Links to earlier posts about confidence of victory.
Data from recent polls.
Boxer vs. Fiorina U.S. Senate (CA)
Date: 10/2
Boxer: 49%
Fiorina: 44%
n = 448
Brown vs. Whitman Governor (CA)
Brown: 50%
Whitman: 43%
n = 448
For both of these polls:
1) Find the 95% confidence interval for both candidates
2) Since the two top candidate poll over 90% total, do the confidence of victory, rounding to the nearest 5% if the value is under 90% and to the nearest 1% if the value of over 90%.
Answers in the comments.
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1) 95% confidence interval
Boxer: 1.96*sqrt(.49*.51/448) = 4.6%
We are 95% confident Boxer's support at the time of the poll was between 44.4% and 53.6% of the likely voters.
Fiorina: 1.96*sqrt(.44*.56/448) = 4.6%
We are 95% confident Fiorina's support at the time of the poll was between 39.4% and 48.6% of the likely voters.
Brown: 1.96*sqrt(.50*.50/448) = 4.6%
We are 95% confident Brown's support at the time of the poll was between 45.4% and 54.6% of the likely voters.
Whitman: 1.96*sqrt(.43*.57/448) = 4.6%
We are 95% confident Whitman's support at the time of the poll was between 38.4% and 47.6% of the likely voters.
Confidence of victory:
f(Boxer) = .49*448 =~ 220
f(Fiorina) = .44*448 =~ 197
new n = 220+197 = 417
p-hat(Boxer) = 220/417 = 52.8%
s_p-hat = sqrt(.528*.472/417) = .0244
z(Boxer) = (.528 - .5)/.0244 = 1.15
z = 1.15 corresponds to .8749, so it barely rounds to 85% as the nearest 5%
If the election were held when the poll was taken, we are 85% confident the lead Boxer shows in the poll is due to her being favored by the underlying population of likely California voters and she would win.
f(Brown) = .50*448 = 224
f(Whitman) = .43*448 =~ 193
new n = 224+193 = 417
p-hat(Brown) = 224/417 = 53.7%
s_p-hat = sqrt(.537*.463/417) = .0244
z(Brown) = (.537 - .5)/.0244 = 1.52
z = 1.52 corresponds to .9357, so it rounds to 94%, since we round to the nearest percent when over 90%.
If the election were held when the poll was taken, we are 94% confident the lead Brown shows in the poll is due to him being favored by the underlying population of likely California voters and he would win.
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