A politician claims that he will gain 50% of votes in a city election and thereby he will win. However, after studying the politician's policies, an expert thinks he will lose the election; i.e. he will gain less than 50% of the votes. To test his conjecture, the expert randomly selected n voters from the city and found Y of them would vote for the candidate. Let p denote the supporting rate for the candidate.

1. State clearly the null and alternative hypotheses for the above research question.

2 As part of your work, you plan to create an asymmetric 95% confidence interval for p with a margin of error of 0.05. What should be then the smallest value of n?

3. What is the exact sampling distribution of Y under the null hypothesis? Explain briefly.

4. Take n = 15 and suppose that for the test in 1.1 your rejection region is Y ≤ 5. Calculate then the probability of Type I error.

5. If the candidate receives 30% of the votes, calculate the power for the test in 1.1 when the critical region is still Y ≤ 5.

6. Now suppose that n = 60 and for the test in 1.1 your rejection region is Y ≤ 20. Which is the asymptotic distribution of Y, under the null hypothesis, due to the Central Limit Theorem? 1.7 For the scenario in 1.6 calculate the probability of Type I error, using the asymptotic distribution of Y

8. For the scenario in 1.6 calculate the probability of Type II error, using the asymptotic distribution of Y if the candidate receives 30% of the votes.