Efron (1982) analyzes data on law school admission, with the object being to examine the correlation between the LSAT (Law School Admission Test) score and the first-year GPA (grade point average). For each of 15 law schools, we have the pair of data points (average LSAT, average GPA):

(a) Calculate the correlation coefficient between LSAT score and GPA.
(b) Use the nonparametric bootstrap to estimate the standard deviation of the correlation coefficient. Use B = 1000 resamples, and also plot them in a histogram.
(c) Use the parametric bootstrap to estimate the standard deviation of the correlation coefficient. Assume that (LSAT, GRE) has a bivariate norma] distribution, and estimate the five parameters. Then generate 1000 samples of 15 pairs from this bivariate normal distribution.
(d) If (X, Y) are bivariate normal with correlation coefficient p and sample correlation r, then the Delta Method can be used to show that

Use this fact to estimate the standard deviation of r. How does it compare to the bootstrap estimates? Draw an approximate pdf of r.
(e) Fisher's z-transformation is a variance-stabilizing transformation for the correlation coefficient. If (X, Y) sire bivariate normal with correlation coefficient e and sample correlation r, then

is approximately normal. Use this fact to draw an approximate pdf of r.
(Establishing the normality result in part (q)d involves some tedious matrix calculations; see Lehmann and Casella 1998, Example 6.5). The z-transformation of part (q)e yields faster convergence to normality that the Delta Method of part (q)d. Diaconis and Holmes 1994 do an exhaustive bootstrap for this problem, enumerating all 77,558,760 correlation coefficients.)

(576,3.39) (580, 3.07) (555, 3.00) (661,3.43) (651,3.36) (605, 3.13) (653,3.12) (575,2.74) (545,2.76) (572,2.88) (594,2.96) (635, 3.30) (558, 2.81) (578,3.03) (666, 3.44) 212