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statistics for engineers and scientists

Introduction To Probability And Statistics For Engineers And Scientists 4th Edition Sheldon M. Ross - Solutions

There is some variability in the amount of phenobarbital in each capsule sold by a manufacturer. However, the manufacturer claims that the mean value is 20.0 mg. To test this, a sample of 25 pills yielded a sample mean of 19.7 with a sample standard deviation of 1.3. What inference would you draw
An advertisement for a new toothpaste claims that it reduces cavities of children in their cavity-prone years. Cavities per year for this age group are normal with mean 3 and standard deviation 1. A study of 2,500 children who used this toothpaste found an average of 2.95 cavities per child. Assume
Consider a test of H0 : μ ≤ 100 versus H1 : μ > 100. Suppose that a sample of size 20 has a sample mean of X = 105. Determine the p-value of this outcome if the population standard deviation is known to equal(a) 5; (b) 10; (c) 15.
The weights of salmon grown at a commercial hatchery are normally distributed with a standard deviation of 1.2 pounds. The hatchery claims that the mean weight of this year’s crop is at least 7.6 pounds. Suppose a random sample of 16 fish yielded an average weight of 7.2 pounds. Is this strong
A British pharmaceutical company, Glaxo Holdings, has recently developed a new drug for migraine headaches. Among the claims Glaxo made for its drug, called somatriptan, was that the mean time it takes for it to enter the bloodstream is less than 10 minutes.To convince the Food andDrug
Verify that the approximation in Equation 8.3.7 remains valid even whenμ1 < μ0.
Suppose in Problem 4 that we wished to design a test so that if the pH were really equal to 8.20, then this conclusion will be reached with probability equal to .95.On the other hand, if the pH differs from 8.20 by .03 (in either direction), we want the probability of picking up such a difference
It is known that the average height of a man residing in the United States is 5 feet 10 inches and the standard deviation is 3 inches. To test the hypothesis that men in your city are “average,” a sample of 20 men have been chosen. The heights of the men in the sample follow:What do you
The mean breaking strength of a certain type of fiber is required to be at least 200 psi. Past experience indicates that the standard deviation of breaking strength is 5 psi. If a sample of 8 pieces of fiber yielded breakage at the following pressures, 210 198 195 202 197.4 196 199 195.5 would you
In a certain chemical process, it is very important that a particular solution that is to be used as a reactant have a pH of exactly 8.20. A method for determining pH that is available for solutions of this type is known to give measurements that are normally distributed with a mean equal to the
A population distribution is known to have standard deviation 20. Determine the p-value of a test of the hypothesis that the population mean is equal to 50, if the average of a sample of 64 observations is(a) 52.5; (b) 55.0; (c) 57.5.
A colony of laboratory mice consists of several thousand mice. The average weight of all the mice is 32 grams with a standard deviation of 4 grams. A laboratory assistant was asked by a scientist to select 25 mice for an experiment.However, before performing the experiment the scientist decided to
Consider a trial in which a jury must decide between the hypothesis that the defendant is guilty and the hypothesis that he or she is innocent.(a) In the framework of hypothesis testing and the U.S. legal system, which of the hypotheses should be the null hypothesis?(b) What do you think would be
In an attempt to show that proofreader A is superior to proofreader B, both proofreaders were given the same manuscript to read. If proofreader A found 28 errors, and proofreader B found 18, with 10 of these errors being found by both, can we conclude that A is the superior proofreader
An industrial concern runs two large plants. If the number of accidents during the past 8 weeks at plant 1 were 16, 18, 9, 22, 17, 19, 24, 8 while the number of accidents during the last 6 weeks at plant 2 were 22, 18, 26, 30, 25, 28, can we conclude, at the 5 percent level of significance, that
Management’s claim that the mean number of defective computer chips produced daily is not greater than 25 is in dispute. Test this hypothesis, at the 5 percent level of significance, if a sample of 5 days revealed 28, 34, 32, 38, and 22 defective chips
Suppose that method 1 resulted in 20 unacceptable transistors out of 100 produced, whereas method 2 resulted in 12 unacceptable transistors out of 100 produced.Can we conclude from this, at the 10 percent level of significance, that the two methods are equivalent?
Historical data indicate that 4 percent of the components produced at a certain manufacturing facility are defective. A particularly acrimonious labor dispute has recently been concluded, and management is curious about whether it will result in any change in this figure of 4 percent. If a random
A computer chip manufacturer claims that no more than 2 percent of the chips it sends out are defective. An electronics company, impressed with this claim, has purchased a large quantity of such chips. To determine if the manufacturer’s claim can be taken literally, the company has decided to
There are two different choices of a catalyst to stimulate a certain chemical process. To test whether the variance of the yield is the same no matter which catalyst is used, a sample of 10 batches is produced using the first catalyst, and 12 using the second.If the resulting data are S2 1= .14 and
A machine that automatically controls the amount of ribbon on a tape has recently been installed. This machine will be judged to be effective if the standard deviationσ of the amount of ribbon on a tape is less than .15 cm. If a sample of 20 tapes yields a sample variance of S2 = .025 cm2, are we
An industrial safety program was recently instituted in the computer chip industry. The average weekly loss (averaged over 1 month) in labor-hours due to accidents in 10 similar plants both before and after the program are as follows:Determine, at the 5 percent level of significance, whether the
Reconsider Example 8.4a, but now suppose that the population variances are unknown but equal.
Twenty-two volunteers at a cold research institute caught a cold after having been exposed to various cold viruses. A random selection of 10 of these volunteers was given tablets containing 1 gram of vitamin C. These tablets were taken four times a day.The control group consisting of the other 12
Two new methods for producing a tire have been proposed. To ascertain which is superior, a tire manufacturer produces a sample of 10 tires using the first method and a sample of 8 using the second. The first set is to be road tested at location A and the second at location B. It is known from past
In a single-server queueing system in which customers arrive according to a Poisson process, the long-run average queueing delay per customer depends on the service distribution through its mean and variance. Indeed, if μ is the mean service time, and σ2 is the variance of a service time, then
The manufacturer of a new fiberglass tire claims that its average life will be at least 40,000 miles. To verify this claim a sample of 12 tires is tested, with their lifetimes(in 1,000s of miles) being as follows:Test the manufacturer’s claim at the 5 percent level of significance. Tire 1 2 Life
A public health official claims that the mean home water use is 350 gallons a day. To verify this claim, a study of 20 randomly selected homes was instigated with the result that the average daily water uses of these 20 homes were as follows:Do the data contradict the official’s claim? 340 344
Among a clinic’s patients having blood cholesterol levels ranging in the medium to high range (at least 220 milliliters per deciliter of serum), volunteers were recruited to test a new drug designed to reduce blood cholesterol. A group of 50 volunteers was given the drug for 1 month and the
All cigarettes presently on the market have an average nicotine content of at least 1.6 mg per cigarette. A firm that produces cigarettes claims that it has discovered a new way to cure tobacco leaves that will result in the average nicotine content of a cigarette being less than 1.6 mg. To test
Suppose in Example 8.3a that we know in advance that the signal value is at least as large as 8. What can be concluded in this case?
For the problem of Example 8.3a, how many signals need be sent so that the .05 level test of H0 : μ = 8 has at least a 75 percent probability of rejection whenμ = 9.2?
It is known that if a signal of value μ is sent from location A, then the value received at location B is normally distributed with mean μ and standard deviation 2.That is, the random noise added to the signal is an N(0, 4) random variable. There is reason for the people at location B to suspect
The breaking strength of a certain type of cloth is to be measured for 10 specimens.The underlying distribution is normal with unknown mean θ but with a standard deviation equal to 3 psi. Suppose also that based on previous experience we feel that the unknown mean has a prior distribution that is
Each item produced will, independently, be defective with probability p. If the prior distribution on p is uniform on (0, 1), compute the posterior probability that p is less than .2 given(a) a total of 2 defectives out of a sample of size 10;(b) a total of 1 defective out of a sample of size
The functional lifetimes in hours of computer chips produced by a certain semiconductor firm are exponentially distributed with mean 1/λ. Suppose that the prior distribution on λ is the gamma distribution with density functionIf the average life of the first 20 chips tested is 4.6 hours, compute
Suppose that the number of accidents occurring daily in a certain plant has a Poisson distribution with an unknown mean λ. Based on previous experience in similar industrial plants, suppose that a statistician’s initial feelings about the possible value of λ can be expressed by an exponential
Consider two estimators d1 and d2 of a parameter θ. If E[d1] = θ, Var(d1) = 6 and E[d2] = θ + 2, Var(d2) = 2, which estimator should be preferred?
Consider two independent samples from normal populations having the same variance σ2, of respective sizes n and m. That is, X1, . . . , Xn and Y1, . . . , Ym are independent samples from normal populations each having variance σ2. Let S2 xand S2 y denote the respective sample variances. Thus both
Let X1, X2, . . . , Xn denote a sample from a population whose mean value θ is unknown. Use the results of Example 7.7b to argue that among all unbiased estimators of θ of the formni=1 λiXi ,ni=1 λi = 1, the one with minimal mean square error has λi ≡ 1/n, i = 1, . . . , n.
Determine 100(1−α) percent one-sided upper and lower confidence intervals forθ in Problem 57.
Suppose the lifetimes of batteries are exponentially distributed with mean θ. If the average of a sample of 10 batteries is 36 hours, determine a 95 percent two-sided confidence interval for θ.
Derive 100(1 − α) percent lower and upper confidence intervals for p, when the data consist of the values of n independent Bernoulli random variables with parameter p.
Of 100 randomly detected cases of individuals having lung cancer, 67 died within 5 years of detection.(a) Estimate the probability that a person contracting lung cancer will die within 5 years.(b) How large an additional sample would be required to be 95 percent confident that the error in
A random sample of 100 items from a production line revealed 17 of them to be defective. Compute a 95 percent two-sided confidence interval for the probability that an item produced is defective. Determine also a 99 percent upper confidence interval for this value. What assumptions are you making?
In a recent study, 79 of 140 meteorites were observed to enter the atmosphere with a velocity of less than 25 miles per second. If we take ˆp = 79/140 as an estimate of the probability that an arbitrary meteorite that enters the atmosphere will have a speed less than 25 miles per second, what can
A market research firm is interested in determining the proportion of households that are watching a particular sporting event. To accomplish this task, they plan on using a telephone poll of randomly chosen households. How large a sample is needed if they want to be 90 percent certain that their
A recent newspaper poll indicated that Candidate A is favored over Candidate B by a 53 to 47 percentage, with a margin of error of ±4 percent. The newspaper then stated that since the 6-point gap is larger than the margin of error, its readers can be certain that Candidate A is the current choice.
An airline is interested in determining the proportion of its customers who are flying for reasons of business. If they want to be 90 percent certain that their estimate will be correct to within 2 percent, how large a random sample should they select?
To estimate p, the proportion of all newborn babies that are male, the gender of 10,000 newborn babies was noted. If 5,106 of them were male, determine (a) a 90 percent and (b) a 99 percent confidence interval estimate of p.
A random sample of 1,200 engineers included 48 Hispanic Americans, 80 African Americans, and 204 females. Determine 90 percent confidence intervals for the proportion of all engineers who are(a) female;(b) Hispanic Americans or African Americans.
A problem of interest in baseball is whether a sacrifice bunt is a good strategy when there is a man on first base and no outs. Assuming that the bunter will be out but will be successful in advancing the man on base, we could compare the probability of scoring a run with a player on first base and
Twoanalysts took repeated readings on the hardness of city water. Assuming that the readings of analyst i constitute a sample from a normal population having variance σ2 i , i = 1, 2, compute a 95 percent two-sided confidence interval for σ2 1 /σ2 2 when the data are as follows: Coded Measures
If X1, . . . , Xn is a sample from a normal population having known mean μ1 and unknown variance σ2 1 , and Y1, . . . , Ym is an independent sample from a normal population having known mean μ2 and unknown variance σ2 2 , determine a 100(1 − α) percent confidence interval for σ2 1 /σ2 2 .
The following are the burning times in seconds of floating smoke pots of two different types.Type I Type II 481 572 526 537 506 561 511 582 527 501 556 605 661 487 542 558 501 524 491 578 Find a 99 percent confidence interval for the mean difference in burning times assuming normality with unknown
Do Problem 42 when it is known in advance that the population variances are 4 and 5.
Independent random samples are taken from the output of two machines on a production line.Theweight of each item is of interest. Fromthe first machine, a sample of size 36 is taken, with sample mean weight of 120 grams and a sample variance of 4. From the second machine, a sample of size 64 is
A civil engineer wishes to measure the compressive strength of two different types of concrete. A random sample of 10 specimens of the first type yielded the following data (in psi)Type 1: 3,250, 3,268, 4,302, 3,184, 3,266 3,297, 3,332, 3,502, 3,064, 3,116 whereas a sample of 10 specimens of the
If X1, . . . , Xn is a sample from a normal population, explain how to obtain a 100(1 − α) percent confidence interval for the population variance σ2 when the population mean μ is known. Explain in what sense knowledge of μ improves the interval estimator compared with when it is
The amount of beryllium in a substance is often determined by the use of a photometric filtration method. If the weight of the beryllium is μ, then the value given by the photometric filtration method is normally distributed with mean μ and standard deviation σ. A total of eight independent
The following are independent samples from two normal populations, both of which have the same standard deviation σ.16, 17, 19, 20, 18 and 3, 4, 8 Use them to estimate σ.
Find a 95 percent two-sided confidence interval for the variance of the diameter of a rivet based on the data given here.Assume a normal population. 6.68 6.66 6.62 6.72 6.76 6.67 6.70 6.72 6.78 6.66 6.76 6.72 6.76 6.70 6.76 6.76 6.74 6.74 6.81 6.66 6.64 6.79 6.72 6.82 6.81 6.77 6.60 6.72 6.74 6.70
The capacities (in ampere-hours) of 10 batteries were recorded as follows:140, 136, 150, 144, 148, 152, 138, 141, 143, 151(a) Estimate the population variance σ2.(b) Compute a 99 percent two-sided confidence interval for σ2.(c) Compute a value v that enables us to state, with 90 percent
Verify the formulas given in Table 7.1 for the 100(1−α) percent lower and upper confidence intervals for σ2.
The daily dissolved oxygen concentration for a water stream has been recorded over 30 days. If the sample average of the 30 values is 2.5 mg/liter and the sample standard deviation is 2.12 mg/liter, determine a value which, with 90 percent confidence, exceeds the mean daily concentration.
National Safety Council data show that the number of accidental deaths due to drowning in the United States in the years from 1990 to 1993 were (in units of one thousand) 5.2, 4.6, 4.3, 4.8. Use these data to give an interval that will, with 95 percent confidence, contain the number of such deaths
Let X1, . . . , Xn, Xn+1 denote a sample from a normal population whose meanμ and variance σ2 are unknown. Suppose that we are interested in using the observed values of X1, . . . , Xn to determine an interval, called a prediction interval, that we predict will contain the value of Xn+1 with
A random sample of 16 full professors at a large private university yielded a sample mean annual salary of $90,450 with a sample standard deviation of $9,400.Determine a 95 percent confidence interval of the average salary of all full professors at that university.
An important issue for a retailer is to decide when to reorder stock from a supplier.A common policy used to make the decision is of a type called s, S: The retailer orders at the end of a period if the on-hand stock is less than s, and orders enough to bring the stock up to S. The appropriate
Suppose that U1,U2, . . . is a sequence of independent uniform (0,1) random variables, and define N by N = min{n : U1 +· · ·+Un > 1}That is, N is the number of uniform (0, 1) random variables that need to be summed to exceed 1. Use random numbers to determine the value of 36 random variables
A set of 10 determinations, by a method devised by the chemist Karl Fischer, of the percentage of water in a methanol solution yielded the following data..50, .55, .53, .56, .54,.57, .52, .60, .55, .58 Assuming normality, use these data to give a 95 percent confidence interval for the actual
Studies were conducted in Los Angeles to determine the carbon monoxide concentration near freeways. The basic technique used was to capture air samples in special bags and to then determine the carbon monoxide concentration by using a spectrophotometer. The measurements in ppm (parts per million)
The range of a new type of mortar shell is being investigated. The observed ranges, in meters, of 20 such shells are as follows:2,100 1,984 2,072 1,898 1,950 1,992 2,096 2,103 2,043 2,218 2,244 2,206 2,210 2,152 1,962 2,007 2,018 2,106 1,938 1,956 Assuming that a shell’s range is normally
Verify the formula given in Table 7.1 for the 100(1−α) percent lower confidence interval for μ when σ is unknown.
In Problem 23, find the smallest value v that “with 90 percent confidence,” exceeds the average debt per cardholder.
A random sample of 300 CitiBank VISA cardholder accounts indicated a sample mean debt of $1,220 with a sample standard deviation of $840. Construct a 95 percent confidence interval estimate of the average debt of all cardholders.
Each of 20 science students independently measured the melting point of lead.The sample mean and sample standard deviation of these measurements were(in degrees centigrade) 330.2 and 15.4, respectively. Construct (a) a 95 percent and (b) a 99 percent confidence interval estimate of the true melting
A standardized test is given annually to all sixth-grade students in the state of Washington. To determine the average score of students in her district, a school supervisor selects a random sample of 100 students. If the sample mean of these students’ scores is 320 and the sample standard
A company self-insures its large fleet of cars against collisions. To determine its mean repair cost per collision, it has randomly chosen a sample of 16 accidents.If the average repair cost in these accidents is $2,200 with a sample standard deviation of $800, find a 90 percent confidence interval
Suppose that a random sample of nine recently sold houses in a certain city resulted in a sample mean price of $222,000, with a sample standard deviation of $22,000.Give a 95 percent upper confidence interval for the mean price of all recently sold houses in this city.
The following are scores on IQ tests of a random sample of 18 students at a large eastern university.130, 122, 119, 142, 136, 127, 120, 152, 141, 132, 127, 118, 150, 141, 133, 137, 129, 142(a) Construct a 95 percent confidence interval estimate of the average IQ score of all students at the
The following data resulted from 24 independent measurements of the melting point of lead.Assuming that the measurements can be regarded as constituting a normal sample whose mean is the true melting point of lead, determine a 95 percent two-sided confidence interval for this value. Also determine
Suppose that when sampling from a normal population having an unknown mean μ and unknown variance σ2, we wish to determine a sample size n so as to guarantee that the resulting 100(1 − α) percent confidence interval for μwill be of size no greater than A, for given values α and A. Explain
In Problem 14, compute a value c for which we can assert “with 99 percent confidence” that c is larger than the mean nicotine content of a cigarette.
A sample of 20 cigarettes is tested to determine nicotine content and the average value observed was 1.2 mg. Compute a 99 percent two-sided confidence interval for the mean nicotine content of a cigarette if it is known that the standard deviation of a cigarette’s nicotine content is σ = .2 mg.
If X1, . . . , Xn is a sample from a normal population whose mean μ is unknown but whose variance σ2 is known, show that (−∞, X + zασ/√n) is a 100(1 − α)percent lower confidence interval for μ.
Let X1, . . . , Xn, Xn+1 be a sample from a normal population having an unknown mean μ and variance 1. Let ¯Xn = ni=1 Xi /n be the average of the first n of them.(a) What is the distribution of Xn+1 − ¯Xn?(b) If ¯Xn = 4, give an interval that, with 90 percent confidence, will contain the
The standard deviation of test scores on a certain achievement test is 11.3. If a random sample of 81 students had a sample mean score of 74.6, find a 90 percent confidence interval estimate for the average score of all students.
The PCB concentration of a fish caught in Lake Michigan was measured by a technique that is known to result in an error of measurement that is normally distributed with a standard deviation of .08 ppm (parts per million). Suppose the results of 10 independent measurements of this fish are 11.2,
An electric scale gives a reading equal to the true weight plus a random error that is normally distributed with mean 0 and standard deviation σ = .1 mg. Suppose that the results of five successive weighings of the same object are as follows: 3.142, 3.163, 3.155, 3.150, 3.141.(a) Determine a 95
A manufacturer of heat exchangers requires that the plate spacings of its exchangers be between .240 and .260 inches. A quality control engineer sampled 20 exchangers and measured the spacing of the plates on each exchanger. If the sample mean and sample standard deviation of these 20 measurements
River floods are often measured by their discharges (in units of feet cubed per second). The value v is said to be the value of a 100-year flood if P{D ≥ v} = .01 where D is the discharge of the largest flood in a randomly chosen year. The following table gives the flood discharges of the largest
Suppose that X1, . . . , Xn are normal with mean μ1; Y1, . . . , Yn are normal with mean μ2; and W1, . . . ,Wn are normal with mean μ1 + μ2. Assuming that all 3n random variables are independent with a common variance, find the maximum likelihood estimators of μ1 and μ2.
The height of a radio tower is to be measured by measuring both the horizontal distance X from the center of its base to a measuring instrument and the vertical angle of the measuring device (see the following figure). If five measurements of the distance L give (in feet) values 150.42, 150.45,
Let X1, . . . , Xn be a sample from a normal μ, σ2 population. Determine the maximum likelihood estimator of σ2 when μ is known. What is the expected value of this estimator?
Determine the maximum likelihood estimator of θ when X1, . . . , Xn is a sample with density function f(x) = e-x-1, -x < x
Let X1, . . . , Xn be a sample from the distribution whose density function isDetermine the maximum likelihood estimator of θ. f(x)= {0 (e-(x-9) x0 0 otherwise
Suppose that if a signal of value s is sent from location A, then the signal value received at location B is normally distributed with mean s and variance 60. Suppose also that the value of a signal sent at location A is, a priori, known to be normally distributed with mean 50 and variance 100. If
Suppose X1, . . . , Xn are independent normal random variables, each having unknown mean θ and known variance σ2 0. If θ is itself selected from a normal population having known mean μ and known variance σ2, what is the Bayes estimator of θ?
Suppose that X1, . . . , Xn are independent Bernoulli random variables, each having probability mass function given bywhere θ is unknown. Further, suppose that θ is chosen from a uniform distribution on (0, 1). Compute the Bayes estimator of θ. f(x)=0 (1-0)1-x, x = 0,1

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