Redo Exercise 61 when the dosage is 20 milligrams. Note the greatly increased width of the confidence limits for the expected number of deaths and of the limits of prediction. This example illustrates that extrapolation, estimating a value of Y for observations outside the range of the data, usually results in a highly inaccurate estimate.

Exercise 61

Use the theory of Exercises 24 and 26, as well as the quantities already calculated in Exercise 51 for the data of Exercise 42, to find

(a) a 99% confidence interval for the expected number of deaths in a group of 25 mice when the dosage is 9 milligrams;

(b) 99% limits of prediction of the number of deaths in a group of 25 mice when the dosage is 9 milligrams.

Exercises:

Derive a (1 - α) 100% confidence interval for μ_Y|x0, the mean of Y at x = x₀, by solving the double inequality -t_{α/2, n-}2 _{α/2, n}-2 with t given by the formula of Exercise 23.

Exercise 23

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

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.

Exercises 26

Solve the double inequality -t_{α/2,n-2 α/2,n-2} with t given by the formula of Exercise 25 so that the middle term is y₀ and the two limits can be calculated without knowledge of y₀. Note that although the resulting double inequality may be interpreted like a confidence interval, it is not designed to estimate a parameter; instead, it provides limits of prediction for a future observation of Y that corresponds to the (given or observed) value x₀

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.

Exercise 51

With reference to Exercise 42, test the null hypothesis β = 1.25 against the alternative hypothesis β > 1.25 at the 0.01 level of significance.

Exercise 42

Various doses of a poisonous substance were given to groups of 25 mice and the following results were observed:

(a) Find the equation of the least squares line fit to these data.

(b) Estimate the number of deaths in a group of 25 mice that receive a 7-milligram dose of this poison.