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- The director of admissions of a small college administered a newly designed entrance test to 20 students selected at random from the new freshman class in a study to determine whether a student’s grade point average (GPA) at the end of the freshman year (Y ) can be predicted from the entrance test score (X). The results of the study follow.
Y 3.1 2.3 3.0 1.9 2.5 3.7 3.4 2.6 2.8 1.
X 5.5 4.8 4.7 3.9 4.5 6.2 6.0 5.2 4.7 4.
Y 2.0 2.9 2.3 3.2 1.8 1.4 2.0 3.8 2.2 1.
X 4.9 5.4 5.0 6.3 4.6 4.3 5.0 5.9 4.1 4.
(a) Obtain the least-square estimate of the regression line when Y regressed on X. What is the point estimate of the change in the mean response when the entrance test score increases by one point?
(b) Plot the estimated regression function and the data. Does the estimated regression line appear to fit the data well?
(c) Obtain a 99% confidence interval for β 1. Interpret your confidence interval. Does it include zero? Why might the director of admissions be interested in whether the confidence interval includes zero?
(d) Obtain a point estimate of the mean freshman GPA for a student with entrance test score X = 5.0.
(e) Estimate σ. In what unit is σ expressed?
(f ) Obtain a 95% interval estimate of the mean freshman GPA for students whose entrance test score is 4.7. Interpret your confidence interval.
(g) Mary Jones obtained a score of 4.7 on the entrance test. Predict her freshman GPA using a 95% prediction interval. Interpret your prediction interval.
(h) Is the prediction interval in part (f ) wider than the confidence interval in part (g)? Should it be?
(i) Test H 0 : β 1 = 0 versus H 1 : β 1 6 = 0, at α = 0.05, by using a t-test.
(j) Test H 0 : β 0 = 0 versus H 1 : β 0 6 = 0, at α = 0.05, by using a t-test.
(k) Would it be more reasonable to consider Xi as known constants or as random variables here? Explain. If the Xi were considered to be random variables, would this have any effect on prediction intervals for new applicants? Explain.
- The susceptibility of catfish to a certain chemical pollutant was determined by immersing individual fish in 2 liters of an emulsion containing the pollutant and measuring the survival time in minutes.^1 The data in the following table give the common log of survival time (Y ) and the common log of concentration (X) of the pollutant in parts per million for 18 fish.
Fish 1 2 3 4 5 6 log 10 Survival time (Y ) 2.516 2.572 2.438 2.621 2.742 2. log 10 Concentration (X) 5.0 5.0 5.0 4.8 4.8 4. Fish 7 8 9 10 11 12 log 10 Survival time (Y ) 2.830 2.910 2.983 3.175 3.056 3. log 10 Concentration (X) 4.6 4.6 4.6 4.4 4.4 4. Fish 13 14 15 16 17 18 log 10 Survival time (Y ) 3.332 3.221 3.293 3.447 3.523 3. log 10 Concentration (X) 4.2 4.2 4.2 4.0 4.0 4.
(a) Determine and draw the estimated straight line of Y regressed on X on the accompanying scatter diagram. Comment on the fit.
(b) Determine a 95% confidence interval for the true mean survival time μY 1 |X (where Y 1 = 10Y^ ) at values of X = 5.
- Fisher (1947)^2 published s data set of H. G. O. Holck giving the heart weights, body weights and sex of 47 female and 97 male adult cats. These has been used in bioassay experiments with the muscle-relaxing drug digitalis. The data sets are given in my web page for male and female cats separately. It appears that cats with a body weight of at least 2 kilogram were selected. Fisher’s interest in the data was to study the relation of heart weight to body weight.
(a) Fit separate (linear) regressions of heart weight on body weight. Can heart
(^1) Adapted from a study by Nagasawa, Osano, and Kondo (1964), “An Analytical Method foe Evaluating the Susceptibility of Fish Species to an Agricultural Chemical”, Japanese Journal of Applied Enterological Zoology, 8 , 118- 122. (^2) See “The Analysis of Covariance Method for the Relation between a Part and the Whole”, Biometrics, 3 , 65-68.
