With reference to Example 4, use the theory of Exercise 22 to test the null hypothesis α = 21.50 against the alternative hypothesis α  21.50 at the 0.01 level of significance.

Exercise 22

Use the result of part (b) of Exercise 20 to show that

 is a value of a random variable having the standard normal distribution. Also, use the first part of Theorem 3 and the fact that  are independent to show that

is a value of a random variable having the t distribution with n - 2 degrees of freedom.

Exercise 20

Under the assumptions of normal regression analysis, show that

(a) the least squares estimate of α in Theorem 2 can be written in the form

(b)  has a normal distribution with

Theorem 2

Example 4

With reference to the data in the table in Section 3,

(a) find the equation of the least squares line that approximates the regression of the test scores on the number of hours studied;

(b) predict the average test score of a person who studied 14 hours for the test.