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6.55. Suppose that n observations, X1, c, Xn, are made from a Poisson distribution with unknown parameter l. Find the maximum likelihood estimate of l. 6.56. A population has a density function given by f (x) 2n !n>px2e nx2, ` x `. For n observations, X1, c, Xn, made from this population, find the maximum likelihood estimate of n. 6.57. A population has a density function given by f (x) e (k 0 1)x k 0 x 1 otherwise
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For n observations X1, c, Xn made from this population, find the maximum likelihood estimate of k.
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CHAPTER 6 Estimation Theory
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6.58. The 99% confidence coefficients (two-tailed) for the normal distribution are given by 2.58. What are the corresponding coefficients for the t distribution if (a) n 4, (b) n 12, (c) n 25, (d) n 30, (e) n 40 6.59. A company has 500 cables. A test of 40 cables selected at random showed a mean breaking strength of 2400 lb and a standard deviation of 150 lb. (a) What are the 95% and 99% confidence limits for estimating the mean breaking strength of the remaining 460 cables (b) With what degree of confidence could we say that the mean breaking strength of the remaining 460 cables is 2400 35 lb
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6.29. (a) 9.5 lb (b) 0.74 lb2 (c) 0.78 and 0.86 lb, respectively.
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6.30. (a) 1200 hours (b) 105.4 hours 6.31. (a) Estimates of population standard deviations for sample sizes 30, 50, and 100 tubes are, respectively, 101.7, 101.0, and 100.5 hours. Estimates of population means are 1200 hours in all cases. 6.32. (a) 11.09 6.33. (a) 0.72642 (b) 0.72642 6.34. (a) 0.72642 0.18 tons (b) 11.09 0.24 tons 0.000072 inch 0.000060 inch
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0.000095 inch (c) 0.72642 0.000085 inch (d) 0.72642 0.000025 inch
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(b) 0.000025 inch
6.35. (a) at least 97 (b) at least 68 (c) at least 167 (d) at least 225 6.36. (a) at least 385 (b) at least 271 (c) at least 666 (d) at least 900 6.37. (a) 7.38 6.39. (a) 0.298 6.40. (a) 0.70 0.82 oz (b) 7.38 0.030 second 0.12, 0.69 1.16 oz 6.38. (a) 7.38 0.049 second 0.15, 0.68 0.15 (c) 0.70 0.18, 0.67 0.17 0.70 oz (b) 7.38 0.96 oz
(b) 0.298 (b) 0.70
6.41. (a) at least 323 (b) at least 560 (c) at least 756 6.43. (a) 1.07 6.44. (a) 0.045 6.45. (a) 63.8 6.46. (a) 1800 0.09 hours (b) 1.07 0.12 hours 0.097 (c) 0.045 0.31 oz 328 lb (c) 1800 382 lb 0.112
0.073 (b) 0.045 0.24 oz (b) 63.8
249 lb (b) 1800
CHAPTER 6 Estimation Theory
6.47. (a) at least 4802 (b) at least 8321 (c) at least 11,250 6.48. (a) 87.0 to 230.9 hours 6.49. (a) 95.6 to 170.4 hours 6.50. (a) 106.1 to 140.5 hours 6.51. (a) 0.269 to 7.70 (b) 78.1 to 288.5 hours (b) 88.9 to 190.8 hours (b) 102.1 to 148.1 hours 6.52. (a) 0.519 to 2.78 (b) 0.673 to 2.14
(b) 0.453 to 4.58 (b) 0.264 to 5.124
6.53. (a) 0.140 to 11.025
6.54. (a) 0.941 to 2.20, 1.067 to 1.944 6.55. l a xk >n 1 6.56. n
(b) 0.654 to 1.53, 0.741 to 1.35 3n c
2(x2 1
x2) n
6.57. k
n ln (x1 c xn) 3.06 (c) 2.79 (d) 2.75 (e) 2.70
6.58. (a)
4.60 (b)
6.59. (a) 2400
45 lb, 2400
59 lb (b) 87.6%
Tests of Hypotheses and Significance
Statistical Decisions
Very often in practice we are called upon to make decisions about populations on the basis of sample information. Such decisions are called statistical decisions. For example, we may wish to decide on the basis of sample data whether a new serum is really effective in curing a disease, whether one educational procedure is better than another, or whether a given coin is loaded.
Statistical Hypotheses. Null Hypotheses
In attempting to reach decisions, it is useful to make assumptions or guesses about the populations involved. Such assumptions, which may or may not be true, are called statistical hypotheses and in general are statements about the probability distributions of the populations. For example, if we want to decide whether a given coin is loaded, we formulate the hypothesis that the coin is fair, i.e., p 0.5, where p is the probability of heads. Similarly, if we want to decide whether one procedure is better than another, we formulate the hypothesis that there is no difference between the procedures (i.e., any observed differences are merely due to fluctuations in sampling from the same population). Such hypotheses are often called null hypotheses or simply hypotheses, are denoted by H0. Any hypothesis that differs from a given null hypothesis is called an alternative hypothesis. For example, if the null hypothesis is p 0.5, possible alternative hypotheses are p 0.7, p 2 0.5, or p 0.5. A hypothesis alternative to the null hypothesis is denoted by H1.
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