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statistics for engineers and scientists

Questions and Answers of Statistics For Engineers And Scientists

9.39 In studying the proportion of water samples containing harmful bacteria, how many samples should be selected before and after a chemical is added if we want to estimate the true difference
9.38 Bacteria in water samples are sometimes difficult to count, but their presence can easily be detected by culturing. In 50 independently selected water samples from a certain lake, 43 contained
9.37 Two different types of coating for pipes are to be compared with respect to their ability to aid in resistance to corrosion. The amount of corrosion on a pipe specimen is quantified by measuring
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
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
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
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%
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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?

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