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inferential statistics
Probability And Statistics For Engineers And Scientists 9th Global Edition Ronald E. Walpole, Raymond Myers, Sharon L. Myers, Keying E. Ye - Solutions
9.68 In the study Germination and Emergence of Broccoli, conducted by the Department of Horticulture at Virginia Tech, a researcher found that at 5◦C, 10 broccoli seeds out of 20 germinated, while at 15◦C, 15 out of 20 germinated. Compute a 95% confidence interval for the difference between the
9.67 A clinical trial was 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. In the group not inoculated,
9.66 In the authorized service center of an internationally reputed automobile company, 16 of the 100 cars entered for service had mechanical defects. In another sample, 6 of the 120 cars had electrical defects.Compute a 95% confidence interval for the difference between the proportions of cars
9.65 In a socio-economic survey conducted among 1200 randomly selected villagers from a coastal village A, 672 earn their living by fishing. From a nearby village B, 744 out of the 1200 villagers earn their living by fishing. Compute a 90% confidence interval for the difference between the
9.64 An opinion poll is to conduct among the residents and shopkeepers of a metropolitan city on whether they agree with the route proposed by the authorities for a metro rail project. How large a sample is needed if one wishes to be at least 98% confident that the estimate is within 0.05% of the
9.63 A study is conducted to estimate the percentage of college students who are fond of musical instruments.How large a sample is needed if one wishes to be at least 90% confident that the estimate is within 1% of the true percentage.
9.62 According to a new bulletin released by the health department, liquor consumption among adolescents of a certain town has increased in recent years.Someone comments: “it is due to the lack of providing awareness on the ill effects of liquor consumption to students from educational
9.61 How large a sample is needed in Exercise 9.52 if we wish to be 98% confident that our sample proportion will be within 0.05 of the true proportion defective?
9.60 How large a sample is needed if we wish to be 99% confident that our sample proportion in Exercise 9.51 will be within 0.05 of the true proportion of homes in the city that are heated by oil?
9.59 How large a sample is needed if we wish to be 96% confident that our sample proportion in Exercise 9.53 will be within 0.02 of the true fraction of the voting population?
9.58 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 for the poll if one wishes to be 95% confident that the estimated percentage will be within 2% of the
9.57 (a) According to a report in the Roanoke Times, approximately 2/3 of 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 program is a good
9.56 An international logistic company decided to conduct a qualifying test on computer skills for all the job applicants to that company. From a random sample of 120 applicants, 90 qualified.(a) Compute a 95% confidence interval for the proportion of qualifying applicants.(b) What can we assert
9.55 Records from a city hospital say that the probability(p) that a confirmed dengue patient stays in the hospital for more than 5 days is 0.7. A new type of medicine claims fast recovery from the fever. A sample of 36 patients admitted with dengue fever is administered with the new medicine and
9.54 A home appliances manufacturing firm producing wet grinders decided to conduct a quality check before marketing their product. From a random sample of 400 units, 18 failed to satisfy the quality requirements.Find a 95% confidence interval for the proportion of wet grinders from the population
9.53 (a) A random sample of 200 voters in a town 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 96% confidence about the possible size of our error if we estimate
9.52 Compute 95% confidence intervals, using both methods on page 317, for the proportion of defective items in a process when it is found that a sample of size 100 yields 8 defectives.
9.51 In a random sample of 1000 homes in a certain city, it is found that 228 are heated by oil. Find 99%confidence intervals for the proportion of homes in this city that are heated by oil using both methods presented on page 317.
9.50 Two types of medicine, X and Y, are tested on two groups of insomnia patients to help them sleep.The time until they fall asleep after taking the medicine is recorded. The results are as follows:Medicine X n1 = 12 ¯x1 = 24 s1 = 4.2 Medicine Y n2 = 16 ¯x2 = 18 s1 = 3.6 Assuming that the
9.49 Two different brands of latex paint are being considered for use. Fifteen specimens of each type of paint were selected, and the drying times, in hours, were as follows:Paint A Paint B 3.5 2.7 3.9 4.2 3.6 4.7 3.9 4.5 5.5 4.0 2.7 3.3 5.2 4.2 2.9 5.3 4.3 6.0 5.2 3.7 4.4 5.2 4.0 4.1 3.4 5.5 6.2
9.48 A toilet soap manufacturing company produces two varieties of soaps, A and B. The sample information for the units of soap sold within the previous month is collected from 20 supermarkets in a city. The summary of the sales data is as follows: ¯xA = 132,¯xB = 123, sA = 23.7, and sB = 31.8.
9.47 An asset management company reported its mutual fund investments growth, through various companies, for the years 2014 and 2015. The figures for 8 of the companies are listed below. Find a 95% confidence interval for the mean change in percentage growth of the asset management company.Total
9.46 The following data represent the running times of films produced by two motion-picture companies.Company Time (minutes)I 103 94 110 87 98 II 97 82 123 92 175 88 118 Compute a 90% confidence interval for the difference between the average running times of films produced by the two companies.
9.45 Two different varieties of paddy seeds are cultivated in an experiment. The yield, in kilograms, from seeds sown in 9 sample plots for each variety, under similar conditions, is recorded as follows:Variety/Plot 1 2 3 4 5 6 7 8 9 I 24 25 26 19 26 28 18 23 20 II 29 32 31 36 27 24 33 32 26 Find a
9.44 Referring to Exercise 9.43, find a 99% confidence interval for μ1 − μ2 if tires of the two brands are assigned at random to the left and right rear wheels of 8 taxis and the following distances, in kilometers, are recorded:Taxi Brand A Brand B 1 34,400 36,700 2 45,500 46,800 3 36,700
9.43 A taxi company is trying to decide whether to purchase brand A 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 A: ¯x1 = 36, 300 kilometers, s1
9.42 An experiment reported in Popular Science compared fuel economies for two types of similarly equipped diesel mini-trucks. Let us suppose that 12 Volkswagen and 10 Toyota trucks were tested in 90-kilometer-per-hour steady-paced trials. If the 12 Volkswagen trucks averaged 16 kilometers per
9.41 The following data represents the total time taken, in days, to deliver books ordered through two online sellers. The sample delivery times are collected and reveal the following information:Seller 1 Seller 2 n1 = 12 n2 = 14¯x1 = 8 ¯x2 = 9 s21= 3.4 s22= 3.2 Find a 98% confidence interval for
9.40 In a study conducted at Virginia Tech 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 northern red oak seedlings exposed to the
9.39 In a common university examination for communicative English, 16 of the mathematics graduates score an average of 78 marks with a standard deviation of 6, and 22 of the physics graduates score an average of 72 marks with a standard deviation of 8. Construct a 95% confidence interval for the
9.38 Two catalysts in a batch chemical process, are 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 prepared using catalyst 2. The 12 batches for which catalyst 1 was used in the reaction
9.37 A study was conducted to determine if a certain 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
9.36 Two kinds of light bulbs are being compared for their longevity. 62 pieces of each type of bulb are tested under similar conditions. Brand A has an average life of 720 hours with a standard deviation of 30 hours, while brand B has an average life of 860 hours with a standard deviation of 42
9.35 A random sample of size n1 = 30 is taken from a normal population with a standard deviation of σ1 = 4 and mean of ¯x1 = 65. A second random sample of size n2 = 48 is taken from a different normal population with a standard deviation of σ2 = 3 and mean of¯x2 = 58. Find a 98% confidence
9.34 Consider Exercise 9.33. Use the MSE discussed in Exercise 9.28 to determine which estimator is more efficient. Write out MSE(S2 )MSE(S2 ) .
9.33 Compare S2 and S2 (see Exercise 9.29), the two estimators of σ2 , to determine which is more efficient. Assume these estimators are found using X1,X2, . . . , Xn , independent random variables from n(x; μ, σ). Which estimator is more efficient considering only the variance of the
9.32 Show that the estimator P of Exercise 9.31(b)becomes unbiased as n→∞.
9.31 If X is a binomial random variable, show that(a) P = X/n is an unbiased estimator of p;(b) P = X +√n/2 n+√n is a biased estimator of p.
9.30 Consider S2 , the estimator of σ2 , from Exercise 9.29. Analysts often use S2 rather than dividingn 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 →∞.
9.29 Let us define S2 =n i=1(Xi − ¯X )2/n. Show that E(S2) = [(n − 1)/n]σ2 , and hence S2 is a biased estimator for σ2 .
9.28 In Section 9.3, we emphasized the notion of“most efficient estimator” by comparing the variance of two unbiased estimators ˆΘ 1 and ˆΘ2 . However, this does not take into account bias in case one or both estimators are not unbiased. Consider the quantity MSE = E(ˆΘ − θ), where MSE
9.27 Consider the situation of Case Study 9.1 on page 301 with a larger sample of metal pieces. The diameters are as follows: 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 1.01, 1.03, 0.99, 1.00, 1.00, 0.99, 0.98, 1.01, 1.02, 0.99 centimeters. Once again the normality assumption may be made. Do the following
9.26 Consider the data in Exercise 9.13. Suppose the manufacturer of the shearing pins insists that the Rockwell hardness of the product be less than or equal to 44.0 only 5% of the time. What is your reaction? Use a tolerance limit calculation as the basis for your judgment.
9.25 Consider the drying time measurements in Exercise 9.14. Suppose the 15 observations in the data set are supplemented by a 16th value of 6.9 hours. In the context of the original 15 observations, is the 16th value an outlier? Show work.
9.24 Refer to Exercise 9.22 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 the manufactured pieces have tensile strength less than 62 kilograms.Is there cause for
9.23 Refer to Exercise 9.22. 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?
9.22 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 strengths, give a
9.21 In a study conducted by the Department of Biological Sciences 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 orthophosphorus in the river. The concentration of the chemical is measured in
9.20 Consider the situation of Exercise 9.11. 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.Find the 95% tolerance limits that contain 95% of the diameters.
9.19 A random sample of 25 tablets of buffered aspirin contains, on average, 325.05 mg of aspirin per tablet, with a standard deviation of 0.5 mg. Find the 95% tolerance limits that will contain 90% of the tablet contents for this brand of buffered aspirin. Assume that the aspirin content is
9.18 Referring to Exercise 9.13, construct a 95% tolerance interval containing 90% of the measurements.
9.17 Consider Exercise 9.9. Compute a 95% prediction interval for the sugar content of the next single serving of Alpha-Bits.
9.16 Consider Exercise 9.10. Compute the 95% prediction interval for the next observed number of words per minute typed by a graduate of the secretarial school.
9.15 Referring to Exercise 9.5, construct a 99% prediction interval for the kilometers traveled annually by an automobile owner in Virginia.
9.14 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 3.6 2.8 3.3 5.6 3.7 2.8 4.4 4.0 5.2 3.0 4.8 Assuming that the measurements represent a random sample from a normal population, find a 95% prediction interval for the drying
9.13 A random sample of 12 shearing pins is taken in a study of the Rockwell hardness of the pin head.Measurements on the Rockwell hardness are made for each of the 12, yielding an average value of 48.50 with a sample standard deviation of 1.5. Assuming the measurements to be normally distributed,
9.12 A random sample of 10 chocolate energy bars of a certain brand has, on average, 230 calories per bar, with a standard deviation of 15 calories. Construct a 99% confidence interval for the true mean calorie content of this brand of energy bar. Assume that the distribution of the calorie content
9.11 A machine produces metal pieces that are cylindrical in shape. A sample of pieces is taken, and the diameters are found to be 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, and 1.03 centimeters. Find a 99% confidence interval for the mean diameter of pieces from this machine, assuming an
9.10 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
9.9 Regular consumption of presweetened cereals contributes to tooth decay, heart disease, and other degenerative diseases, according to studies conducted by Dr.W. H. Bowen of the National Institute of Health and Dr. J. Yudben, Professor of Nutrition and Dietetics at the University of London. In a
9.8 A marketing agency wishes to determine the average time, in days, that it takes to sell a product in various stores in a city. How large a sample will they need in order to be 98% confident that their sample mean will be within 2 days of the true mean? Assume that σ = 5 days.
9.7 How large a sample is needed in Exercise 9.3 if we wish to be 95% confident that our sample mean will be within 0.0005 inch of the true mean?
9.6 How large a sample is needed in Exercise 9.2 if we wish to be 96% confident that our sample mean will be within 10 hours of the true mean?
9.5 A random sample of 100 automobile owners in the state of Virginia shows that an automobile is driven on average 23,500 kilometers per year with a standard deviation of 3900 kilometers. Assume the distribution of measurements to be approximately normal.(a) Construct a 99% confidence interval for
9.4 The prices of a particular variety of rice, per kilogram, collected from 48 local stores in the suburban areas of a district vary with a mean of $3 and a standard deviation of $1.6.(a) Construct a 95% confidence interval for the mean price.(b) With 95% confidence, what can we assert about the
9.3 Many cardiac patients wear an implanted pacemaker to control their heartbeat. A plastic connector module mounts on the top of the pacemaker. Assuming a standard deviation of 0.0015 inch and an approximately normal distribution, find a 95% confidence interval for the mean of the depths of all
9.2 An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If a sample of 30 bulbs has an average life of 780 hours, find a 96% confidence interval for the population mean of all bulbs produced by this
9.1 As part of a research on nutrition, a particular protein diet was tried out on a large group of mice.Researchers claimed that the new diet led to an increase in the weight of the mice. Assuming that it is known from previous studies that σ = 4.5 grams, how many mice should be included in our
8.76 Group Project: The class should be divided into groups of four people. The four students in each group should go to the college gym or a local fitness center. The students should ask each person who comes through the door his or her height in inches.Each group will then divide the height data
8.75 Consider the situation in Review Exercise 8.74.Suppose a considerable effort is conducted to “tighten”the variability in the system. Following the effort, a random sample of size 40 is taken from the new assembly line and the sample variance is s2 = 0.188 ounces2 .Do we have strong
8.74 Suppose a filling machine is used to fill cartons with a liquid product. The specification that is strictly enforced for the filling machine is 9 ± 1.5 oz. If any carton is produced with weight outside these bounds, it is considered by the supplier to be defective. It is hoped that at least
8.73 In Chapter 9, the concept of parameter estimation will be discussed at length. Suppose X is a random variable with mean μ and variance σ2 = 1.0.Suppose also that a random sample of size n is to be taken and ¯x is to be used as an estimate of μ. When the data are taken and the sample mean
8.72 Given a normal random variable X with mean 20 and variance 9, and a random sample of size n taken from the distribution, what sample size n is necessary in order that P(19.9 ≤ ¯X ≤ 20.1) = 0.95?
8.71 From the information in Review Exercise 8.70, compute (assuming μB = 65%) P(¯XB ≥ 70).
8.70 The concentration of an active ingredient in the output of a chemical reaction is strongly influenced by the catalyst that is used in the reaction. It is felt that when catalyst A is used, the population mean concentration exceeds 65%. The standard deviation is known to be σ = 5%. A sample of
8.69 Two distinct solid fuel propellants, type A and type B, are being considered for a space program activity.Burning rates of the propellant are crucial. Random samples of 20 specimens of the two propellants are taken with sample means 20.5 cm/sec for propellant A and 24.50 cm/sec for propellant
8.68 Consider the situation of Review Exercise 8.62.If the population from which the sample was taken has population mean μ = 53, 000 kilometers, does the sample information here seem to support that claim? In your answer, compute t =¯x − 53, 000 s/√10 and determine from Table A.4 (with 9
8.67 The breaking strength X of a certain rivet used in a machine engine has a mean 5000 psi and standard deviation 400 psi. A random sample of 36 rivets is taken. Consider the distribution of ¯X, the sample mean breaking strength.(a) What is the probability that the sample mean falls between 4800
8.66 Consider Review Exercise 8.56. Comment on any outliers in the data.
8.65 Consider Example 1.5 on page 45. Comment on any outliers.
8.64 If S2 1 and S2 2 represent the variances of independent random samples of size n1 = 16 and n2 = 27, taken from normal populations with variances σ2 1 = 8 and σ2 2 = 14, respectively, find P(S2 1 /S2 2 > 1.873).
8.63 Consider the data of Exercise 1.19 on page 51.Construct a box-and-whisker plot. Comment. Compute the sample mean and sample standard deviation.
8.62 A taxi company tests a random sample of 10 steel-belted radial tires of a certain brand and records the following tread wear: 48,000, 53,000, 45,000, 61,000, 59,000, 56,000, 63,000, 49,000, 53,000, and 54,000 kilometers. Use the results of Exercise 8.14 on page 251 to find the standard
8.61 The number of destructive tsunami that originate in the Pacific basin each year is a random variable having a Poisson distribution, with μ = 2 per year.Find the probability that this area will be hit by(a) exactly 5 tsunami in 2 years;(b) at most 3 tsunami in 2 years.
8.60 The price quotations for a home appliance is collected randomly from five different retail shops in a city.These are $305, $312, $296, $304, and $307. Find the variance of the quoted prices.
8.58 In testing for carbon monoxide in a certain brand of cigarette, the data, in milligrams per cigarette, were coded by subtracting 12 from each observation.Use the results of Exercise 8.14 on page 251 to find the standard deviation for the carbon monoxide content of a random sample of 15
8.57 If X1,X2, . . . , Xn are independent random variables having identical exponential distributions with parameter θ, show that the density function of the random variable Y = X1+X2+· · ·+Xn is that of a gamma distribution with parameters α = n and β = θ.
8.56 Consider the data displayed in Exercise 1.20 on page 51. Construct a box-and-whisker plot and comment on the nature of the sample. Compute the sample mean and sample standard deviation.
8.55 Construct a normal quantile-quantile plot of these data, which represent the diameters of 36 rivet heads in 1/100 of an inch:6.72 6.77 6.82 6.70 6.78 6.70 6.62 6.75 6.66 6.66 6.64 6.76 6.73 6.80 6.72 6.76 6.76 6.68 6.66 6.62 6.72 6.76 6.70 6.78 6.76 6.67 6.70 6.72 6.74 6.81 6.79 6.78 6.66 6.76
8.54 Construct a quantile plot of these data, which represent the lifetimes, in hours, of fifty 40-watt, 110-volt internally frosted incandescent lamps taken from forced life tests:919 1196 785 1126 936 918 1156 920 948 1067 1092 1162 1170 929 950 905 972 1035 1045 855 1195 1195 1340 1122 938 970
8.53 Consider the following measurements of the heat-producing capacity of the coal produced by two mines (in millions of calories per ton):Mine 1: 8260 8130 8350 8070 8340 Mine 2: 7950 7890 7900 8140 7920 7840 Can it be concluded that the two population variances are equal?
8.52 Pull-strength tests on 10 soldered leads for a semiconductor device yield the following results, in pounds of force required to rupture the bond:19.8 12.7 13.2 16.9 10.6 18.8 11.1 14.3 17.0 12.5 Another set of 8 leads was tested after encapsulation to determine whether the pull strength had
8.50 A transport company claims that the average running time for a bus on a particular route is 300 minutes. Six buses are randomly observed and their running times are recorded as follows: 320, 310, 295, 312, 302, and 308 minutes. Would you agree with the transport company’s claim? Assume a
8.49 A normal population with unknown variance has a mean of 15. Is one likely to obtain a random sample of size 16 from this population with a mean of 18 and a standard deviation of 3.8? If not, what conclusion would you draw?
8.48 A technical training firm claims that the average time for their trainees to master a particular skill is 40 hours. To maintain this average, 24 students are tested every month. If the computed t-value falls between−t0.025 and t0.025 , the firm is satisfied with its claim. What conclusion
8.47 Given a random sample of size 30 from a normal distribution, find k such that,(a) P(−1.699 < T < k) = 0.945;(b) P(k < T < 2.045) = 0.025;(c) P(−k < T < k) = 0.98.
8.46 (a) Find P(−t0.01 < T < t0.05 ).(b) Find P(−t0.05 > T > t0.025 ).
8.45 (a) Find P(T < 2.65), when v = 13.(b) Find P(T > 2.06), when v = 25.(c) Find P(−1.74 < T < 2.567), when v = 17.(d) Find P(T > −2.06), when v = 25.
8.44 (a) Find t0.05, when v = 16.(b) Find −t0.05, when v = 12.(c) Find t0.95, when v = 9.
8.43 Show that the variance of S2 for random samples of size n from a normal population decreases as n becomes large. [Hint: First find the variance of(n − 1)S2/σ2 .]
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