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midterm2-solutions

# midterm2-solutions - Statistics 5021 Midterm 2 Name...

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Statistics 5021 Midterm 2 Name: Internet-ID: There are 3 questions, with point values given in parentheses for each part of each question. Show all work to receive credit. You are allowed two 8.5x11 inch sheets of paper with your notes written on both sides. You are permitted to use a calculator. Question Points Earned Points Possible 1 6 2 7 3 7 Total 20 1

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1. Suppose that a candidate in the upcoming 2012 presidential election is interested in estimating the proportion θ of US voters that would vote for him or her. In particular, this candidate wishes to obtain a conservative approximate 95% confidence interval for θ , with a margin of error of at most 5%. Assume that the yet-to-be recorded responses: X 1 , . . . , X n are a random sample from Bern( θ ). Note that z 0 . 975 = 1 . 959964 (a) (1 point) What sample size n is needed for this candidate to to obtain this conservative approximate confidence interval (described above)? Note that the sample size must be a positive integer. Solution: The required sample size, with m e = 0 . 05 and α = 0 . 05. n z 1 - α/ 2 0 . 5 m e 2 = 1 . 959964 * 0 . 5 0 . 05 2 = 384 . 1459 Implying that 385 individuals must be included in the sample. (b) (1 point) A sample of the size computed in part 1a was obtained and this candidate computed the observed conservative approximate 95% confidence interval for θ to be: (0 . 3708338 , 0 . 4707246) Interpret this interval. Solution: We are approximately 95% confident that θ , the population proportion of voters that will vote for this candidate is in the interval (0 . 3708338 , 0 . 4707246). (c) (2 points) We wish to carry out the hypothesis test: H 0 : θ = 0 . 5 H a : θ < 0 . 5 at the α = 0 . 01 significance level. Using the confidence interval given in part 1b and the sample size computed in part 1a, compute the test statistic realization for this large sample Z-test.
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