1. (i) For a binary response y , let y be the proportion of ones in the sample (which is equal to the sample average of the yj ). Let q ^ 0 be the percent correctly predicted for the outcome y 5 0 and...

1. (i) For a binary response
y, let
y
be the proportion of ones in the sample (which is equal to the sample average of the
yj). Let
q^ 0 be the percent correctly predicted for the outcome
y
5 0 and let
q^ 1 be the percent correctly predicted for the outcome
y
5 1. If
p^ is the overall percent correctly predicted, show that
p^ is a weighted average of
q^ 0 and
q^ 1:
p^ 5 11 2
y2q^ 0 1
yq^ 1.

(ii) In a sample of 300, suppose that
y
5 .70, so that there are 210 outcomes with
yi
5 1 and 90

with
yi
5 0. Suppose that the percent correctly predicted when
y
5 0 is 80, and the percent

correctly predicted when
y
5 1 is 40. Find the overall percent correctly predicted.