Use the results obtained for the data of example 9 and the

Question

Use the results obtained for the data of Example 9 and the result of part (b) of Exercise 40 to construct 99% limits of prediction for the sales price of a three-bedroom house with two baths in the given housing development.

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.

Example 9

The following data show the number of bedrooms, the number of baths, and the prices at which a random sample of eight one-family houses sold in a certain large housing development:

Use the method of least squares to fit a linear equation that will enable us to predict the average sales price of a one-family house in the given housing development in terms of the number of bedrooms and the number of baths.