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Applied Statistics And Probability For Engineers 6th Edition Douglas C. Montgomery, George C. Runger - Solutions
A random sample of 12 graduates of a certain secretarial school typed an average of 79.3 words per minute with a standard deviation of 7.8 words per minute. Assuming a normal distribution for the number of words typed per minute find a 95% confidence interval for the average number of words typed
A random sample of 25 bottles of buffered aspirin contain, on average, 325.05 mg of aspirin with a standard deviation of 0.5 mg. Find the 95% tolerance limits that will contain 90% of the aspirin contents for this brand of buffered aspirin. Assume that the aspirin content is normally distributed.
The following measurements were recorded for the drying time, in hours, of a certain brand of latex paint:3.4 2.5 4.8 2.9 3.62.8 3.3 5.6
Referring to Exercise 9.7, construct a 99% tolerance interval containing 99% of the miles traveled by automobiles annually in Virginia.
Referring to Exercise 9.15, construct, a 95% tolerance interval containing 90% of the measurements
In Section 9.3wc emphasized the notion of "most efficient estimator" by comparing the variance of two unbiased estimators Θ1 and Θ2. However this does not take into account bias in case one or both estimators are not unbiased. Consider the quantity MSE = E(Θ - θ), where MSE denotes mean squared
Consider Exercise 9.1 and S2, the estimator of σ2. Analysts often use S2 rather than dividing ΣN i=1 (Xi — X)2 by n — 1, the degrees of freedom in the sample.(a) What is the bias of S’2?(b) Show that the bias of S'2 approaches zero as n → ∞.
Compare S2 and S'2 (see Exercise 9.1), the two estimators of a2, to determine which is more efficient, assume they are estimators found using X1, X2,...,Xn, independent random variables from n(∞: μ, σ). Which estimator is more efficient considering only the variance of the estimators?
Consider Exercise 9.23. Use the MSE discussed in Exercise 9.21 to determine which estimator is more efficient. In fact, write out MSE (S2)/MSE (S’2)
Consider Exercise 9.12. Compute a 95% prediction interval for the sugar content for the next single serving of Alpha-Bits.
Consider Exercise 9.16. Compute the 95% prediction interval for the next observed number of words per minute typed by a member of the secretarial school.
Consider Exercise 9.18. Compute a 95% prediction interval on a new observed measured drying time for the latex paint.
Consider the situation of Exercise 9.13. Estimation of the mean diameter, while important, is not nearly as important as trying to "pin down" the location of the majority of the distribution of diameters. To that end find the 95% tolerance limits that contain 95% of the diameters.
In a study conducted by the Department of Zoology at Virginia Tech. fifteen "samples" of water were collected from a certain station in the James River in order to gain some insight regarding the amount of ortho phosphorous in the river. The concentration of the chemical is measured in milligrams
A type of thread is being studied for its tensile strength properties. Fifty pieces were tested under similar conditions and the results showed an average tensile strength of 78.3 kilograms and a standard deviation of 5.6 kilograms. Assuming a normal distribution of tensile strength, give a lower
Refer to Exercise 9.30. Why are the quantities requested in the exercise likely to be more important to the manufacturer of the thread than say a confidence interval on the mean tensile strength?
Refer to Exercise 9.30 once again. Suppose that specifications by a buyer of the thread are that the tensile strength of the material must be at least 62 kilograms. The manufacturer is satisfied if at most 5% of manufactured pieces have tensile strength less than 62 kilograms. Is there cause for
Consider the drying time measurements in Exercise 9.18. Suppose the 15 observations in the data set also included a 16th value of 6.9 hours. In the context of the original 15 observations, is the 16th value an outlier? Show work.
Consider the data in Exercise 9.15. Suppose the manufacturer of the shearing pins insists that the Rockwell hardness of the product will be as low as or lower than a value: of 44.0 only 5%, of the time. What is your reaction? Use a tolerance limit calculation to determine your judgment.
A random sample of size n1 = 25 taken from a normal population with a standard deviation σ1 = 5 has a mean x1 = 80. A second random sample of size n2 = 36, taken from a different normal population with a standard deviation σ2 = 3, has a mean x2 = 75. Find a 94% confidence interval for μ1 – μ2.
Two kinds of thread are being compared for strength. Fifty pieces of each type of thread are tested under similar conditions. Brand A had an average: tensile strength of 78.3 kilograms with a standard deviation of 5.6 kilograms, while brand B had an average tensile strength of 87.2 kilograms with a
A study was conducted to determine if a certain metal treatment has any effect on the amount of metal removed in a pickling operation. A random sample of 100 pieces was immersed in a bath for 24 hours without the treatment, yielding an average of 12.2 millimeters of metal removed and a sample
In a hatch chemical process, two catalysts arc being compared for their effect on the output of the process reaction. A sample of 12 batches was prepared using catalyst 1 and a sample of 10 batches was obtained using catalyst 2. The 12 batches for which catalyst 1 was used gave an average yield of
Students may choose between a 3-semcstcr-hour course in physics without, labs and a 4-semester-liour course with labs. The final written examination is the same for each section. If 12 students in the section with labs made an average examination grade of 84 with a standard deviation of 4, and 18
In a study conducted at (lie Virginia Polytechnic: Institute: and State University on the development of ectomycorrhizal. a symbiotic relationship between the roots of trees and a fungus in which minerals are transferred from the fungus to the trees and sugars from the trees to the fungus, 20
The following data, recorded in days, represent the length of time to recovery for patients randomly treated with one of two medications to clear up severe bladder infections: Find a 99% confidence interval for the difference ?2 ???1?in the mean recovery time for the two medications, assuming
An experiment reported in Popular Science compared fuel economics for two types of similarly equipped diesel mini-trucks. Let us suppose that 12 Volkswagen and 10 Toyota trucks are: used in 90-kilometer per hour steady-spaced tests. If the 12 Volkswagen trucks average 16 kilometers per liter with a
A taxi company is trying to decide: whether to purchase brand .4 or brand B tires for its fleet of taxis. To estimate the difference in the two brands, an experiment is conducted using 12 of each brand. The tires are run until they wear out. The results are Brand .4:
Referring to Exercise 9.43, find a 99% confidence interval for ?1 ? ?2 if a tire from each company is assigned at random to the rear wheels of 8 taxis and the following distance, in kilometers, recorded: Assume that the differences of the distances are approximately normally distributed.
The government awarded grants to the agricultural departments of 9 universities to test the yield capabilities of two new varieties of wheat. Each variety was planted on plots of equal area at each university and the yields, in kilograms per plot, recorded as follows. Find a 95% confidence interval
The following data represent the running times of films produced by two motion-picture companies. Compute a 90% confidence interval for the difference between the average running times of films produced by the two companies. Assume that the running-time differences are approximately normally
From Fortune magazine (March 1997), 10 of the 431 companies studied are listed below. The total returns for the 10 year's prior to 1996 and also for 1996 are listed. Find a 95% confidence interval for the mean change in percent return to investors.
An automotive company is considering two types of batteries for its automobile. Sample information on the life of the battery is being used. Twenty batteries of type A and twenty batteries of type B are being used. The summary statistics are XA = 32.91, xB = 30.47, SA = 1-57, and SB – 1.74.
Two different brands of latex paint are being considered for use. Drying time in hours is being measured on specimen samples of the use of the two paints. Fifteen specimens for each were selected and the drying times are as follows: Assume the drying time is normally distributed with σA = σB.
Two levels (low and high) of insulin doses are given to two groups of diabetic rats tea check the insulin binding capacity, yielding the following data:Low dose: n1 = 8 x1 — 1.98 S1 = 0.51High dose: n2 = 13 x2 = 1.30 8a = 0.35Assume
(a) A random .sample of 200 voters is selected and 114 are found to support, an annexation suit. Find the 96% confidence interval for the fraction of the voting population favoring the suit.(b) What can we assert with 90% confidence about the possible size of our error if we estimate the fraction
A manufacturer of compact disk players uses a set of comprehensive tests to access the electrical function of its product. All compact disk players must pass all tests prior to being sold. A random sample of 500 disk players resulted in 15 failing one or more tests. Find a 90% confidence interval
In a random sample of 1000 homes in a certain city, it is found that 228 are heated by oil. Find the 99%, confidence interval for the: proportion of homes in this city that are heated by oil.
Compute a 98% confidence interval for the proportion of defective items in a process when it is found that a sample of size 100 yields 8 defectives.
A new rocket-launching system is being considered for deployment of small, short-range rockets. The existing system has p = 0.8 as the probability of a successful launch. A sample of 40 experimental launches is made with the new system and 34 are successful.(a) Construct a 95% confidence interval
A geneticist is interested in the proportion of African males that have a certain minor blood disorder. In a. random sample of 100 African males, 24 are found to be afflicted.(a) Compute a 99% confidence interval for the proportion of African males that have this blood disorder.(b) What can we
(a) According to a report in the Roanoke Times & World-News, approximately 2/3 of the 1600 adults polled by telephone said they think the space shuttle program is a good investment for the country. Find a 95% confidence interval for the proportion of American adults who think the space shuttle
In the newspaper article referred to in Exercise 9.57, 32% of the 1600 adults polled said the U.S. space program should emphasize scientific exploration. How large a sample of adults is needed in the poll if one wishes to be 95% confident that the estimated percentage will be within 2% of the true
How large a sample is needed in Exercise 9.51 if we wish to be 96% confident that our sample proportion will be within 0.02 of the true fraction of the voting population?
How large a sample is needed in Exercise 9.53 if we wish to be 99% confident that our sample proportion will be within 0.05 of the true proportion of homes in this city that are heated by oil?
How large a sample is needed in Exercise 9.54 if we wish to be 98% confident that our sample proportion will be within 0.05 of the true proportion defective?
A study is to be made to estimate the percentage of citizens in a town who favor having their water fluoridated. How large a sample is needed if one wishes to be at least 95% confident that our estimate is within 1% of the true percentage?
A conjecture by a faculty member in the microbiology department at Washington University School of Dental Medicine in St. Louis states that a couple of cups of either green or oolong tea each day will provide sufficient fluoride to protect your teeth from decay. How large a sample is needed to
A study is to be made to estimate the proportion of residents in a certain city and its suburbs who favor the construction of a nuclear power plant. How large a sample is needed if one wishes to be at least 95% confident that the estimate is within 0.04 of the true proportion of residents in this
A certain geneticist is interested in the proportion of males and females in the population that have a certain minor blood disorder. In a random sample of 1000 males, 250 are found to be afflicted, whereas 275 of 1000 females tested appear to have the disorder. Compute a 95% confidence interval
Ten engineering schools in the United States were surveyed. The sample contained 250 electrical engineers, 80 being women; 175 chemical engineers, 40 being women. Compute a 90% confidence interval for the difference between the proportion of women in these two fields of engineering. Is there a
A clinical trial is conducted to determine if a certain type of inoculation has an effect on the incidence of a certain disease. A sample of 1000 rats was kept in a controlled environment for a period of 1 year and 500 of the rats were given the inoculation. Of the group not given the drug, there
In a study Germination and Emergence of Broccoli, conducted by the Department of Horticulture at Virginia Polytechnic Institute and State University, a researcher found that at 5CC, 10 seeds out of 20 germinated, while at 15CC, 15 out of 20 seeds germinated. Compute a 95% confidence interval for
A survey of 1000 students concluded that 274 students chose a professional baseball team. .4. as his or her favorite team. In 1991, the same survey was conducted involving 760 students. It concluded that 240 of them also chose team A as their favorite. Compute a 95% confidence interval for the
According to USA Today (March 17. 1997), women made up 33.7% of the editorial staff at. Local TV stations in 1990 and 36.2% in 1994 assume 20 new employees were: hired as editorial staff.(a) Estimate the number that would have been women in each year, respectively.(b) Compute a 95% confidence
A manufacturer of car batteries claims that his batteries will last, on average, 3 years with a variance of 1 year. If 5 of these batteries have lifetimes of 1.9, 2.4, 3.0, 3.5, and 4.2 years, construct a 95% confidence interval for σ2 = and decide if the manufacturer's claim that σ2 = 1 is
A random sample of 20 students obtained a mean of x = 72 and a variance of s~ = 16 on a college placement, test in mathematics. Assuming the scores to be normally distributed, construct a 98% confidence interval for σ2.
Construct a 95% confidence interval for σ2 in Exercise 9.12.
Construct a 99% confidence interval for σ2 in Exercise 9.13.
Construct a 99% confidence interval for σ in Exercise 9.14.
Construct a 90% confidence interval for σ in Exercise 9.15.
Construct a 98% confidence interval for σ1/σ2 in Exercise 9.42, where σ1 and σ2 are, respectively, the standard deviations for the distances obtained per liter of fuel by the Volkswagen and Toyota mini-trucks.
Construct a 90% confidence interval for σ21\ σ22 in Exercise 9.43 were we justified in assuming that σ21 ≠ σ22 when we constructed our confidence interval for μ1 – μ2?
Construct a 90% confidence interval for σ21/ σ211 in Exercise 9.46 should we have assumed σ21 = σ211 in constructing our confidence interval for μ21 – μ11?
Construct a 95% confidence interval for σ2A/ σ2B in Exercise 9.49 should the equal-variance assumption be used?
Suppose that there are n trials x1, x2,... , xn, from a Bernoulli process with parameter p, the probability of a success. That is, the probability of r successes is given by (nr)pr (l – p)n-r. Work out the maximum likelihood estimator for the parameter p.
Consider a sample of x1, x2,... ,?xn,, observations from a Weibull distribution with parameters a and ? and density function for a, 3 > 0. (a) Write out the likelihood function. (b) Write out the equations which when solved give the maximum likelihood estimators of a and, 3. 0, elsewhere, fix)
Consider the lognormal distribution with the density function given in Section 6.9. Suppose we have a sample x\, x2,,... , xn, from a lognormal distribution. (a) Write out the likelihood function.(b) Develop the maximum likelihood estimators of p and a2.
Consider the observations x1, x2,...,xn from the gamma distribution discussed in Section 6.6.(a) Write out the likelihood function.(b) Write out a set of equations which when solved give the maximum likelihood estimators of a and 8.
Consider a hypothetical experiment where a man with a fungus uses an antifungal drug and is cured. Consider this, then, a sample of one from a Bernoulli distribution with probability function fix) = p∞ q L–∞ x = 0, 1, where p is the probability of a success (cure) and q = 1 — p. Now, of
Consider the observation X from the negative binomial distribution given in Section 5.5. Find the maximum likelihood estimator for p, given k known.
It is claimed that a new diet will reduce a person's weight by 4.5 kilograms on the average in a period of 2 weeks. The weights of 7 women who followed this diet were recorded before and after a 2-week period. Test a manufacturer's claim by computing a 95% confidence interval for the mean
According to the Roanoke Times (March 16, 1997), McDonald's sold 42.1% of the market share of hamburgers. A random sample of 75 burgers sold resulted in 28 of them by McDonald's. Use material in Section 9.10 to determine if this information supports the claim in the Roanoke Times.
A study was undertaken at the Virginia Polytechnic Institute and State University to determine if fire can be used as a viable management tool to increase the amount of forage available to deer during the critical months in late winter and early spring. Calcium is a required element for plants and
A health spa claims that a new exercise program will reduce a person's waist size by 2 centimeters on the average over a 5-day period. The waist sizes of 6 men who participated in this exercise program are recorded before and after the 5-day period in the following table: By computing a 95%
The Department of Civil Engineering at the Virginia Polytechnic Institute and State University compared a modified (M-5 hr) assay technique for recovering fecal coli forms in storm water runoff from an urban area to a most probable number (MPN) technique. A total of 12 runoff samples were collected
An experiment was conducted to determine whether surface finish has an effect on the endurance limit of steel. An existing theory says that polishing increases the average endurance limit (reverse bending). From a practical point of view, polishing should not have any effect on the standard
An anthropologist is interested in the proportion of individuals in two Indian tribes with double occipital hair whorls. Suppose that independent samples are taken from each of the two tribes, and it is found that 24 of 100 Indians from tribe A and 36 of 120 Indians from tribe B possess this
A manufacturer of electric irons produces these items in two plants. Both plants have the same suppliers of small parts. A saving can be made by purchasing thermostats for plant B from a local supplier. A single lot was purchased from the local supplier and it was desired to test whether or not
It is argued that the resistance of wire A is greater than the resistance of wire B. An experiment on the wires shows the following results (in ohms): Assuming equal variances, what conclusions do you draw? Justify youranswer.
An alternative form of estimation is accomplished through the method of moments. The method involves equating the population mean and variance to the corresponding sample mean x and sample variance s2 and solving for the parameter, the result being the moment estimators. In the case of a single
Specify the moment estimators for μ and s2 for the normal distribution.
Specify the moment estimators for the p and er2 for the lognormal distribution.
Specify the moment estimators for α and β for the gamma distribution.
A survey was done with the hope of comparing salaries of chemical plant managers employed in two areas of the country, the northern and west central regions. Independent random samples of 300 plant managers were selected for each of the two regions. These managers were asked for their annual
Consider Review Exercise 9.101. Let us assume that the data have not been collected yet. Let us also assume that previous statistics suggest that σ1 = σ2 = $4000. Are the sample sizes in Review Exercise 9.101 sufficient to produce a 95% confidence interval on p\ —μ2 having a width of only
A specific labor union is becoming defensive about gross absenteeism by its members. The union decided to check this by monitoring a random sample of its members. The union leaders had always claimed that in a typical month, 95% of its members are absent less than 10 hours per month. Use the data
A random sample of 30 firms dealing in wireless products were selected to determine the proportion of such firms that have implemented new software to improve productivity. It turned out that eight of the 30 have implemented such software. Find a 95% confidence interval on p, the true proportion of
Refer to Review Exercise 9.104. Suppose there is concern about whether or not the point estimates p = 8/30 is accurate enough because the confidence interval around p is not sufficiently narrow. Using p as our estimate of p, how many companies would need to be sampled in order to have a 95%
A manufacturer turns out a product item that is labeled either "defective" or "not defective." In order to estimate the proportion defective, a random sample of 100 items is taken from production and 10 are found to be defective. Following the implementation of a quality improvement program, the
A machine is used to fill boxes of product in an assembly line operation. Much concern centers around the variability in the number of ounces of product in the box. The standard deviation in weight of product is known to be 0.3 ounces. An improvement is implemented after which a random sample of 20
A consumer group is interested in comparing operating costs for two different types of automobile engines. The group is able to find 15 owners whose cars have engine type A and 15 who have engine type B. All 30 owners bought their cars at roughly the same time and all have kept good records for a
Consider the statistic S2p, the pooled estimate of σ2. The estimator is discussed in Section 9.8. It is used when one is willing to assume that σ21 = σ22 = a2. Show that the estimator is unbiased for a2 (i.e., show that E(S2) = a2).
A group of human factor researchers are concerned about reaction to a stimulus by airplane pilots with a certain cockpit arrangement. An experiment was conducted in a simulation laboratory and 15 pilots were used with average reaction time 3.2 seconds and sample standard deviation 0.6 seconds, it
A certain supplier manufactures a type of rubber at that is sold to automotive companies. In the application, the pieces of material must have certain hardness characteristics. Defective mats are occasionally discovered and rejected. The supplier claims that the proportion defective is 0.05. A
Ten samples were taken from a plating bath used in an electronics manufacturing process, and the bath pH was determined. The sample pH values are 7.91, 7.85, 6.82, 8.01, 7.46, 6.95, 7.05, 7.35, 7.25, 7.42. Manufacturing engineering believes that pH has a median value of 7.0. Do the sample data
The titanium content in an aircraft-grade alloy is an important determinant of strength. A sample of 20 test coupons reveals the following titanium content (in percent): 8.32, 8.05, 8.93, 8.65, 8.25, 8.46, 8.52, 8.35, 8.36, 8.41, 8.42, 8.30, 8.71, 8.75, 8.60, 8.83, 8.50, 8.38, 8.29, 8.46 The median
The impurity level (in ppm) is routinely measured in an intermediate chemical product. The following data were observed in a recent test: 2.4, 2.5, 1.7, 1.6, 1.9, 2.6, 1.3, 1.9, 2.0, 2.5, 2.6, 2.3, 2.0, 1.8, 1.3, 1.7, 2.0, 1.9, 2.3, 1.9, 2.4, 1.6 Can you claim that the median impurity level is less
Consider the data in Exercise 15-1. Use the normal approximation for the sign test to test H0: = 70 versus H0: 7.0. What is the P-value for this test?
Consider the compressive strength data in Exercise 8-26.(a) Use the sign test to investigate the claim that the median strength is at least 2250 psi. Use a = 0.05. (b) Use the normal approximation to test the same hypothesis that you formulated in part (a). What is the P-value for this test?
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