Exercise 8.25

Stat 121

KJ Wang

Votes for Bush and Buchanan in all Florida Counties

0 50000 100000 150000 200000 250000 300000

bush2000

0

1000

2000

3000

buch

anan

2000

Quantiles of Standard Normal

Res

idua

ls

-2 -1 0 1 2-1

000

010

0020

00

36

13

67

Palm Beach County (outlier)

Log Transformation

7 8 9 10 11 12 13

logbush

2

4

6

8

logb

ucha

nan

Data excluding Palm Beach County

0 50000 100000 150000 200000 250000 300000

bush2000

0

200

400

600

800

1000bu

chan

an20

00

S Plus Model for data w/o Palm Beach County

• *** Linear Model ***

• Call: lm(formula = buchanan2000 ~ bush2000, data = ex0825, na.action = na.exclude)

• Residuals:• Min 1Q Median 3Q Max • -512.4 -47.97 -17.09 41.78 305.5

• Coefficients:• Value Std. Error t value Pr(>|t|) • (Intercept) 65.5735 17.3304 3.7837 0.0003 • bush2000 0.0035 0.0003 13.9226 0.0000

• Residual standard error: 112.5 on 64 degrees of freedom• Multiple R-Squared: 0.7518 • F-statistic: 193.8 on 1 and 64 degrees of freedom, the p-value is 0

CI Attempt 1• Formula: Buchanan Votes = (0.0035)Bush Votes + 66• 95% CI = fitted +/- t*SE(fitted)• Intercept = 66 +/- 17.3*(1.99773) = 100.5, 31.5• Coefficient = 0.0035 +/- 0.0003(1.99773) = 0.0041, 0.0029• Bush = 152846

– Predicted High with 95% CI: 152846 *0.0041 + 100.5 = 727

– Predicted Low with 95% CI: 152846 * 0.0029 + 31.5 = 475

• THEREFORE: Predicted number of votes Buchanan should have received in Palm Beach County based on the number of votes Bush received is 475-727

• Buchanan actually received 3407 votes in Palm Beach County• IF the statistically extra Buchanan votes were intended for Gore, then Gore should

have received at least (3407-727 =) 2680 more votes.

CI Attempt 2• Formula: Buchanan Votes = (0.0035)Bush Votes + 66• 95% CI = fitted +/- t*SE(predicted)• SE(predicted) = square root(σ^2 + SE(fit)^2)• σ = Residual Standard Error = 112.5• SE(fit) = 31.05• Buchanan(predicted) =597.77• SE(pred) = root(112.5^2 + 31.05^2) = 116.7058• qt(.975, 64) = 1.99773; qt(.975,64)*SE(pred) = 233.0965• Buchanan(predict) = 597.77 +/- 233.0965

• THEREFORE: Predicted number of votes Buchanan should have received in Palm Beach County based on the number of votes Bush received is 365 - 831

• Buchanan actually received 3407 votes in Palm Beach County• IF the statistically extra Buchanan votes were intended for Gore, then Gore should

have received at least (3407-831 =) 2576 more votes.

FINAL CONCLUSION

• Gore would have won the election if the voters in Palm Beach followed the same statistical model as the rest of the state

• 2576 extra votes is more than Bush’s initial margin of victory of 1738 votes

• BUT TOO BAD!