The slope of the line drawn tangent to the probit regression curve at a particular x value equals (0.40)β exp[−(α + βx)2/2].

a. Show this is highest when x = −α/β, where it equals 0.40β. At this point,

π(x) = 0.50.

b. The fit of the probit model to the horseshoe crab data using x = width is probit[ ˆ π(x)] = −7.502 + 0.302x. At which x-value does the estimated probability of a satellite equal 0.50?

c. Find the rate of change in πˆ (x) per 1 cm increase in width at the xvalue found in (b). Compare the results with those obtained with logistic regression in Section 4.1.3, for which πˆ (x) = 1/2 at x = 24.8, where the rate of change is 0.12. (Probit and logistic models give very similar fits to data.)