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Mathematical Statistics with Applications, Seventh Edition

www.downloadslide.com Exercises 375 the population standard deviation of pH measurements is not known, past experience indicates that most soils have a pH value of between 5 and 8. If the scientist selects n = 40 samples, find the approximate probability that the sample mean of the 40 pH measurements will be within .2 unit of the true average pH for the field. [Hint: See Exercise 1.17.] 7.47 Suppose that the scientist of Exercise 7.46 would like the sample mean to be within .1 of the true mean with probability .90. How many core samples should the scientist take? 7.48 An important aspect of a federal economic plan was that consumers would save a substantial portion of the money that they received from an income tax reduction. Suppose that early estimates of the portion of total tax saved, based on a random sampling of 35 economists, had mean 26% and standard deviation 12%. a b What is the approximate probability that a sample mean estimate, based on a random sample of n = 35 economists, will lie within 1% of the mean of the population of the estimates of all economists? Is it necessarily true that the mean of the population of estimates of all economists is equal to the percent tax saving that will actually be achieved? 7.49 The length of time required for the periodic maintenance of an automobile or another machine usually has a mound-shaped probability distribution. Because some occasional long service times will occur, the distribution tends to be skewed to the right. Suppose that the length of time required to run a 5000-mile check and to service an automobile has mean 1.4 hours and standard deviation .7 hour. Suppose also that the service department plans to service 50 automobiles per 8-hour day and that, in order to do so, it can spend a maximum average service time of only 1.6 hours per automobile. On what proportion of all workdays will the service department have to work overtime? 7.50 Shear strength measurements for spot welds have been found to have standard deviation 10 pounds per square inch (psi). If 100 test welds are to be measured, what is the approximate probability that the sample mean will be within 1 psi of the true population mean? 7.51 Refer to Exercise 7.50. If the standard deviation of shear strength measurements for spot welds is 10 psi, how many test welds should be sampled if we want the sample mean to be within 1 psi of the true mean with probability approximately .99? 7.52 Resistors to be used in a circuit have average resistance 200 ohms and standard deviation 10 ohms. Suppose 25 of these resistors are randomly selected to be used in a circuit. a b What is the probability that the average resistance for the 25 resistors is between 199 and 202 ohms? Find the probability that the total resistance does not exceed 5100 ohms. [Hint: see Example 7.9.] 7.53 One-hour carbon monoxide concentrations in air samples from a large city average 12 ppm (parts per million) with standard deviation 9 ppm. a b Do you think that carbon monoxide concentrations in air samples from this city are normally distributed? Why or why not? Find the probability that the average concentration in 100 randomly selected samples will exceed 14 ppm. 7.54 Unaltered bitumens, as commonly found in lead–zinc deposits, have atomic hydrogen/carbon (H/C) ratios that average 1.4 with standard deviation .05. Find the probability that the average H/C ratio is less than 1.3 if we randomly select 25 bitumen samples.

www.downloadslide.com 376 Chapter 7 Sampling Distributions and the Central Limit Theorem 7.55 The downtime per day for a computing facility has mean 4 hours and standard deviation .8 hour. a Suppose that we want to compute probabilities about the average daily downtime for a period of 30 days. b i ii What assumptions must be true to use the result of Theorem 7.4 to obtain a valid approximation for probabilities about the average daily downtime? Under the assumptions described in part (i), what is the approximate probability that the average daily downtime for a period of 30 days is between 1 and 5 hours? Under the assumptions described in part (a), what is the approximate probability that the total downtime for a period of 30 days is less than 115 hours? 7.56 Many bulk products—such as iron ore, coal, and raw sugar—are sampled for quality by a method that requires many small samples to be taken periodically as the material is moving along a conveyor belt. The small samples are then combined and mixed to form one composite sample. Let Y i denote the volume of the ith small sample from a particular lot and suppose that Y 1 , Y 2 ,...,Y n constitute a random sample, with each Y i value having mean µ (in cubic inches) and variance σ 2 . The average volume µ of the samples can be set by adjusting the size of the sampling device. Suppose that the variance σ 2 of the volumes of the samples is known to be approximately 4. The total volume of the composite sample must exceed 200 cubic inches with probability approximately .95 when n = 50 small samples are selected. Determine a setting for µ that will allow the sampling requirements to be satisfied. 7.57 Twenty-five heat lamps are connected in a greenhouse so that when one lamp fails, another takes over immediately. (Only one lamp is turned on at any time.) The lamps operate independently, and each has a mean life of 50 hours and standard deviation of 4 hours. If the greenhouse is not checked for 1300 hours after the lamp system is turned on, what is the probability that a lamp will be burning at the end of the 1300-hour period? 7.58 Suppose that X 1 , X 2 ,...,X n and Y 1 , Y 2 ,...,Y n are independent random samples from populations with means µ 1 and µ 2 and variances σ1 2 and σ 2 2 , respectively. Show that the random variable U n = (X − Y ) − (µ 1 − µ 2 ) √ (σ 2 1 + σ2 2)/n satisfies the conditions of Theorem 7.4 and thus that the distribution function of U n converges to a standard normal distribution function as n →∞.[Hint: Consider W i = X i − Y i , for i = 1, 2,...,n.] 7.59 An experiment is designed to test whether operator A or operator B gets the job of operating a new machine. Each operator is timed on 50 independent trials involving the performance of a certain task using the machine. If the sample means for the 50 trials differ by more than 1 second, the operator with the smaller mean time gets the job. Otherwise, the experiment is considered to end in a tie. If the standard deviations of times for both operators are assumed to be 2 seconds, what is the probability that operator A will get the job even though both operators have equal ability? 7.60 The result in Exercise 7.58 holds even if the sample sizes differ. That is, if X 1 , X 2 ,...,X n1 and Y 1 , Y 2 ,...,Y n2 constitute independent random samples from populations with means µ 1 and µ 2 and variances σ1 2 and σ 2 2 , respectively, then X − Y will be approximately normally distributed, for large n 1 and n 2 , with mean µ 1 − µ 2 and variance (σ1 2/n 1) + (σ2 2/n 2). The flow of water through soil depends on, among other things, the porosity (volume proportion of voids) of the soil. To compare two types of sandy soil, n 1 = 50 measurements are to be taken on the porosity of soil A and n 2 = 100 measurements are to be taken on soil B.

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