With reference to exercise 78 use the result of part b of

Question

With reference to Exercise 78, use the result of part (b) of Exercise 40 to construct 98% limits of prediction for 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 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 40

With x₀₁, x₀₂, ... , x_0k and X₀ as defined in Exercise 39 and Y0 being a random variable that has a normal distribution with the mean β₀ + β_1x01 +···+ β_kx0k and the variance σ², 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 25.

(b) Derive a formula for (1 - α)100% limits of prediction for a future observation of Y₀.

Exercise 25

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

is a random variable having a normal distribution with zero mean and the variance

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.