When > 0, the logistic regression curve (4.1) has the shape of the cdf of a

When β > 0, the logistic regression curve (4.1) has the shape of the cdf of a logistic distribution with mean μ = −α/β and standard deviation σ =

1.814/β. Section 4.1.3 showed that the horseshoe crab data with x = width has fit, logit[ ˆ π(x)] = −12.351 + 0.497x.

a. Show that the curve for πˆ (x) has the shape of a logistic cdf with mean 24.8 and standard deviation 3.6.

b. Since about 95% of a bell-shaped distribution occurs within two standard deviations of the mean, argue that the probability of a satellite increases from near 0 to near 1 as width increases from about 17 to 32 cm.