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The Islamic University of Gaza Faculty of Engineering Department of Civil Engineering 12/12/2009 Statistics and Probability for Engineering Applications ENGC 6310 Dr. Samir Safi Practice Midterm Exam #2 9.2 X is a binomial random variable, show that (a ) Pˆ = X / n is an biased estimator of p; X+ n 2 (b) Pˆ = is a biased estimator of p; n+ n 9- 3 Show that the estimator P' of Exercise: 9.2(b) becomes unbiased as n → ∞ 9.4 An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If a sample of 30 bulbs has an average life of 780 hours, find a 96% confidence interval for the population mean of all bulbs produced by this firm. 9.6 The heights of a random sample of 50 college students showed a mean of 174.5 centimetres and a standard deviation of 6.9 centimetres. (a) Construct a 98% confidence interval for the mean height of all college students. (b) What can we assert with 98% confidence about the possible size of our error it we estimate the mean height of all college students to be 174.5 centimetres? 9.8 How large a sample is needed in Exercise 9.4 if we wish to be 96% confident that our sample mean will be within 10 hours of the true mean? 9.10 An efficiency expert wishes to determine the average time that it takes to drill three holes in a certain metal clamp. How large a sample will he need to be 95% confident that his sample mean will be within 15 seconds of the true mean? Assume that it is known from previous studies that a = 40 seconds. ١ 9.13 A machine is producing metal pieces that are cylindrical in shape. A sample of pieces is taken and the diameters are 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, and 1.03 centimetres. Find a 99% confidence interval for the mean diameter of pieces from this machine, assuming an approximate normal distribution. 9.16 A random sample of 12 graduates of a certain secretarial school typed an average of 79.3 words per minute with a standard deviation of 7.8 words per minute. Assuming a normal distribution for the number of words typed per minute; find a 95% confidence interval for the average number of words typed by all graduates of this school. 9.36 Two kinds of thread are being compared for strength. Fifty pieces of each type of thread are tested under similar conditions. Brand A had an average: tensile strength of 78.3 kilograms with a standard deviation of 5.6 kilograms, while brand B had an average tensile strength of 87.2 kilograms with a standard deviation of 6.3 kilograms. Construct a 95% confidence interval for the difference of the population means. 9.38 In a hatch chemical process, two catalysts arc being compared for their effect on the output of the process reaction. A sample of 12 batches was prepared using catalyst 1 and a sample of 10 batches was obtained using catalyst 2. The 12 batches for which catalyst 1 was used gave an average yield of 85 with a sample standard deviation of 4, and the second sample gave an average of 81 and a sample standard deviation of 5. Find a 90% confidence interval for the difference between the population means, assuming that the populations are approximately normally distributed with equal variances. 9.43 A taxi company is trying to decide: whether to purchase brand .4 or brand B tires for its fleet of taxis. To estimate the difference in the two brands, an experiment is conducted using 12 of each brand. The tires are run until they wear out. The results are ٢ Brand A: x 1 = 36 , 300 36,300 kilometres, s1 = 5, 000 kilometres. Brand B: x 2 = 38,100 kilometres, s 2 = 6 ,100 kilometres. Compute a 95% confidence interval for µ A − µ B assuming the populations to be approximately normally distributed. You may not assume that the variances are equal. 9.44 Referring to Exercise 9.43, find a 99% confidence interval for pi — p2 if a tire from each company is assigned at random to the rear wheels of 8 taxis and the following distance, in kilometres, recorded: Assume that the differences of the distances are approximately normally distributed. 9.51 (a) A random .sample of 200 voters is selected and 114 are found to support, an annexation suit. Find the 96% confidence interval for the fraction of the voting population favouring the suit. (b) What can we assert with 96% confidence about the possible size of our error if we estimate the fraction of voters favouring the annexation suit to be 0.57? 9.52 A manufacturer of compact disk players uses a set of comprehensive tests to access the electrical function of its product. All compact disk players must pass all tests prior to being sold. A random sample of ٣ 500 disk players resulted in 15 failing one or more tests. Find a

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