A random sample of 415potential voters was interviewed 3 weeks before the start

Question

A random sample of 415potential voters was interviewed 3 weeks before the start of astate-wide campaign for governor; 223 of the 415 said they favoredthe new candidate over the incumbent. However, the new candidatemade several unfortunate remarks one week before the election.Subsequently, a new random sample of 630 potential voters showedthat 317 voters favored the new candidate.

Do these data support the conclusion that there was a decrease invoter support for the new candidate after the unfortunate remarkswere made? Give appropriate statistical evidence to support youranswer. A random sample of 415potential voters was interviewed 3 weeks before the start of astate-wide campaign for governor; 223 of the 415 said they favoredthe new candidate over the incumbent. However, the new candidatemade several unfortunate remarks one week before the election.Subsequently, a new random sample of 630 potential voters showedthat 317 voters favored the new candidate.

Do these data support the conclusion that there was a decrease invoter support for the new candidate after the unfortunate remarkswere made? Give appropriate statistical evidence to support youranswer.

Explanation / Answer

Large Sample Hypothesis Test for the Difference in Proportions Let X be the number of success in nx independent and identicallydistributed Bernoulli trials, i.e., X ~ Binomial(nx, px) Let Y be the number of success in ny independent and identicallydistributed Bernoulli trials, i.e., Y ~ Binomial(ny, py) Let pxHat = X / nx pyHat = Y / ny pHat = (X + Y) / (ny + ny) Assuming that (nx + ny)*pHat > 10 and (nx + ny)*(1-pHat) > 10(some will say the necessary condition here is > 5, I preferthis more conservative assumption so that the approximations in thetail of the distribution are more accurate), then to test the nullhypothesis H0: px - py = or H0: px - py = or H0: px - py = Find the test statistic z = ((pxHat - pyHat) - ) /(sqrt(pHat * (1 - pHat) * (1/nx + 1/ny)) The p-value of the test is the area under the normal curve that isin agreement with the alternate hypothesis. H1: px - py > ; p-value is the area to the right of z H1: px - py < ; p-value is the area to the left of z H1: px - py ; p-value is the area in the tailsgreater than |z| If the p-value is less than or equal to the significance level, i.e., p-value , then we reject the nullhypothesis and conclude the alternate hypothesis is true. If thep-value is greater than the significance level, i.e., p-value >, then we fail to reject the null hypothesis and concludethat the null is plausible. Note that we can conclude the alternateis true, but we cannot conclude the null is true, only that it isplausible. The hypothesis test in this question is: px = 223 / 415 py = 317 / 630 p = (223 + 317) / (415 + 630) H0: px - py 0 vs. H1: px - py > 0 The test statistic is: z = (( 0.5373494 - 0.5031746 ) - 0 ) / ( ( 0.5167464 * (1 -0.5167464 ) * ( 1 / 415 + 1 / 630 ) z = 1.081722 The p-value = P( Z > z ) = P( Z > 1.081722 ) = 0.1396880 Since the p-value is greater than the significance level of 0.05 wefail to reject the null hypothesis and conclude px - py 0is plausible.

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