The common brushtail possum of the Australia region is a bit cuter than its distant cousin, the American opossum (see Figure 8.4 on page 307). We consider 104 brushtail possums from two regions in Australia, where the possums may be considered a random sample from the population. The first region is Victoria, which is in the eastern half of Australia and traverses the southern coast. The second region consists of New South Wales and Queensland, which make up eastern and northeastern Australia. We use logistic regression to differentiate between possums in these two regions. The outcome variable, called population, takes value 1 when a possum is from Victoria and 0 when it is from New South Wales or Queensland. We consider five predictors: sex male (an indicator for a possum being male), head length, skull width, total length, and tail length. Each variable is summarized in a histogram. The full logistic regression model and a reduced model after variable selection are summarized in the table.

(a) Examine each of the predictors. Are there any outliers that are likely to have a very large influence on the logistic regression model?

(b) The summary table for the full model indicates that at least one variable should be eliminated when using the p-value approach for variable selection: head length. The second component of the table summarizes the reduced model following variable selection. Explain why the remaining estimates change between the two models.

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Frequency Frequency 60 40 20- 0 10 5- 0 r 75 0 (Female) sex_male 80 T 85 1 (Male) T 90 total_length (in cm) 95 Frequency Frequency 15 10 20 15- 10 5 32 85 90 95 head_length (in mm) 34 100 1 T T 36 38 40 tail_length (in cm) m □ 42 Frequency Frequency 15 10 5- 0 60 40 20 0 50 55 60 skull_width (in mm) 0 (Not Victoria) 65 population ▬ 1 (Victoria)