Question

Question 1

Suppose we wanted to predict the selling price of a house using its size in a certain area of a city. A random sample of six houses were selected from the area. The data is presented in the following table with size given in hundreds of square feet, and sale price in thousands of dollars:

Size (Xi)	12	15	18	21	24	27
Price (Yi)	60	85	75	105	120	110

We are interested in fitting the following simple linear regression model:

This can also be written in matrix form as Y = Xβ + ε

a)  Calculate X′X, (X′X)^-1  and X′Yand then calculate the least squares estimates of β₀ andβ_1.

b)   Calculate the variance-covariance matrix of b, and use it to perform a t-test to test the null hypothesis that β₁ = 0. Use α = 0.05.

Use the matrix approach to calculate the following:

c)  Calculate a 95% confidence interval for the mean of Ywhen X= 20.

d)   Calculate a 95% prediction interval for an individual new value of Ywhen X= 20.

Answer 1