52 'Suppose that the linear regression equation E(Y)=a+BX with normality and constant standard deviation a is truly appropriate for the relationship between Y and X Then, the interval of numbers 1 + "1 (X-X) (X-X) predicts where a new observation on Y will fall at that value of X. This interval is called a prediction interval for Y. To make an inference about the mean of Y (rather than a single value of ) at that value of X. one can use the confidence interval + (X-X) - )2 The 7-value in these intervals is based on df = -2. Most software has options for cal- culating these formulas (e.g., the options CLI and CLM in PROC REG in SAS). Refer to the housing data in Table 9.4, at house size X = 2.0

a) Show that = 126.0 and a 95% prediction interval is (87.0, 165.0).

b) Show that a 95% confidence interval for the mean selling price is (121.2. 130.8).

c) Explain intuitively why a prediction interval for a single observation is much wider than a confidence interval for the mean.

d) Results using these formulas are typically overly optimistic. because the model as- sumptions never hold exactly. Explain how prediction intervals would likely he in error il, in fact, the variability in housing prices tends to increase as house size increases.