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Probability And Statistics For Engineers 5th Edition Richard L Scheaffer, Madhuri Mulekar, James T McClave, Cecie Starr - Solutions
9.36 Refer to Exercise 9.35. Suppose a similar experiment is to be run again with an equal number of specimens from each type of coupling agent.How many specimens should be used if we want to estimate the true difference between mean resistances to within 1 unit, with a confidence coefficient of
9.35 The abrasive resistance of rubber is increased by adding a silica filler and a coupling agent to chemically bond the filler to the rubber polymer chains.Fifty specimens of rubber made with a type I coupling agent gave a mean resistance measure of 92, the variance of the measurements being
9.34 A. C. Nielsen Co. has electronic monitors hooked up to about 1,200 of the 80 million American homes. The data from the monitors provide estimates of the proportion of homes tuned to a particular TV program. Nielsen offers the following defense of this sample size:Mix together 70,000 white
9.33 The results of a Louis Harris poll state that 36% of Americans list football as their favorite sport. A note then states: “In a sample of this size (1,091 adults) one can say with 95% certainty that the results are within plus or minus three percentage points of what they would be if the
9.32 The quantity used in constructing confidence intervals for(or p) is sometimes called the sampling error.A Time magazine (April 5, 1993) article on religion in America reported that 54% of those between the ages of 18 and 26 found religion to be a “very important” part of their lives. The
9.31 Using the data given in Exercise 9.30, construct a 90% confidence interval for the variance of the number of cycles to failure for beams of this type.What assumptions are necessary for your answer to be valid?
9.30 Fatigue behavior of reinforced concrete beams in seawater was studied by Hodgkiess et al.(Materials Performance, July 1984, pp. 27–29).The number of cycles to failure in seawater for beams subjected to certain bending and loading stress was as follows (in thousands):774, 633, 477, 268, 407,
9.29 In the article quoted in Exercise 9.28, a sample of 12 measurements on the tensile strength of steel bars resulted in a mean of and a standard deviation of 38. Estimate the true mean tensile strength of bars of this size and type using a 95% confidence interval. List any assumptions you need
9.28 The yield stress in steel bars (Grade FeB 400 HWL) reported by Booster (J. Quality Technology, 15, no. 4, 1983, p. 191) gave the following data for one test:Estimate the true yield stress for bars of this grade and size in a 90% confidence interval. Are any assumptions necessary for your
9.27 The increased use of wood for residential heating has caused some concern over the pollutants and irritants released into the air. In one reported study in the northwestern United States, the total suspended particulates (TSP) in nine air samples had a mean of 72 and a standard deviation of
9.26 In producing resistors, variability of the resistances is an important quantity to study, as it reflects the stability of the manufacturing process. Estimate , the true variance of the resistance measurements, in a 90% confidence interval, if a sample of 15 resistors showed resistances with a
9.25 The variance of LC50 measurements is important because it may reflect an ability (or inability)to reproduce similar results in identical experiments.Find a 95% confidence interval for , the true variance of the LC50 measurements for DDT, using the data in Exercise 9.21.
9.24 Fifteen resistors were randomly selected from the output of a process supposedly producing 10-ohm resistors. The 15 resistors actually showed a sample mean of 9.8 ohms and a sample standard deviation of 0.5 ohm. Find a 95%confidence interval for the true mean resistance of the resistors
9.23 Answer Exercise 9.22 if the same sample data resulted from a sample of a 10 specimens b 100 specimens
9.22 The warpwise breaking strength measured on five specimens of a certain cloth gave a sample mean of 180 psi and a standard deviation of 5 psi. Estimate the true mean warpwise breaking strength for cloth of this type in a 95% confidence interval. What assumption is necessary for your answer to
9.21 The Environmental Protection Agency has collected data on the LC50 (concentration killing 50% of the test animals in a specified time interval)measurements for certain chemicals likely to be found in freshwater rivers and lakes. For a certain species of fish, the LC50 measurements(in parts per
9.20 In conducting an inventory and audit of parts in a certain stockroom, it was found that, for 60 items sampled, the audit value exceeded the book value on 45 items. Estimate, with confidence coefficient 0.90, the true fraction of items in the stockroom for which the audit value exceeds the book
9.19 Refer to Exercise 9.18. Suppose it is desired to estimate the true proportion of cracked supports to within 0.1, with confidence coefficient 0.98.How many supports should be sampled to achieve the desired accuracy?
9.18 Careful inspection of 70 precast concrete supports to be used in a construction project revealed 28 with hairline cracks. Estimate the true proportion of supports of this type with cracks in a 98% confidence interval.
9.17 Refer to Exercise 9.16. If it is desired to estimate the true proportion failing to meet tolerance specifications to within 0.05, with confidence coefficient 0.95, how many resistors should be tested?
9.16 Upon testing 100 resistors manufactured by Company A, it is found that 12 fail to meet the tolerance specifications. Find a 95% confidence interval for the true fraction of resistors manufactured by Company A that fail to meet the tolerance specification. What assumptions are necessary for
9.15 Refer to Exercise 9.12. How many specimens should be tested if it is desired to estimate the percent of shrinkage to within 0.2 with confidence coefficient 0.98?
9.14 In the setting of Exercise 9.11, how many engineers should be sampled if it is desired to estimate the mean number of hours worked to within 0.5 hour with confidence coefficient 0.95?
9.13 The breaking strength of threads has a standard deviation of 18 grams. How many measures on breaking strength should be used in the next experiment if the estimate of the mean breaking strength is to be within 4 grams of the true mean breaking strength, with confidence coefficient 0.90?
9.12 An important property of plastic clays is the percent of shrinkage on drying. For a certain type of plastic clay, 45 test specimens showed an average shrinkage percentage of 18.4 and a standard deviation of 1.2. Estimate the true average percent of shrinkage for specimens of this type in a 98%
9.11 A random sample of 40 engineers was selected from among the large number employed by a corporation engaged in seeking new sources of petroleum.The hours worked in a particular week were determined for each engineer selected. These data had a mean of 46 hours and a standard deviation of 3
9.10 For a random sample of 50 measurements on the breaking strength of cotton threads, the mean breaking strength was found to be 210 grams and the standard deviation 18 grams.Obtain a confidence interval for the true mean breaking strength of cotton threads of this type, with confidence
9.9 USA TODAY (August 3, 2004) reported results of USA/TODAY/CNN/Gallop Poll from 1,129 likely voters. Before the Democratic Convention, 47%of likely voters opted for Kerry. After the Democratic Convention, 45% of likely voters opted for Kerry. The results are reported with a margin of error of 3
9.7 For a certain new model of microwave oven, it is desired to set a guarantee period so that only 5% of the ovens sold will have a major
9.6 Suppose X1,…, Xn is a random sample from a normal distribution with mean and variance.a Show that is a biased estimator of .b Adjust S to form an unbiased estimator of .
9.5 Refer to Exercise 9.2. Find MSE when is used to estimate .
9.4 The bias B of an estimator is given by The mean squared error, or MSE, of an estimator is given by Show that[Note: if is an unbiased estimator of . Otherwise, .]
9.3 The number of breakdowns per week for a certain minicomputer is a random variable X having a Poisson distribution with mean . A random sample X1, . . . ,Xn of observations on the number of breakdowns per week is available.a Find an unbiased estimator of .b The weekly cost of repairing these
9.2 The reading on a voltage meter connected to a test circuit is uniformly distributed over the interval( , 1), where is the true but unknown voltage of the circuit. Suppose X1, . . . ,Xn denotes a random sample of readings from this voltage meter.a Show that is a biased estimator of .b Find a
9.1 Suppose X1, X2, X3 denotes a random sample from the exponential distribution with density function Consider the following four estimators of :a Which of the above estimators are unbiased for ?b Among the unbiased estimators of , which has the smallest variance?
8.77 If Y has a distribution, then Y can be represented as where has a distribution and the ’s are independent. It is not surprising, then, that Y will be approximately normally distributed for large n. Use this fact in the solution of the following problem.A machine in a heavy-equipment factory
8.76 A plant supervisor is interested in budgeting for weekly repair costs (in dollars) for a certain type of machine. These repair costs over the past 10 years tend to have an exponential distribution with a mean of 20 for each machine studied. Let denote the repair costs for five of these
8.75 If Y has an exponential distribution with mean , show that has a distribution with 2 degrees of freedom.
8.74 A study of the effects of copper on a certain species (say, species A) of fish shows the variance of LC50 measurements (in milligrams per liter) to be 1.9. The effects of copper on a second species (say, species B) of fish produce a variance of LC50 measurements of 0.8. If the population means
8.73 Suppose that and are independent random samples, with the ’s having mean and variance and the ’s having mean and variance . The difference between the sample means, , will again be approximately normally distributed, because the Central Limit Theorem will apply to this difference.a Find .b
8.72 Suppose that denotes a random sample of measurements on the proportion of impurities in samples of iron ore. Suppose that each has the probability density function The ore is to be rejected by a potential buyer if . Find the approximate probability that the ore will be rejected, based on the
8.71 Twenty-five lamps are connected so that when one lamp fails, another takes over immediately.(Only one lamp is on at any one time.) The lamps operate independently, and each has a mean life of 50 hours and a standard deviation of 4 hours. If the system is not checked for 1,300 hours after the
8.70 The U.S. Census Bureau reports median ages of the population and median incomes for households.On the other hand, the College Board reports average scores on the Scholastic Aptitude Test. Do you think each group is using an appropriate measure? Why or why not?
8.69 For a random sample of size n from a normal distribution, show that E(S) c4 where[Hint: Make use of proportions of the gamma distribution from Chapter 6.]
8.68 a Show that Cp Cpk for a centered process.b Construct an example to show that Cp can be a poor index of capability for a process that is not centered.
8.67 Refer to the capability index discussion in Example 8.19. Suppose the process can be“centered” so that is midway between LSL and USL. How will this change Cpk and the proportion out of specification?
8.65 Suppose variation among defect counts is important in a process like that in Exercise 8.63. Is it necessary to have a separate method to construct control limits for variation?Discuss.
8.63 Refer to Exercise 8.62. Adjust the control limits so that a count for a process in control will fall outside the limits, with an approximate probability of only 0.01.
8.60 Refer to Exercise 8.59. Adjust the control limits so that a sample fraction will fall outside the limits with probability 0.05 when the process is in control.
8.57 Refer to Exercises 8.54 and 8.55. The specifications for these ammeters require individual readings on the test circuit to be within 15.0 0.4.Do the ammeters seem to be meeting this specification?Calculate Cpk and the proportion out of specification.
8.56 Refer to Exercise 8.54, part (a). How would the control limits change if the probability of the sample mean falling outside the limits when the process is in control is to be 0.01?
8.55 Refer to Exercise 8.54. Repeat parts (a), (b), and(c) using sample ranges instead of sample standard deviations.
8.53 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 75 independent trials involving the performance of a certain task on the machine. If the sample means for the 75 trials differ by more than 5 seconds, the
8.52 The service times for customers coming through a checkout counters in a retail store are independent random variables with a mean of 15 minutes and a variance of 4. At the end of the work day, the manager selects independently a random sample of 100 customers each served by checkout counters A
8.51 The strengths of a type of thread manufactured by two machines are random variables with the same mean. The floor supervisor takes 10 random thread samples from each machine independently. The standard deviations of measured strengths are 0.2 pound and 0.18 pound, respectively. It is
8.50 The downtime per day for the university’s computing server averages about 40 minutes and for the email server averages about 20 minutes. The downtimes were recorded for five randomly selected days, which showed standard deviations of 8 and 10 minutes, respectively. The director of computer
8.49 One-hour carbon monoxide concentrations in air samples from a large city average 12 ppm, with a standard deviation of 9 ppm. The same from its twin city averages about 10 ppm, with a standard deviation of 9 ppm.a Find the probability that the average concentration in 10 samples selected
8.48 Soil acidity is measured by a quantity called pH. A scientist wants to estimate the difference in the average pH for two large fields using pH measurements from randomly selected core samples. If the scientist selects 20 core samples from field I and 15 core samples from field 2, independently
8.47 Shear-strength measurements for spot welds of two different types have been found to have approximate standard deviations of 10 and 12 psi, respectively. If 100 test welds are to be measured for each type, find the approximate probability that the two sample means will differ by at most 1 psi.
8.46 For an aptitude test for quality-control technicians in an electronics firm, history shows scores to be normally distributed with a variance of 225. If 20 applicants are to take the test, find an interval in which the sample variance of test scores should lie with probability 0.90.
8.45 In constructing an aptitude test for a job, it is important to plan for a fairly large variance in test scores so the best applicants can be easily identified. For a certain test, scores are assumed to be normally distributed with a mean of 80 and a standard deviation of 10. A dozen applicants
8.44 Answer the questions posed in Exercise 8.43 if the sample size is 25 rather than 15.
8.43 A certain type of resistor is marketed with the specification that the variance of resistances produced is around 50 ohms. A sample of 15 of these resistors is to be tested for resistances produced.a Find the approximate probability that the sample variance S2 will exceed 80.b Find the
8.42 Ammeters produced by a certain company are marketed under the specification that the standard deviation of gauge readings be no larger than 0.2 amp. Ten independent readings on a test circuit of constant current, using one of these ammeters, gave a sample variance of 0.065. Does this suggest
8.41 Suppose n 20 observations are to be taken on normally distributed LC50 measurements, with 1.9. Find two numbers a and b such that. (S2 is the sample variance of the 20 measurements.)
8.40 If, in Exercise 8.39, it is desired that the sample mean differ from the population mean by no more than 0.5 with probability 0.95, how many tests should be run?
8.39 The Environmental Protection Agency is concerned with the problem of setting criteria for the amount of certain toxic chemicals to be allowed in freshwater lakes and rivers. A common measure of toxicity for any pollutant is the concentration of the pollutant that will kill half of the test
8.38 In the setting of Exercise 8.37, what should the mean efficiency per bulb equal if the specification for the room is to be met with probability approximately 0.90? (Assume the standard deviation of efficiency measurements remains at 0.5.)
8.37 The efficiency ratings (in lumens per watt) of light bulbs of a certain type have a population mean of 9.5 and a standard deviation of 0.5, according to production specifications. The specifications for a room in which eight of these bulbs are to be installed call for the average efficiency of
8.36 An auditor samples 100 of a firm’s travel vouchers to check on how many of these vouchers are improperly documented. Find the approximate probability that more than 30% of the sampled vouchers will be found to be improperly documented if, in fact, only 20% of all the firm’s vouchers are
8.35 A large construction firm has won 60% of the jobs for which it has bid. Suppose this firm bids on 25 jobs next month.a Approximate the probability that it will win at least 20 of these.b Find the exact binomial probability that it will win at least 20 of these. Compare this probability to your
8.34 Waiting times at a service counter in a pharmacy are exponentially distributed with a mean of 10 minutes. If 100 customers come to the service counter in a day, approximate the probability that at least half of them must wait for more than 10 minutes.
8.33 At a specific intersection, vehicles entering from the east are equally likely to turn left, turn right, or proceed straight ahead. If 500 vehicles enter this intersection from the east tomorrow, what is the approximate probability that a 150 or fewer turn right?b At least 350 turn?
8.32 The daily water demands for a city pumping station exceed 500,000 gallons with probability only 0.15. Over a 30-day period, find the approximate probability that demand for over 500,000 gallons per day occurs no more than twice.
8.31 The capacitances of a certain type of capacitor are normally distributed with a mean of (microfarads)and a standard deviation of If 64 such capacitors are to be used in an electronic system, approximate the probability that at least 12 of them will have capacitances below
8.30 The quality of computer disks is measured by the number of missing pulses. For a certain brand of disk, 80% are generally found to contain no missing pulses. If 100 such disks are inspected, find the approximate probability that 15 or fewer contain missing pulses.
8.29 Of the customers entering a showroom for stereo equipment, only 30% make purchases. If 40 customers enter the showroom tomorrow, find the approximate probability that at least 15 make purchases.
8.28 A lot acceptance sampling plan for large lots, similar to that of Example 8.6, calls for sampling 50 items and accepting the lot if the number of nonconformances is no more than 5. Find the approximate probability of acceptance if the true proportion of nonconformances in the lot is a 10%b
8.27 The median age of residents of the United States is 33 years. If a survey of 100 randomly selected U.S. residents is taken, find the approximate probability that at least 60 of them will be under 33 years of age.
8.26 The service times for customers coming through a checkout counter in a retail store are independent random variables with a mean of 1.5 minutes. The standard deviation of a random sample of 12 customers served is estimated to be 1.0 minute.Approximate the probability that 12 customers can be
8.25 The downtime per day for a certain computing facility has averaged 4.0 hours for the last year.The standard deviation of downtimes of the last 20 failures is 0.8 hours.a Find the probability that the average daily downtime for a period of 20 days is between 1 and 5 hours.b Find the probability
8.24 One-hour carbon monoxide concentrations in air samples from a large city average 12 ppm. The standard deviation of carbon monoxide concentrations measured from the last five air samples was 9 ppm. Find the probability that the average concentration in next five samples selected randomly will
8.23 Resistors of a certain type have resistances that are approximately normally distributed with mean 200 ohms. Fifteen of these resistors are to be used in a circuit. The standard deviation of 12 measurements is 10 ohms.a Find the probability that the average resistance of the 15 resistors is
8.22 It is known from past samples that pH of water in Bolton Creek tends to be approximately normally distributed. The average water pH level of water in the creek is estimated regularly by taking 12 samples from different parts of the creek.Assuming they represent random samples from the creek,
8.21 Shear-strength measurements for spot welds of a certain type have been found to have an approximate normal distribution with standard deviation 10 psi. If 10 test welds are to be measured, find the probability that the sample mean will be within 1 psi of the true population mean.
8.20 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 on the machine. If the sample means for the 50 trials differ by more than 1 second, the
8.19 Refer to Exercise 8.18. Suppose samples are to be selected with n1 n2 n. Find the value of n that will allow the difference between the sample means to be within 0.04 unit of ( ) with probability approximately 0.90.
8.18 Suppose that X1, …, and Y1, …, constitute independent random samples from populations with means and and variances and , respectively. Then the Central Limit Theorem can be extended to show that is approximately normally distributed for large n1 and n2, with mean (m1 - m2) and
8.17 Refer to Exercise 8.16. Find the number of customers n such that the probability of servicing all n customers in less than 2 hours is approximately 0.1.
8.16 The service times for customers coming through a checkout counter in a retail store are independent random variables with a mean of 1.5 minutes and a variance of 1.0. Approximate the probability that 100 customers can be serviced in less than 2 hours of total service time by this one checkout
8.15 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 aggregated and mixed to form one composite sample. Let Yi denote
8.14 The strength of a thread is a random variable with mean 0.5 pound and standard deviation 0.2 pound. Assume the strength of a rope is the sum of the strengths of the threads in the rope.a Find the probability that a rope consisting of 100 threads will hold 45 pounds.b How many threads are
8.13 The downtime per day for a certain computing facility averages 4.0 hours, with a standard deviation of 0.8 hour.P Aaq i=1Xi 7 aB = P(nX 7a) = P(X 7 a>n)a Find the probability that the average daily downtime for a period of 30 days is between 1 and 5 hours.b Find the probability that the total
8.12 Unaltered bitumens, as commonly found in leadzinc deposits, have atomic hydrogen/carbon (H/C)ratios that average 1.4, with a standard deviation of 0.05. Find the probability that 25 samples of bitumen have an average H/C ratio below 1.3.
8.11 One-hour carbon monoxide concentrations in air samples from a large city average 12 ppm, with a standard deviation of 9 ppm. Find the probability that the average concentration in 100 samples selected randomly will exceed 14 ppm.
8.10 Resistors of a certain type have resistances that average 200 ohms, with a standard deviation of 10 ohms. Twenty-five of these resistors are to be used in a circuit.a Find the probability that the average resistance of the 25 resistors is between 199 and 202 ohms.b Find the probability that
8.9 Suppose the scientist of Exercise 8.8 would like the sample mean to be within 0.1 of the true mean with probability 0.90. How many core samples should she take?
8.8 The soil acidity is measured by a quantity called pH, which may range from 0 to 14 for soils ranging from low to high acidity. Many soils have an average pH in the 5–8 range. A scientist wants to estimate the average pH for a large field from n randomly selected core samples by measuring the
8.7 If shear-strength measurements have a standard deviation of 10 psi, how many test welds should be used in the sample if the sample mean is to be within 1 psi of the population mean with probability approximately 0.95?
8.6 Shear-strength measurements for spot welds of a certain type have been found to have a standard deviation of approximately 10 psi. If 100 test welds are to be measured, find the approximate probability that the sample means will be within 1 psi of the true population mean.
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