In a recent election, candidate Q received 40% of the vote. Candidate Q believes that the voting results are manipulated and more people have voted for him.

Candidate Q hires a consulting agency to randomly sample 1000 voters and record whether or not each person voted for him. Then a judge needs to use the statistical evidence and hypothesis testing to decide whether or not there is election fraud.

i. Let denote the population proportion of people who voted for candidate Q, state the null and the alternative hypotheses.

ii. Interpret type 1 and type 2 errors in this case.

iii. Let p denote the proportion of people who voted for candidate Q from the sample of 1000

voters. Write down and interpret the sampling distribution of p.

iv. From the survey, 420 out of 1000 had voted for a candidate Q. Suppose that the significance level is set at 5%. Is there enough statistical evidence to reject the null? Define the test statistic, write down its null distribution, and use the p-value approach to make the decision.

v. Suppose that if there is enough statistical evidence to reject the null, the judge will declare the original election fraud and rerun the election. The judge chose the 5% significance level when she thought the cost to rerun the election is $3,000. It turns out that the cost of a rerun is $300,000. Should the judge change the significance level she uses to make the decision?