Chapter 13.2, Problem 27E

Exercise 13.18 described a regression analysis with y = Sales revenue and x = Advertising expenditure. Summary quantities given there result in

$n=15b=52.27s_b=8.05$

1. a. Test the hypothesis H₀: β = 0 versus H_a: β ≠ 0 using a significance level of 0.05. What does the conclusion say about the nature of the relationship between x and y?
2. b. Consider the hypothesis H₀: β = 40 versus H_a: β > 40. The null hypothesis states that the average change in sales revenue associated with a 1-unit increase in advertising expenditure is (at most) $40,000. Carry out a test using significance level 0.01.

13.18 A simple linear regression model was used to describe the relationship between sales revenue y (in thousands of dollars) and advertising expenditure x (also in thousands of dollars) for fast-food outlets during a 3-month period. A random sample of 15 outlets resulted in the accompanying summary quantities.

$\begin{array}{l}\sum x=14.10\sum y=1438.50\sum x^2=13.92\\ \sum y^2=140,354\sum xy=1387.20\\ \sum {(}^y=2401.85\sum {(}^y=561.46\end{array}$

1. a. What proportion of observed variation in sales revenue can be attributed to the linear relationship between revenue and advertising expenditure?
2. b. Calculate s_e and s_b.
3. c. Calculate a 90% confidence interval for β, the average change in revenue associated with a $1000 (that is, 1 unit) increase in advertising expenditure.