Probability and Statistics for Engineering and the Sciences/Book and Disk

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CIV 245, Princeton University Fall 1997 Example Midterm #2 Problems Here are a collection of questions that I deem suitable to ask on midterm exams in the future. They can be used to get a general feel for what sort of questions one might expect on a midterm. Note, however, that my experience has been that students almost always feel that the actual midterm questions are harder than those shown here. By and large, this is not usually the case (but there have been exceptions), questions often seem more difficult when they are new and when there is a time limit! In general it is safest to use the following questions to expose to yourself which areas you need more work on. The questions are largely arranged by concept, progressing from issues of bias to confidence intervals to hypothesis tests. If you are going to look at a handful of questions, make sure you do some from each concept. The second midterm will cover all of these concepts. Solutions are provided for some of the followingquestions - the later questions are missing solutions because I haven’t had time to get around to solving them. 1) Let 1 2 be a random sample of size from an exponential distributionwith parameter . Show that is an unbiased estimator of 1 given that is the average of the sample. 2) If 1 2 16 is a random sample of size = 16 from a normal distribution with mean 50 and variance 100, determine P 796 2 16 =1 ( 50) 2 2630 3) Suppose that is a random variable following some distribution and that 1 2 is a random sample from this distribution. One possible estimator of E 2 is ¯ 2 where ¯ = 1 =1 Show whether ¯ 2 is a biased estimator of E 2 or not. 4) Suppose that 1 2 form a random sample from a normal distributionfor which both the mean and the variance are unknown. Estimate the point such that P = 0 95. 5) The mass in milligrams of a certain pharmaceutical product is distributed normally with a variance of 0.0025 ( 2 ). Find the minimum number of samples which must be taken in order to estimate the mean mass to within 0.025 at a 95% confidence level. 6) A metal tube is produced to have a mean wall thickness of 0.35 inches and a standard deviation of 0.025 inches. If 4 samples of the tube are taken and the wall thickness measured, find a lower limit below which the sample average will not fall more than 2 times out of a 1000 on average. 7) Let denote the number of hours per week that a fourth year TUNS student spends studying probability and statistics. Suppose that ( 4) and the unknown mean is to be estimated by ¯ , the mean of a random sample of size . a) Determine the minimum number of samples required to ensure that the true value of is estimated to within 0.5 hours with probability 0.95. b) If a random sample of size 10 is to be taken, 1 2 10 , what type of distributiondoes the statistic 10 =1 ( ¯ ) 2 2
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CIV 245, Princeton University Example Midterm #2 Problems Fall 1997: pg 2 (where 2 = 4) follow?
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