The data set has three predictors: X1 = achievement test score #1; X2 = achievement test score #2; and, Gender where 0 = females and 1 = males.The dependent variable is success in calculus: student who passed calculus I was coded Y = 1, while a student who failed calculus I (or received a WD) was coded Y = 0.

Multiple logistic regression techniques with three-predictor variables is assumed to be appropriate.

(a)Use R to find the maximum likelihood estimates of b₀, b₁, b₂, and b₃.

(b)Calculate and interpret exp(b₁), exp(b₂), and exp(b₃).

(c)What is the estimated probability that a student who scored 55 on achievement test #1 and a 60 on achievement test #2 will pass calculus?

(d)Based on the fitted regression function obtained in part (a), calculate a 95% confidence interval for the student who scored 55 on achievement test #1 and a 60 on achievement test #2.

(e)Find the total error rate for the following cutoffs: .40, .45, .50, .55, and .60.

(f)Based on your results from part (e) only, which of the 5 cutoffs minimizes the total error rate?

(g)Obtain the AUC and interpret.

(h)Provide a plot of the ROC.

Y X1 X2 Gender 0 59 52 0 0 61 55 1 1 82 51 0 0 51 70 0 0 53 70 0 0 62 49 1 0 51 69 1 0 70 54 1 0 71 65 1 0 55 58 1 0 58 48 0 0 53 58 1 0 72 65 0 0 56 68 0 0 56 83 0 0 81 68 0 0 62 44 0 0 49 70 0 0 56 69 1 0 50 74 0 0 53 57 0 0 56 64 1 0 56 67 1 0 50 83 1 0 52 48 1 0 52 81 0 0 67 53 1 0 51 61 0 0 70 51 0 0 64 51 0 0 61 65 1 0 53 51 0 0 77 54 1 0 73 64 1 0 67 69 0 0 50 71 0 1 80 38 0 1 75 51 0 0 65 54 1 0 60 59 1 1 68 57 1 0 61 63 0 1 62 48 0 0 53 58 0 0 72 56 0 0 54 59 0 1 59 75 0 0 61 48 0 0 50 79 1 0 48 66 0 0 52 57 1 0 54 68 0 0 62 48 0 0 71 60 0 1 65 63 0 0 49 61 0 0 58 57 0 0 62 69 0 0 69 38 1 1 56 50 1 1 76 45 1 0 51 72 0 0 64 51 0 0 57 62 1 0 51 81 0 0 81 55 1 0 50 77 0 0 64 65 1 0 64 53 1 1 59 49 1 0 53 65 0 0 63 58 0 0 59 60 1 1 70 57 1 0 72 37 0 0 68 49 0 0 75 55 1 0 57 60 0 0 67 57 1 0 59 56 1 0 55 58 0 1 75 64 1 0 66 51 0 0 67 59 0 0 59 61 0 0 78 49 1 0 59 49 0 0 68 55 0 1 59 61 1 0 68 50 1 1 78 47 1 0 55 73 1 1 71 45 1 0 51 45 0 0 65 59 0 0 54 61 1 0 79 52 0 0 64 50 0 1 82 46 1 0 64 67 0 0 70 56 1 1 59 50 0 0 59 56 1 0 63 61 1 0 48 74 0 0 61 78 0 0 51 68 0 0 48 71 0 1 71 58 1 0 51 57 0 0 57 51 1 0 49 74 0 0 67 56 1 0 73 57 0 0 73 65 0 0 56 47 0 0 48 69 1 0 50 71 0 0 50 76 1 0 66 60 1 0 53 75 1 0 50 65 1 1 51 42 0 0 68 66 1 1 72 49 1 0 51 58 1 0 62 61 1 0 60 55 0 0 67 60 1 0 70 54 1 0 55 63 1 0 66 56 0 0 65 59 1 0 84 52 1 0 58 63 0 1 68 57 1 0 51 59 1 0 67 53 1 0 52 67 0 0 68 62 0 0 76 63 1 0 54 62 1 0 50 52 1 0 63 58 0 0 77 49 1 0 60 65 1 0 51 55 0 0 51 60 1 0 66 51 1 0 52 67 0 0 66 64 1 0 56 55 1 0 49 58 0 0 67 66 0 0 57 64 1 0 56 66 0 1 76 22 1 1 68 32 0 1 73 56 1