hypothesis

[tex]H_{0}: \mu = 300[/tex] vs [tex]H_{1}: \mu > 300[/tex] (upper-tail alternative)

We will use the test statistic

[tex]T = \frac{\bar{X}-300}{S/\sqrt{25}}[/tex] and the observed value is

[tex]t_{0} = \frac{315.4 - 300}{43.20/\sqrt{25}} = 1.7824[/tex].

If [tex]H_{0}[/tex] is true, then T has a t distribution with n-1 = 24 degrees of freedom.

a. The rejection region is given by RR = {t | t > [tex]t_{0.9}[/tex]} where [tex]t_{0.9} = 1.3178[/tex] is the 90th quantile of the t distribution with 24 df, so, RR = {t | t > 1.3178}. Because the observed value satisty 1.7824 > 1.3178, there is evidence that the population mean is above $300.

b. If s = 75, then the observed value is [tex]t_{0} = \frac{315.4 - 300}{75/\sqrt{25}} = 1.0267[/tex]. The rejection region for a 0.05 level of significance is RR = {t | t > [tex]t_{0.95}[/tex]} where [tex]t_{0.95} = 1.7108[/tex] is the 95th quantile of the t distribution with 24 df, so, RR = {t | t > 1.7108}. Because the observed value does not fall inside the rejection region, there is no evidence that the population mean is above $300.

c. If [tex]\bar{x} = 305.11[/tex] and s = 43.20, the observed value is [tex]t_{0} = \frac{305.11 - 300}{43.20/\sqrt{25}} = 0.5914[/tex]. For RR = {t | t > 1.3178} we have that the observed value does not fall inside RR, therefore, there is no evidence that the population mean is above $300.

d. Because the director of admissions is concerned about the high cost of textbooks, and there is evidence that the population mean of costs is above $300, the director could ask for cheaper similar books.