Question

Chapter 9, Section 3, Exercise 057

Two intervals are given, A and B, for the same value of the explanatory variable. A: 3.9 to 6.1; B: 2.6 to 7.4

(a) Which interval is the confidence interval for the mean response? A or B? Which interval is the prediction interval for the response? A or B?

(b) What is the predicted value of the response variable for this value of the explanatory variable?

Enter the exact answer.

The predicted value is

Answer 1

Solution

We assume that the underlying prediction model is: Y = β₀ + β₁X + ε and

the least square estimated Regression is: Y_cap = β_0cap + β_1capX

Back-up Theory

100(1 - α)% Confidence Interval (CI) for y_cap at x = x₀ is: (β_0cap + β_1capx₀) ± t_{n – 2, α/2}xs√[(1/n) + {(x₀ – Xbar)²/Sxx}]

100(1 - α)% Prediction Interval (PI) for y_cap at x = x₀ is: (β_0cap + β_1capx₀) ± {t_{n – 2, α/2} x s√[1 + (1/n) + {(x₀ – Xbar)²/Sxx}]}

Now to work out the solution,

Part (a)

A close examination of the above two formulae would reveal that the two formulae are virtually identical except that PI has an additional 1 (indicated in bold) in the ± part.

This would imply that ± part would be larger in PI, which in turn implies that the PI is wider than CI.

Thus, A is the CI and B is the PI. ANSWER

Part (b)

A close examination of the above two formulae would also reveal that the average of lower bound and upper bound is the same in both formulae and that common average is: (β_0cap + β_1capx₀), which is nothing but the predicted value, Y_cap.

In the given case, the common average is: {(3.9 + 6.1)/2} = 5 = {(2.6 + 7.4)/2}

Thus, predicted value of the response variable for this value of the explanatory variable is 5 ANSWER