312-midterm1 and soln

# Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat312: Sample Midterm I. Moo K. Chung [email protected] September 30, 2004 Answer all questions clearly and circle your final an- swer. If you submit two contradicting computations and answers, the grade will be the minimum of two. Correct answers without valid computation or explanation will not get full credit. If you used a wrong formula without much explanation of what you are doing and ended up submitting a wrong solution, you will get almost noth- ing. 1. Let X 1 , · · · , X n be a random sample from a nor- mal distribution with mean μ and variance σ 2 . (a) What is E ( S 2 2 ) ? S 2 is the sample variance. Explain your result (5 points). (b) Among all estimators of the form aX 1 + bX 2 , find the minimum variance unbiased estimator of 2 μ (10 points). 2. Consider the sample of fat content of 10 randomly selected dogs: 25, 21, 22, 17, 29, 25, 16, 20, 19, 22. Suppose that these are from a normal population. (a) Find the sample standard deviation using two numbers 216 and 4806 in R output. No point will be given if you do not use these two numbers (5 points). (b) Find a 95% confidence interval of the mean fat
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