With reference to Exercise 78, use the result of part (b) of Exercise 39 to construct a 99% confidence interval for the mean weekly net profit of restaurants with a seating capacity of 210 at a location where the daily traffic count averages 14,000 cars.

Exercise 78

The following are data on the average weekly profits (in $1,000) of five restaurants, their seating capacities, and the average daily traffic (in thousands of cars) that passes their locations:

(a) Assuming that the regression is linear, estimate β₀, β₁, and β₂.

(b) Use the results of part (a) to predict the average weekly net profit of a restaurant with a seating capacity of 210 at a location where the daily traffic count averages 14,000 cars

Exercise 39

If x₀₁, x₀₂, ... , x_0k are given values of x₁, x₂, ... , x_k and X₀ is the column vector

it can be shown that

is a value of a random variable having the t distribution with n - k - 1 degrees of freedom.

(a) Show that for k = 1 this statistic is equivalent to the one of Exercise 23.

(b) Derive a (1 - α)100% confidence interval formula for

Exercise 23

Use the results of Exercises 20 and 21 and the fact that

Exercises 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

Exercises 21

This question has been intentionally omitted for this edition.