In a survey of 400 likely voters, 215 responded that they would vote for the incumbent and 185 responded that they would vote for the challenger. Let p denote the fraction of all likely voters who preferred the incumbent at the time of the survey, and let p̂ be the fraction of survey respondents who preferred the incumbent.
Give an estimate of p.
p̂ = the sample mean = 215 / 400 = 0.5375
This is also the estimate of the population mean, p.
Calculate the Standard Error (SE) of the estimate, p̂.
This is a Bernoulli sample. So, the sample variance for a Bernoulli sample is:
sx2 = p̂ * (1- ?p̂ )
sx2 = 0.5375*(1-0.5375)
sx2 = 0.2486
So, the SE of the estimated mean is:
SE = sqrt(sx2 / n)
SE = sqrt(0.2486 / 400)
SE = 0.0249
What is the p-value for the test H0 : p = 0.5 vs. H1 : p != 0.5?
The first thing to note is that, because the alternate hypothesis is !=, this is a two-sided hypothesis.
Next, compute the t-statistic assuming the null hypothesis:
t = ( (Sample Mean) – (Null Hypothesis) / SE )
t = (0.5375 – 0.5) / 0.0249
t = 1.506
The p-value for the two sided hypothesis is, therefore,
p = 2 * NORMDIST(-1.506, 0, 1, 1)
p = .1321
Note that we use the negative t-statistic in this calculation of area under the curve because we are interested in the area that is inconsistent with the null hypothesis.
What is the p-value for the test H0 : p = 0.5 vs. H1 : p > 0.5?
This time, we have a one-sided hypothesis test. So, the answer is half of what it was for the two sided test:
p = NORMDIST(-1.506, 0, 1, 1)
p = 0.066
The results of the one-sided hypothesis test differs from the two sided test because in the one-sided test, we are only interested in the case where the average is > the null hypothesis. In this case, we’re interested in the case where the incumbent may actually have less than 50% of the population support. In the two-sided case, we were interested in the case where the incumbent had support from less than 50% of the population as well as the case where the incumbent had more than 57.5% of the population’s support.
Do the survey results contain statistically significant evidence that the incumbent was ahead of the challenger at the time of the survey?
It depends on the chosen significance level. For significance levels > 0.066, there is statistical evidence that the incumbent is ahead.