Consider again the conditions of Exercise 7. If a

Show that the M.L.E. of 2 is given by Eq. (11.2.3).

Show that E( 1) = 1.

. Show that E( 0) = 0.

Show that Var( 0) is as given in Eq. (11.2.5).

Show that Cov( 0, 1) is as given in Eq. (11.2.6). Hint: Use the result in Exercise 8 in Sec. 4.6

Show that in a problem of simple linear regression, the estimators 0 and 1 will be independent if x = 0.

Consider a problem of simple linear regression in which a patients reaction Y to a new drug B is to be related to his reaction X to a standard drug A. Suppose that the 10 pairs of observed values given in Table 11.1 are obtained. a. Determine the values of the M.L.E.s 0, 1, and 2. b. Determine the values of Var( 0) and Var( 1). c. Determine the value of the correlation of 0 and 1

Consider again the conditions of Exercise 7, and suppose that it is desired to estimate the value of = 30 21 + 5. Determine an unbiased estimator of and find its M.S.E

Consider again the conditions of Exercise 7, and let = 30 + c11, where c1 is a constant. Determine an unbiased estimator of . For what value of c1 will the M.S.E. of be smallest?

Consider again the conditions of Exercise 7. If a particular patients reaction to drug A has the value x = 2, what is the predicted value of his reaction to drug B, and what is the M.S.E. of this prediction?

Consider again the conditions of Exercise 7. For what value x of a patients reaction to drug A can his reaction to drug B be predicted with the smallest M.S.E.

Consider a problem of simple linear regression in which the durability Y of a certain type of alloy is to be related to the temperature X at which it was produced. Suppose that the eight pairs of observed values given in Table 11.3 are obtained. Determine the values of the M.L.E.s 0, 1, and 2, and also the values of Var( 0) and Var( 1).

For the conditions of Exercise 12, determine the value of the correlation of 0 and 1

Consider again the conditions of Exercise 12, and suppose that it is desired to estimate the value of = 5 40 + 1. Find an unbiased estimator of . Determine the value of and the M.S.E. of .

Consider again the conditions of Exercise 12, and let = c11 0, where c1 is a constant. Determine an unbiased estimator of . For what value of c1 will the M.S.E. of be smallest?

Consider again the conditions of Exercise 12. If a specimen of the alloy is to be produced at the temperature x = 3.25, what is the predicted value of the durability of the specimen, and what is the M.S.E. of this prediction?

Consider again the conditions of Exercise 12. For what value of the temperature x can the durability of a speci

Moore and McCabe (1999, p. 174) report prices paid for several species of seafood in 1970 and 1980. These values are in Table 11.6. If we were interested in trying to predict 1980 seafood prices from 1970 prices, a linear regression model might be used. a. Find the least-squares regression coefficients for predicting 1980 prices from 1970 prices. b. If an additional species sold for 21.4 in 1970, what would you predict for the 1980 selling price? c. What is the M.S.E. for predicting the 1980 price of a species that sold for 21.4 in 1970? Table 11.6 Fish prices in 1970 and 1980 for Exercise 18 1970 1980 1970 1980 13.1 27.3 26.7 80.1 15.3 42.4 47.5 150.7 25.8 38.7 6.6 20.3 1.8 4.5 94.7 189.7 4.9 23 61.1 131.3 55.4 166.3 135.6 404.2 39.3 109.7 47.6 149

In the 1880s, Francis Galton studied the inheritance of physical characteristics. Galton found that the sons of tall men tended to be taller than average, but shorter than their fathers. Similarly, sons of short men tended to be shorter than average, but taller than their fathers. Thus, the average heights of the sons were closer to the mean height of the population, regardless of whether the fathers were taller or shorter than average. From these observations, one might conclude that the variability of height decreases over successive generations, both tall persons and short persons tend to be eliminated, and the population regresses toward some average height. This conclusion is an example of the regression fallacy. In this problem you will prove that the regression fallacy arises in the bivariate normal distribution even when both coordinates have the same variance. In particular, assume that the vector (X1, X2) has the bivariate normal distribution with common mean , common variance 2, and positive correlation < 1. Prove that E(X2|x1) is closer to than x1 is to for every value x1. (This occurs despite the fact that X1 and X2 have the same mean and the same variance.)