Suppose we're looking at a two-candidate mayoral race in Providence. Candidate A is declared winner with 55 percent of the vote (p(hat) = 0.55). However, candidate B is suspicious of these results. Having his group of close friends take a random sample of 17 voters from Providence, he finds that 7 voted for candidate A while 10 voted for him.
On the basis of this study performed at the alpha = 0.05 level of significance, should candidate B demand a recount?
Formulate the null and alternative hypothesis and perform the test in order to respond to this question.
For hypothesis testing, I only know how to solve these problems if we have 4 parameters: Xn the sample mean, u the population mean, sd the standard deviation and n the sample size. But here I don't have the sample mean nor the standard deviation. What is the formula used to find the normal distribution t given a 95% confidence interval when doing hypothesis testing when n is not large? Since we don't have the sample mean nor the standard deviation, what is the process of hypothesis testing here?
H0: u = 0.55
H1: u < 0.55 since p = 7/14 = 0.41
z = p(hat) - p / sqrt(p(1-p)/n) = 0.55 - 0.41 / sqrt(0.41(1-0.41)/17) = 1.17
p(z < 1.17) = 0.879 p(z > 1.17) = 1 - 0.879 = 0.123
Since 0.123 > 0.05, we don't reject the null hypothesis and candidate B shouldn't demand a recount.
Is this correct?