Consider a linear model and y₁ = ẞ1 + ẞ2 (1 + α²) + ɛi, i = 1,3,5, ..., 2n - 1 yi = ß₁ + ß2 (1 − α²) + ɛi, i = 2,4,6, ...,2n where a is assumed to be a known constant and ɛi 's are independent and identically distributed N(0, 2) random variables. (i) Let B1 and B2 be the least squares estimators of ẞ₁ and ẞ2, respectively. Find the Var (C1ẞ₁ + C2ẞ2) for any constant C1 and C2 (ii) Find Var(B2 - B1). (iii) Find Var(B2 + ß1). (iv) Comment on the effect of a on the estimability of ẞ2 - ẞ₁ and ẞ2 + ẞ1.

A scientist has measured quantities y, x1,and x,. She believes that y is related to x, and x2 through the equation y = ae+Px2 8, where & is a random error that is always positive. Find a transformation of the data that will enable her to use a linear model to estimate B, and B,.

)A county real estate appraiser wants to develop a statistical model to predict the appraised value of 3) houses in a section of the county called East Meadow. One of the many variables thought to be an important predictor of appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the simple linear regression model: E(u) = Bo + Bix, where y = appraised value of the house (in thousands of dollars) and x = number of rooms. Using data collected for a sample of n = 73 houses in Fast Meadow, the following results were obtained: y = 73.80 + 19.72x What are the properties of the least squares line, y = 73.80 + 19.72x? A) Average error of prediction is 0, and SSE is minimum. B) It will always be a statistically useful predictor of y. C) It is normal, mean 0, constant variance, and independent. D) All 73 of the sample y-values fall on the line.