Question

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?

Answer 1

a. There is evidence that the population mean is above $300.

b. There is no evidence that the population mean is above $300.

c. There is no evidence that the population mean is above $300.

d. The director could ask for cheaper similar books.

Explanation:

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

We have the following null and alternative hypothesis

`H_(0): \mu = 300` vs
`H_(1): \mu > 300` (upper-tail alternative)

We will use the test statistic

`T = \frac{\bar{X}-300}{S/√(25)}` and the observed value is

`t_(0) = (315.4 - 300)/(43.20/√(25)) = 1.7824`.

If
`H_(0)` 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 >
`t_(0.9)`} where
`t_(0.9) = 1.3178` is the 90th quantile of the t distribution with 24 df, so, RR = t . 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
`t_(0) = (315.4 - 300)/(75/√(25)) = 1.0267`. The rejection region for a 0.05 level of significance is RR = {t | t >
`t_(0.95)`} where
`t_(0.95) = 1.7108` is the 95th quantile of the t distribution with 24 df, so, RR = t . Because the observed value does not fall inside the rejection region, there is no evidence that the population mean is above $300.

c. If
`\bar{x} = 305.11` and s = 43.20, the observed value is
`t_(0) = (305.11 - 300)/(43.20/√(25)) =  0.5914`. For RR = t 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.