The well-known psychologist Dr. Elbod has established what he calls his Generalized Anxiety (GA) scale. The GA scale, which is a scale from 0 to 10, measures the "general anxiety" of an individual, with higher GA scores corresponding to more anxiety. (Dr. Elbod's assessment of anxiety is based on a variety of measurements, both physiological and psychological.) We'd like to make predictions about individuals' sleep behavior based on their GA scores. We've collected bivariate data that give the GA score (denoted by x) and the number of hours of sleep last night (denoted by y) for each of the 9 adults participating in a study. The least-squares regression equation for our data is y =8.02 -0.28x. We have used this equation to predict tonight's sleep time for a woman whose GA score is 6.2. We're now interested in both a prediction interval for her sleep time and a confidence interval for the mean sleep time of individuals with her GA score. We have computed the following for our data. . mean square error (MSE) = 0.606 1 , (6.2-X) 0.1356, where X1, X2, ...; X9 denote the GA scores in the sample, and x denotes their mean 1=1 Based on this information, and assuming that the regression assumptions hold, answer the questions below. (If necessary, consult a list of formulas.) (a) What is the 90% prediction interval for an individual value for sleep time (in hours) when the GA score is 6.2? (Carry your intermediate computations to at least four decimal places, and round your answer to at least two decimal places. X ? Lower limit: Upper limit: (b) Consider (but do not actually compute) the 90% confidence interval for the mean sleep time when the GA score is 6.2. How would this confidence interval compare to the prediction interval computed above (assuming that both intervals are X ? computed from the same sample data)? The confidence interval would have the same center as, but would be wider than, the prediction interval. The confidence interval would be positioned to the right of the prediction interval. The confidence interval would be identical to the prediction interval. The confidence interval would be positioned to the left of the prediction interval. The confidence interval would have the same center as, but would be narrower than, the prediction interval. (c) For the GA score values in this sample, 8.5 is more extreme than 6.2 is, that is, 8.5 is farther from the sample mean GA score than 6.2 is. How would the 90% prediction interval for the mean sleep time when the GA score is 8.5 compare to X ? the 90% prediction interval for the mean sleep time when the GA score is 6.2? The interval computed from a GA score of 8.5 would be wider but have the same center. The interval computed from a GA score of 8.5 would be wider and have a different center. The intervals would be identical. The interval computed from a GA score of 8.5 would be narrower and have a different center. B Pic Stitchroma GA score of 8.5 would be narrower but have the same center
