3 of 3 (c) Another 503 students are selected at random from Florida. They are given a three-hour preparation course before the test is administered. Their average test score is 1019 with a standard deviation of 95 (a) Construct a 95% confidence interval for the change in average test score associated with the pre course (b) Is there statistically significant evidence that the prep course helped? (d) The original 453 students are given the prep course and then asked to take the test a second time. The average change in their test scores in 9 points, and the standard deviation of the change is 60 points. (a) Construct a 95% confidence interval for the change in average test scores (b) Is there statistically significant evidence that students will perform better on their second attempt after taking the prep course? (c) Students may have performed better in their second attempt be- cause of the pre course or because they gained test-taking exper rience in their first attempt. Describe an experiment that would quantify the two effects.20f3 (c) Explain the difference between an conditional and an unconditional expectation. ((1) Explain why an unbiased estimator need not necessarily be a good estimator. Question 3 (30 points) 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 3 denote the fraction of all likely voters who preferred the incumbent at the time of the survey, and let s be the fraction of survey respondents who preferred the incumbent. (3.) Use the survey results to estimate 8. (b) Use the estimator of the variance of , (1 ) to calculate the standard error of your estimator. (c What is the pvalue for the test H0 :17 = 0.5 vs H1 :10 75 0.5 ? (d What is the pvalue for the test H0 :1) = 0.5 vs H1 :p > 0.5 ? ) ) e) Why do the results from (c) and (d) differ? f) ( ( Did the survey contain statistically signicant evidence that the incum- bent was ahead of the challenger at the time of the survey? Explain. Question 4 (35 points) Grades on a standardized test are known to have a mean of 1000 for students in the United States. The test is administered to 453 randomly selected stu- bent was ahead of the challenger at the time of the survey? Explain. 2 of 3 Question 4 (35 points) Grades on a standardized test are known to have a mean of 1000 for students in the United States. The test is administered to 453 randomly selected stu- dents in Florida; in this sample, the mean is 1013 and the standard deviation (s) is 108. (a) Construct a 95% confidence interval for the average test score for Florida students. (b) Is there statistically significant evidence that Florida students perform differently than other students in the United States? 2Question 1 (15 points) Let {Y1, Y2, ...Yn} be a random sample from a population with a normal distribution with mean / and variance o2 (i.e., Y ~ N(u, 2 )). Consider the following two alternative estimators for M: M = (n - 1)y n and 12 = Y + - n. Where Y = 1 ELYi. Compare the small-sample properties of these estimators (i.e. check whether they are unbiased and efficient). Question 2 (20 points) (a) Explain the difference between a Type I and a Type II error in hypoth- esis testing. (b) Explain the difference between an interval estimate and a point esti- mate