Let ( x 2 , x 3 , x 4 ) = the probability a UF undergraduate attended a home foo

Question

Let ( x2, x3, x4 ) = the probability a UF undergraduate attended a home football game this past year, x2= 1 if a UF undergraduate is a sophomore and 0 otherwise, x3 = 1 if a UF undergraduate is a junior and 0 otherwise,    and x3 = 1 if a UF undergraduate is a senior and 0 otherwise. Suppose a sample of UF undergraduates is taken and each is asked if they attended a home football game this past year. The proportions who answered yes in this sample were .25 for freshmen, .23 for sophomores, .18 for juniors, and .27 for seniors.

a. Consider the model   ( x2, x3, x4 ) = + 2x2 +   3x3 +   4x4 . Give the values for , 2, 3, and 4   that would match this saturated model to the sample information given above. Note that this model uses the identity link.

b. Repeat part a for the logit model      logit[( x2, x3, x4 ) } = + 2x2 +   3x3 +   4x4 .

Explanation / Answer

a. For freshmen, x2 = x3 = x4 =0

So, pi = alpha = 0.25

x2=1 for sophomore,

pi = alpha + beta2 = 0.23, beta2 = -0.02

x3 =1 for jumior

pi = alpha + beta3 = 0.18, beta3 = -0.07

For senior year, pi = alpha + beta4 = 0.27, beta4 = 0.02

b.

logit(pi = ln(pi/(1-pi))

Logit:

for freshmen = ln(0.25/0.75) = -1.10 = alpha

for sophomore = ln(0.23/0.77) = -1.21 = alpha + beta2, beta2 = -0.11

for junior = ln(0.18/0.82) = -1.52 = alpha + beta3 , beta3 = -0.42

for senior = ln(0.27/0.73) = -0.99 = alpha + beta4, beta4 = 0.11

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