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# Hypothesis Testing

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Of all the presidents since WWII, Jimmy Carter suffered the worst overall approval rating at 45%. George W. Bush's approval rating varied widely. It reached a high of 90% just after 9/11, but in mid-July, it stood at 49% as measured by CNN/USA Today/Gallup poll of 1000 randomly selected adults. Is there evidence to suggest that Bush's July 2004 rating reflects higher public approval than the 45% that Jimmy Carter averaged over his presidency?

Conditions met to use the Normal Model:
1. Independence - yes
2. Randomness - yes
3. n (1000) < 10% of the entire adult population
4. np = (1000)(.45) = 450; nq = (1000)(.55) = 550; both are > 10

Ho : p = .45
Ha : p > .45

p = .45
q = .55
p^ = .49
n = 1000

SD(p^) = sqrt of pq/n = sqrt of [(.45)(.55) / 1000] = 0.0157

z = (p^ - p) / SD(p^) = (.49 - .45) / 0.0157 = 2.55

P-value = 0.0054

The P-value of 0.0054 indicates that if the null hypothesis were true, i.e., that the true proportion of Bush's approval rating in July 2004 was .45 (no higher than Carter's overall approval rating), then an observed proportion of .49 would occur at random less than 6 time in 1000. This P-value of much less than 5% provides strong evidence to reject the null hypothesis that Bush's approval rating in July 2004 was no higher than Carter's overall approval rating; instead it lends strong support to the alternative hypothesis that Bush's July 2004 rating reflects a higher public approval rating than Carter's overall approval rating of 45%.