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CHAPTER 7 POINT ESTIMATION OF PARAMETERS
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7-8. Let three random samples of sizes n1 20, n2 10, and n3 8 be taken from a population with mean and variance 2. Let S 2, S 2 , and S 2 be the sample variances. 1 2 3 Show that S 2 120S 2 10S 2 8S 2 2 38 is an unbiased 1 2 3 estimator of 2. 7-9.
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(a) Show that g i 1 1Xi X 2 2 n is a biased estimator of 2 . (b) Find the amount of bias in the estimator. (c) What happens to the bias as the sample size n increases
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7-10. Let X1, X2, p , Xn be a random sample of size n from a population with mean and variance 2 . (a) Show that X 2 is a biased estimator for 2 . (b) Find the amount of bias in this estimator. (c) What happens to the bias as the sample size n increases 7-11. Data on pull-off force (pounds) for connectors used in an automobile engine application are as follows: 79.3, 75.1, 78.2, 74.1, 73.9, 75.0, 77.6, 77.3, 73.8, 74.6, 75.5, 74.0, 74.7, 75.9, 72.9, 73.8, 74.2, 78.1, 75.4, 76.3, 75.3, 76.2, 74.9, 78.0, 75.1, 76.8. (a) Calculate a point estimate of the mean pull-off force of all connectors in the population. State which estimator you used and why. (b) Calculate a point estimate of the pull-off force value that separates the weakest 50% of the connectors in the population from the strongest 50%. (c) Calculate point estimates of the population variance and the population standard deviation. (d) Calculate the standard error of the point estimate found in part (a). Provide an interpretation of the standard error. (e) Calculate a point estimate of the proportion of all connectors in the population whose pull-off force is less than 73 pounds. 7-12. Data on oxide thickness of semiconductors are as follows: 425, 431, 416, 419, 421, 436, 418, 410, 431, 433, 423, 426, 410, 435, 436, 428, 411, 426, 409, 437, 422, 428, 413, 416. (a) Calculate a point estimate of the mean oxide thickness for all wafers in the population. (b) Calculate a point estimate of the standard deviation of oxide thickness for all wafers in the population. (c) Calculate the standard error of the point estimate from part (a). (d) Calculate a point estimate of the median oxide thickness for all wafers in the population. (e) Calculate a point estimate of the proportion of wafers in the population that have oxide thickness greater than 430 angstrom. 7-13. X1, X2, p , Xn is a random sample from a normal distribution with mean and variance 2 . Let Xmin and Xmax be the smallest and largest observations in the sample. (a) Is 1Xmin Xmax 2 2 an unbiased estimate for (b) What is the standard error of this estimate (c) Would this estimate be preferable to the sample mean X
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7-14. Suppose that X is the number of observed successes in a sample of n observations where p is the probability of success on each observation. (a) Show that P X n is an unbiased estimator of p. (b) Show that the standard error of P is 1p11 p2 n. How would you estimate the standard error 2 7-15. X 1 and S 1 are the sample mean and sample variance from a population with mean and variance 2. Similarly, X2 2 and S 2 are the sample mean and sample variance from a sec2 ond independent population with mean 1 and variance 2 . 2 The sample sizes are n1 and n2 , respectively. (a) Show that X 1 X 2 is an unbiased estimator of 1 2. (b) Find the standard error of X1 X2. How could you estimate the standard error 7-16. Continuation of Exercise 7-15. Suppose that both pop2 ulations have the same variance; that is, 2 1 2 2 . Show that S2 p 1n1 12 S 2 1 n1 n2 1n2 12 S 2 2 2
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is an unbiased estimator of 2. 7-17. Two different plasma etchers in a semiconductor factory have the same mean etch rate . However, machine 1 is newer than machine 2 and consequently has smaller variability in etch rate. We know that the variance of etch rate for machine 1 is 2 and for machine 2 it is 2 a 2. Suppose that we have 1 2 1 n1 independent observations on etch rate from machine 1 and n2 independent observations on etch rate from machine 2. (a) Show that ) X 2 is an unbiased estimaX 1 (1 tor of for any value of between 0 and 1. (b) Find the standard error of the point estimate of in part (a). (c) What value of would minimize the standard error of the point estimate of (d) Suppose that a 4 and n1 2n2. What value of would you select to minimize the standard error of the point estimate of . How bad would it be to arbitrarily choose 0.5 in this case 7-18. Of n1 randomly selected engineering students at ASU, X1 owned an HP calculator, and of n2 randomly selected engineering students at Virginia Tech X2 owned an HP calculator. Let p1 and p2 be the probability that randomly selected ASU and Va. Tech engineering students, respectively, own HP calculators. (a) Show that an unbiased estimate for p1 p2 is (X1 n1) (X2 n2). (b) What is the standard error of the point estimate in part (a) (c) How would you compute an estimate of the standard error found in part (b) (d) Suppose that n1 200, X1 150, n2 250, and X2 185. Use the results of part (a) to compute an estimate of p1 p2. (e) Use the results in parts (b) through (d) to compute an estimate of the standard error of the estimate.
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