Final answer:
To perform the generalized likelihood ratio test, we compute the test statistics by comparing the log-likelihood ratios of the alternative and null hypotheses. The test statistic is not a uniformly most powerful test. Using Wilks' theorem, we can find the critical value of the test statistic. Using the given test statistics, p-value, and central limit theorem, we can determine the p-value and make a decision. If the second candidate wins the election, it suggests that there may be problems with the statistical analysis.
Step-by-step explanation:
(a) To compute the test statistics of the generalized likelihood ratio test, we need to calculate the log-likelihood ratio statistic between the two hypotheses:
LR = 2 * [log(L(H1)) - log(L(H0))]
where L(H1) and L(H0) are the likelihoods under the alternative and null hypotheses, respectively. In this case, the test statistic is approximately 22.1365.
No, this test is not a uniformly most powerful test. A uniformly most powerful test is a test that has the highest power for any alternative hypothesis among all possible tests of the same size. It is not guaranteed that the generalized likelihood ratio test is the uniformly most powerful test for the given hypotheses.
(b) Wilks' theorem states that, under certain regularity conditions, the distribution of the test statistic -2 * log(LR) is approximately chi-squared with degrees of freedom equal to the difference in the number of parameters between the alternative and null hypotheses. Here, since we have 1 parameter difference (p), the critical value of the test at a 0.05 level is approximately 3.841.
Since the test statistic value (22.1365) is greater than the critical value (3.841), we reject the null hypothesis and conclude that there is evidence that the first candidate wins more than half of the votes.
(c) The test statistic given in the question, p - po/√(po(1 - po)/n), follows a standard normal distribution under the null hypothesis. Here, po = 0.5. To compute the p-value, we need to find the probability of observing a test statistic value as extreme as the one obtained, assuming the null hypothesis is true. Using the central limit theorem, the p-value can be calculated as the area under the standard normal curve to the right of the test statistic value. In this case, the p-value is extremely small, indicating strong evidence against the null hypothesis.
Since the p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that there is evidence that the first candidate wins more than half of the votes.
(d) If the second candidate wins the election, it suggests that the statistical analysis may have some problems. The analysis assumes a binomial model, where the responses are independent and follow a binomial distribution. If the second candidate wins, it could indicate that the assumptions of the model are violated, leading to biased results. It is important to carefully examine the data and assumptions made in the analysis to understand the potential issues.