Question: 13.

Car purchase. A marketing research firm was engaged by an automobile manufacturer to conduct a pilot study to examine the feasibility of using logistic regression for ascertaining the likelihood that a family will purchase a new car during the next year. A random sample of 33 suburban families was selected. Data on annual family income (X₁, in thousand dollars) and the current age of the oldest family automobile (X₂, in years) were obtained. A follow?up interview conducted 12 months later was used to determine whether the family actually purchased a new car (Y = 1) or did not purchase a new car (Y = 0) during the year.

i: 1 2 3 . . . 31 32 33

X_i1: 30 45 60 . . . 21 32 17

X_i2: 2 2 2 . . . 3 5 1

Y_j: 1 0 1 . . . 0 1 0

Multiple logistic regression model: E{Y_i} = π_i = exp(X’_iβ) / 1 + exp(X’_iβ) with two predictor variables in first?order terms is assumed to be appropriate.

a. Find the maximum likelihood estimates of b_o, b₁, and b₂. State the fitted response function.

b. Obtain exp(b₁) and exp(b₂) and interpret these numbers.

c. What is the estimated probability that a family with annual income of $50 thousand and an oldest car of 3 years will purchase a new car next year?

e. Which independent variable is most strongly related to Y? How do you know?