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6.40. An urn contains red and white marbles in an unknown proportion. A random sample of 60 marbles selected with replacement from the urn showed that 70% were red. Find (a) 95%, (b) 99%, (c) 99.73% confidence limits for the actual proportion of red marbles in the urn. Present the results using both the approximate formula and the more exact formula of Problem 6.27. 6.41. How large a sample of marbles should one take in Problem 6.40 in order to be (a) 95%, (b) 99%, (c) 99.73% confident that the true and sample proportions do not differ more than 5% 6.42. It is believed that an election will result in a very close vote between two candidates. Explain by means of an example, stating all assumptions, how you would determine the least number of voters to poll in order to be (a) 80%, (b) 95%, (c) 99% confident of a decision in favor of either one of the candidates.
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6.43. Of two similar groups of patients, A and B, consisting of 50 and 100 individuals, respectively, the first was given a new type of sleeping pill and the second was given a conventional type. For patients in group A the mean number of hours of sleep was 7.82 with a standard deviation of 0.24 hour. For patients in group B the mean number of hours of sleep was 6.75 with a standard deviation of 0.30 hour. Find (a) 95% and (b) 99% confidence limits for the difference in the mean number of hours of sleep induced by the two types of sleeping pills.
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6.44. A sample of 200 bolts from one machine showed that 15 were defective, while a sample of 100 bolts from another machine showed that 12 were defective. Find (a) 95%, (b) 99%, (c) 99.73% confidence limits for the difference in proportions of defective bolts from the two machines. Discuss the results obtained. 6.45. A company manufactures ball bearings having a mean weight of 0.638 oz and a standard deviation of 0.012 oz. Find (a) 95%, (b) 99% confidence limits for the weights of lots consisting of 100 ball bearings each.
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6.46. The standard deviation of the breaking strengths of 100 cables tested by a company was 1800 lb. Find (a) 95%, (b) 99%, (c) 99.73% confidence limits for the standard deviation of all cables produced by the company. 6.47. How large a sample should one take in order to be (a) 95%, (b) 99%, (c) 99.73% confident that a population standard deviation will not differ from a sample standard deviation by more than 2% 6.48. The standard deviation of the lifetimes of 10 electric light bulbs manufactured by a company is 120 hours. Find (a) 95%, (b) 99% confidence limits for the standard deviation of all bulbs manufactured by the company. 6.49. Work Problem 6.48 if 25 electric light bulbs show the same standard deviation of 120 hours. 6.50. Work Problem 6.48 by using the x2 distribution if a sample of 100 electric bulbs shows the same standard deviation of 120 hours.
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6.51. The standard deviations of the diameters of ball bearings produced by two machines were found to be 0.042 cm and 0.035 cm, respectively, based on samples of sizes 10 each. Find (a) 98%, (b) 90% confidence intervals for the ratio of the variances. 6.52. Determine the (a) 98%, (b) 90% confidence intervals for the ratio of the standard deviations in Problem 6.51. 6.53. Two samples of sizes 6 and 8, respectively, turn out to have the same variance. Find (a) 98%, (b) 90% confidence intervals for the ratio of the variances of the populations from which they were drawn. 6.54. Work (a) Problem 6.51, (b) Problem 6.53 if the samples have sizes 120 each.
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