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statistics for engineers and scientists

Introduction To Probability And Statistics For Engineers And Scientists 6th Edition Sheldon M Ross - Solutions

17. In a 10-year study of the dispersal patterns of beavers (Sun, L. and Muller-Schwarze, D., “Statistical resampling methods in biology: A case study of beaver dispersal patterns,” American Journal of Mathematical and Management Sciences, 16, pp. 463–502, 1996) a total of 332 beavers were
16. In a 1943 experiment (Whitlock, H. V., and Bliss, D. H., “A bioassay technique for antihelminthics,” Journal of Parasitology, 29, pp. 48–58, 10), albino rats were used to study the effectiveness of carbon tetrachloride as a treatment for worms. Each rat received an injection of worm
15. The following are the weights of random samples of adult males from different political affiliations.Republicans: 204, 178, 195, 187, 240, 182, 152, 166 Democrats: 175, 200, 168, 192, 156, 164, 180, 138 We want to use these data to test the null hypothesis that the two distributions are
14. Determine the p-value in Problem 13 bya. using the normal approximation;b. using a simulation study.
13. Fifteen cities, of roughly equal size, are chosen for a traffic safety study.Eight of them are randomly chosen, and in these cities a series of newspaper articles dealing with traffic safety is run over a 1-month period. The number of traffic accidents reported in the month following this
12. In a study of bilingual coding, 12 bilingual (French and English) college students are divided into two groups. Each group reads an article written in French, and each answers a series of 25 multiple-choice questions covering the content of the article. For one group the questions are written
11. Let X1, . . . , Xn be a sample from the continuous distribution F having median m; and suppose we are interested in testing the hypothesis H0 : m = m0 against the one-sided alternative H1 : m > m0. Present the one-sided analog of the signed rank test. Explain how the p-value would be computed.
10. Ten pairs of duplicate spectrochemical determinations for nickel are presented below. The readings in column 2 were taken with one type of measuring instrument and those in column 3 were taken with another type.Sample Duplicates 1 1.94 2.00 2 1.99 2.09 3 1.98 1.95 4 2.07 2.03 5 2.03 2.08 6 1.96
9. An engineer claims that painting the exterior of a particular aircraft affects its cruising speed. To check this, the next 10 aircraft off the assembly line were flown to determine cruising speed prior to painting, and were then painted and reflown. The following data resulted.Do the data uphold
8. Twelve patients having high albumin content in their blood were treated with a medicine. Their blood content of albumin was measured before and after treatment. The measured values are shown in the table.Is the effect of the medicine significant at the 5 percent level?a. Use the sign test.b. Use
7. Determine the p-value when using the signed rank statistic in Problems 1 and 2.
6. An experiment was initiated to study the effect of a newly developed gasoline detergent on automobile mileage. The following data, representing mileage per gallon before and after the detergent was added for each of eight cars, resulted.Car Mileage without Additive Mileage with Additive 1 24.2
5. In 2004, the national median salary of all U.S. financial accountants was$124,400. A recent random sample of 14 financial accountants showed 2007 incomes of (in units of $1000)125.5, 130.3, 133.0, 102.6, 198.0, 232.5, 106.8, 114.5, 122.0, 100.0, 118.8, 108.6, 312.7, 125.5 Use these data to test
4. To test the hypothesis that the median weight of 16-year-old females from Los Angeles is at least 110 pounds, a random sample of 200 such females was chosen. If 120 females weighed less than 110 pounds, does this discredit the hypothesis? Use the 5 percent level of significance. What is the
3. The published figure for the median systolic blood pressure of middleaged men is 128. To determine if there has been any change in this value, a random sample of 100 men has been selected. Test the hypothesis that the median is equal to 128 ifa. 60 men have readings above 128;b. 70 men have
2. An engineering firm is involved in selecting a computer system, and the choice has been narrowed to two manufacturers. The firm submits eight problems to the two computer manufacturers and has each manufacturer measure the number of seconds required to solve the design problem with
1. A new medicine against hypertension was tested on 18 patients. After 40 days of treatment, the following changes of the diastolic blood pressure were observed.−5, −1, +2, +8, −25, +1, +5, −12, −16−9, −8, −18, −5, −22, +4, −21, −15, −11 Use the sign test to determine if
25. Data are said to be from a lognormal distribution with parameters μ and σif the natural logarithms of the data are normally distributed with meanμ and standard deviation σ. Use the Kolmogorov–Smirnov test with significance level .05 to decide whether the following lifetimes (in days) of a
24. In a famous article (S. Russell, “A red sky at night. . . ,” Metropolitan Magazine London, 61, p. 15, 1926) the following data set of frequencies of sunset colors and whether each was followed by rain was presented.Sky Color Number of Observations Number Followed by Rain Red 61 26 Mainly
23. To determine if a malpractice lawsuit is more likely to follow certain types of surgeries, random samples of three different types of surgeries were studied, and the following data resulted.Type of Operation Number Sampled Number Leading to a Lawsuit Heart surgery 400 16 Brain surgery 300 19
22. To study the effect of fluoridated water supplies on tooth decay, two communities of roughly the same socioeconomic status were chosen. One of these communities had fluoridated water while the other did not. Random samples of 200 teenagers from both communities were chosen, and the numbers of
21. A sample of 300 cars having cellular phones and one of 400 cars without phones were tracked for 1 year. The following table gives the number of these cars involved in accidents over that year.Accident No Accident Cellular phone 22 278 No phone 26 374 Use the above to test the hypothesis that
20. The following table shows the number of defective, acceptable, and superior items in samples taken both before and after the introduction of a modification in the manufacturing process.Defective Acceptable Superior Before 25 218 22 After 9 103 14 Is this change significant at the .05 level?
19. An experiment designed to study the relationship between hypertension and cigarette smoking yielded the following data.Test the hypothesis that whether or not an individual has hypertension is independent of how much that person smokes. Nonsmoker Moderate Smoker Heavy Smoker Hypertension 20 No
18. The number of infant mortalities as a function of the baby’s birthweight(in grams) for 72,730 live white births in New York in 1974 is as follows:Test the hypothesis that the birthweight is independent of whether or not the baby survives its first year. Birthweight Outcome at the End of 1
17. Repeat Problem 16 with all of the data values doubled — that is, with these data:20 80 30 270
16. The following data relate themother’s age and the birthweight (in grams)of her child.Test the hypothesis that the baby’s birthweight is independent of the mother’s age. Maternal Age Birthweight 20 years or less Greater than 20 Less Than 2500 Grams More Than 2500 Grams 10 15 40 135
15. A random sample of 500 migrant families was classified by region and income (in units of $1000). The following data resulted.Income South North 0–10 42 53 10–20 55 90 20–30 47 88>30 36 89 Determine the p-value of the test that a family’s income and region are independent.
14. In Problem 4, test the hypothesis that the daily number of failures has a Poisson distribution.
13. A sample of size 120 had a sample mean of 100 and a sample standard deviation of 15. Of these 120 data values, 3 were less than 70; 18 were between 70 and 85; 30 were between 85 and 100; 35 were between 100 and 115; 32 were between 115 and 130; and 2 were greater than 130. Test the hypothesis
12. ] Use simulation to determine the p-value and compare it with the result you obtained using the chi-square approximation in Problem 1. Let the number of simulation runs bea. 1000;b. 5000;c. 10,000.
11. Sometimes reported data fit a model so well that it makes one suspicious that the data are not being accurately reported. For instance, a friend of mine has reported that he tossed a fair coin 40,000 times and obtained 20,004 heads and 19,996 tails. Is such a result believable? Explain your
10. A study was instigated to see if southern California earthquakes of at least moderate size (having values of at least 4.4 on the Richter scale) are more likely to occur on certain days of the week than on others. The catalogs yielded the following data on 1100 earthquakes.Day Sun Mon Tues Wed
9. In a certain region, insurance data indicate that 82 percent of drivers have no accidents in a year, 15 percent have exactly 1 accident, and 3 percent have 2 or more accidents. In a random sample of 440 engineers, 366 had no accidents, 68 had exactly 1 accident, and 6 had 2 or more. Can you
8. Neutrino radiation was observed over a certain period and the number of hours in which 0, 1, 2,. . . signals were received was recorded.Test the hypothesis that the observations come from a population having a Poisson distribution with mean .3. Number of Signals per Hour Number of Hours with
7. The neutrino radiation from outer space was observed during several days. The frequencies of signals were recorded for each sidereal hour and are as given below in Table 11.3:Test whether the signals are uniformly distributed over the 24-hour period. Table 11.3 Frequency of Neutrino Radiation
6. The past output of a machine indicates that each unit it produces will be top grade with probability .40 high grade with probability .30 medium grade with probability .20 low grade with probability .10 A new machine, designed to perform the same job, has produced 500 units with the following
5. Among 100 vacuum tubes tested, 41 had lifetimes of less than 30 hours, 31 had lifetimes between 30 and 60 hours, 13 had lifetimes between 60 and 90 hours, and 15 had lifetimes of greater than 90 hours. Are these data consistent with the hypothesis that a vacuum tube’s lifetime is exponentially
4. It is believed that the daily number of electrical power failures in a certain Midwestern city is a Poisson random variable with mean 4.2. Test this hypothesis if over 150 days the number of days having i power failures is as follows:Failures Number of Days 0 0 1 5 2 22 3 23 4 32 5 22 6 19 7 13
3. Determine the birth and death dates of 100 famous individuals and, using the four-category approach of Example 11.2.a, test the hypothesis that the death month is not affected by the birth month. Use the chi-square approximation.
2. To ascertain whether a certain die was fair, 1000 rolls of the die were recorded, with the following results.Outcome Number of Occurrences 1 158 2 172 3 164 4 181 5 160 6 165 Test the hypothesis that the die is fair (that is, that pi = 1 6 , i = 1, . . . , 6) at the 5 percent level of
1. According to the Mendelian theory of genetics, a certain garden pea plant should produce either white, pink, or red flowers, with respective probabilities 14 , 1 2 , 1 4 . To test this theory, a sample of 564 peas was studied with the result that 141 produced white, 291 produced pink, and 132
30. Suppose that μ,α1, . . . , αm,β1, . . . , βn and μ,α1, . . . , αm,β1, . . . , βn are such that +; + ;=' + + B'; for all i, j ==; = ; =0 Show that =', =, ; = B'; for all i and j. This shows that the parameters , 1,..., am. B1, Bn in our representation of two-factor ANOVA are uniquely
29. In a study of platelet production, 16 rats were put at an altitude of 15,000 feet, while another 16 were kept at sea level (Rand, K., Anderson, T., Lukis, G., and Creger, W., “Effect of hypoxia on platelet level in the rat,”Clinical Research, 18, p. 178, 1970). Half of the rats in both
28. An experiment has been devised to test the hypothesis that an elderly person’s memory retention can be improved by a set of “oxygen treatments.”A group of scientists administered these treatments to men and women. The men and women were each randomly divided into 4 groups of 5 each, and
27. Suppose, in Problem 23, that there has been some controversy about the assumption of no interaction between gasoline and additive used. To allow for the possibility of an interaction effect between gasoline and additive, it was decided to run 36 motors — 4 in each grouping. The following data
26. A study was made as to how the concentration of a certain drug in the blood, 24 hours after being injected, is influenced by age and gender. An analysis of the blood samples of 40 people given the drug yielded the following concentrations (in milligrams per cubic centimeter).a. Test the
25. A researcher is interested in comparing the breaking strength of different laminated beams made from 3 different types of glue and 3 varieties of wood. To make the comparison, 5 beams of each of the 9 combinations were manufactured and then put under a stress test. The following table indicates
24. Suppose in Problem 6 that the 10 people placed on each diet consisted of 5 men and 5 women, with the following data.a. Test the hypothesis that there is no interaction between gender and diet.b. Test the hypothesis that the diet has the same effect on men and women. Diet 1 Diet 2 Women 7.6 19.5
23. An experiment was devised to test the effects of running 3 different types of gasoline with 3 possible types of additives. The experiment called for 9 identical motors to be run with 5 gallons for each of the pairs of gasoline and additives. The following data resulted.a. Test the hypothesis
22. Three different washing machines were employed to test four different detergents. The following data give a coded score of the effectiveness of each washing.a. Estimate the improvement in mean value when using detergent 1 over using detergents (i) 2; (ii) 3; (iii) 4.b. Estimate the improvement
21. For the model of Problem 19:a. Do the methods of extraction appear to differ?b. Do the storage conditions affect the content? Test at the α = .05 level of significance.
20. The following data refer to the number of deaths per 10,000 adults in a large Eastern city in the different seasons for the years 1982 to 1986.a. Assuming a two-factor model, estimate the parameters.b. Test the hypothesis that death rates do not depend on the season.Use the 5 percent level of
19. A study has been made on pyrethrum flowers to determine the content of pyrethrin, a chemical used in insecticides. Four methods of extracting the chemical are used, and samples are obtained from flowers stored under three conditions: fresh flowers, flowers stored for 1 year, and flowers stored
18. If xij = ai +bj , show that mn m n =+m bj i=1 j=1 i=1 j=1
17. If xij = i +j 2, determine a. j=1- b. =1. j=1 Xij
15. Test the hypothesis that the following three independent samples all come from the same normal probability distribution.Sample 1 Sample 2 Sample 3 35 29 44 37 38 52 29 34 56 27 30 30 32!~!~!~!16. For data xij , i = 1, . . . , m,j = 1, . . . , n, show that m n x= =xi./m = xj/n i=1 j=1
14. A nutritionist randomly divided 15 bicyclists into 3 groups of 5 each.The first group was given a vitamin supplement to take with each of their meals during the next 3 weeks. The second group was instructed to eat a particular type of high-fiber whole-grain cereal for the next 3 weeks. The
13. Five servings each of three different brands of processed meat were tested for fat content. The following data (in fat percentage per gram) resulted.Brand 1 2 3 Fat content 32 41 36 34 32 37 31 33 30 35 29 28 33 35 33a. Does the fat content differ depending on the brand?b. Find confidence
12. An emergency room physician wanted to know whether there were any differences in the amount of time it takes for three different inhaled steroids to clear a mild asthmatic attack. Over a period of weeks she randomly administered these steroids to asthma sufferers, and noted the time it took for
11. A study of the trunk flexor muscle strength of 75 girls aged 3 to 7 was reported by Baldauf, K., Swenson, D., Medeiros, J., and Radtka, S., “Clinical assessment of trunk flexor muscle strength in healthy girls 3 to 7,”Physical Therapy, 64, pp. 1203–1208, 1984. With muscle strength graded
10. Plasma bradykininogen levels are related to the body’s ability to resist inflammation. In a 1968 study (Eilam, N., Johnson, P. K., Johnson, N. L., and Creger,W., “Bradykininogen levels in Hodgkin’s disease,” Cancer, 22, pp. 631–634), levels were measured in normal patients, in
9. The following data relate to the ages at death of a certain species of rats that were fed 1 of 3 types of diets. Thirty rats of a type having a short life span were randomly divided into 3 groups of size 10 each. The sample means and sample variances of the ages at death (measured in months)of
8. In the one-factor analysis of variance model with n observations per sample, let S2 i , i = 1, . . . , m denote the sample variances for the m samples.Show that m SSW = (n - 1) S i=1
In a test of the ability of a certain polymer to remove toxic wastes from water, experiments were conducted at three different temperatures. The data below give the percentages of the impurities that were removed by the polymer in 21 independent attempts.Low Temperature Medium Temperature High
Twenty overweight individuals, each more than 40 pounds overweight, were randomly assigned to one of two diets. After 10 weeks, the total weight losses (in pounds) of the individuals on each of the diets were as follows:Weight Loss Diet 1 Diet 2 22.2 24.2 23.4 16.8 24.2 14.6 16.1 13.7 9.4 19.5 12.5
Four standard chemical procedures are used to determine the magnesium content in a certain chemical compound. Each procedure is used four times on a given compound with the following data resulting.Method 1 2 3 4 76.42 80.41 74.20 86.20 78.62 82.26 72.68 86.04 80.40 81.15 78.84 84.36 78.20 79.20
A machine shop contains 3 ovens that are used to heat metal specimens.Subject to random fluctuations, they are all supposed to heat to the same temperature. To test this hypothesis, temperatures were noted on 15 separate heatings. The following data resulted.Oven Temperature 1 492.4, 493.6, 498.5,
Explain why we cannot efficiently test the hypothesis H0 : μ1 = μ2 =· · ·=μm by running t-tests on all of them2pairs of samples.
2. We want to know what type of filter should be used over the screen of a cathode-ray oscilloscope in order to have a radar operator easily pick out targets on the presentation. A test to accomplish this has been set up. A noise is first applied to the scope to make it difficult to pick out a
1. A purification process for a chemical involves passing it, in solution, through a resin on which impurities are adsorbed. A chemical engineer wishing to test the efficiency of 3 different resins took a chemical solution and broke it into 15 batches. She tested each resin 5 times and then
58. Twelve first-time heart attack victims were given a test that measures internal anger. The following data relates their scores and whether they had a second heart attack within 5 years.Anger Score Second Heart Attack 80 yes 77 yes 70 no 68 yes 64 no 60 yes 50 yes 46 no 40 yes 35 no 30 no 25
57. A study of 64 prematurely born infants was interested in the relation between the gestational age (in weeks) of the infant at birth and whether the infant was breast-feeding at the time of release from the birthing hospital. The following data resulted:In the preceding, the frequency column
56. For the logistics regression model, find the value x such that p(x) = .5.
55. Suppose in Problem 54 that job satisfaction was related solely to years on the job, with the following data resulting.Years on the Job Job Satisfaction 8 5.6 4 6.3 12 6.8 9 6.7 16 7.0 14 7.7 10 7.0 15 8.0 22 7.8a. Estimate the regression parameters α and β.b. What is the qualitative
54. A recently completed study attempted to relate job satisfaction to income(in 1000s) and seniority for a random sample of 9 municipal workers.The job satisfaction value given for each worker is his or her own assessment of such, with a score of 1 being the lowest and 10 being the highest.The
53. The following data relate the systolic blood pressure to the age (x1) and weight (x2) of a set of individuals of similar body type and lifestyle.a. Test the hypothesis that, when an individual’s weight is known, age gives no additional information in predicting blood pressure.b. Determine an
52. The cost of producing power per kilowatt hour is a function of the load factor and the cost of coal in cents per million Btu. The following data were obtained from 12 mills.a. Estimate the relationship.b. Test the hypothesis that the coefficient of the load factor is equal to 0.c. Determine a
51. Consider the following data set:a. Fit a linear relationship between y and x1, x2.b. Determine the variance of the error term.c. Determine an interval that, with 95 percent confidence, will contain the response when the inputs are x1 = 10.2 and x2 = 17. x1 %2 5.1 2 55.42 5.4 8 100.21 5.9 -2
50. Explain why, for the same data, a prediction interval for a future response always contains the corresponding confidence interval for the mean response.
49. The time to failure of a machine component is related to the operating voltage (x1), the motor speed in revolutions per minute (x2), and the operating temperature (x3).A designed experiment is run in the research and development laboratory, and the following data, where y is the time to failure
48. The tensile strength of a certain synthetic fiber is thought to be related to x1, the percentage of cotton in the fiber, and x2, the drying time of the fiber. A test of 10 pieces of fiber produced under different conditions yielded the following results.a. Fit a multiple regression equation.b.
47.a. Fit a multiple linear regression equation to the following data set.b. Test the hypothesis that β0 = 0.c. Test the hypothesis that β3 = 0.d. Test the hypothesis that the mean response at the input levels x1 = x2 = x3 = 1 is 8.5. x1 12 13 y 7.1 .68 4 41.53 9.9 .64 1 63.75 3.6 .58 1 16.38 9.3
46. The following data refer to Stanford heart transplants. It relates the survival time of patients that have received heart transplants to their age when the transplant occurred and to a so-called mismatch score that is supposed to be an indicator of how well the transplanted heart should fit the
45. Fit a multiple linear regression equation to the following data set. y x2 13 X4 11 16 4 275 10 9 3 183. 4 2 140 1 82 1 97 -1 122 -2 146 -3 246 7 8 9 9 16 -4 359 10 25 -5 482 12140 D98765432 1123456TOS
44. The sediment load in a stream is related to the size of the contributing drainage area (x1) and the average stream discharge (x2). Estimate this relationship using the following data. Area Discharge (x103 mi) (ft/sec) Sediment Yield (millions of tons/yr) 8 65 1.8 19 625 6.4 31 1450 3.3 16 2400
43. The peak discharge of a river is an important parameter for many engineering design problems. Estimates of this parameter can be obtained by relating it to the watershed area (x1) and watershed slope (x2). Estimate the relationship based on the following data. 11 x2 Peak Discharge (m) (ft/ft)
42. The following data set refers to Example 9.8.c.Number of Cars(Daily)Number of Accidents(Monthly)2000 15 2300 27 2500 20 2600 21 2800 31 3000 16 3100 22 3400 23 3700 40 3800 39 4000 27 4600 43 4800 53a. Estimate the number of accidents in a month when the number of cars using the highway is
41. The following data relate the proportions of coal miners who exhibit symptoms of pneumoconiosis to the number of years of working in coal mines.Years Working Proportion Having Penumoconiosis 5 0 10 .0090 15 .0185 20 .0672 25 .1542 30 .1720 35 .1840 40 .2105 45 .3570 50 .4545 Estimate the
40. Redo Problem 5 under the assumption that the variance of the gain in reading speed is proportional to the number of weeks in the program.
39. The following data represent the relation between the number of cans damaged in a boxcar shipment of cans and the speed of the boxcar at impact.Speed Number of Cans Damaged 3 54 3 62 3 65 5 94 5 122 5 84 6 142 7 139 7 184 8 254a. Analyze as a simple linear regression model.b. Plot the
38. A new drug was tested on mice to determine its effectiveness in reducing cancerous tumors. Tests were run on 10 mice, each having a tumor of size 4 grams, by varying the amount of the drug used and then determining the resulting reduction in the weight of the tumor. The data were as
37. The following data yield the amount of hydrogen present (in parts per million) in core drillings of fixed size at the following distances (in feet)from the base of a vacuum-cast ingot.Distance 1 2 3 4 5 6 7 8 9 10 Amount 1.28 1.50 1.12 .94 .82 .75 .60 .72 .95 1.20a. Draw a scatter diagram.b.
36. The following data represent the bacterial count of five individuals at different times after being inoculated by a vaccine consisting of the bacteria.Days Since Inoculation Bacterial Count 3 121,000 6 134,000 7 147,000 8 210,000 9 330,000a. Fit a curve.b. Estimate the bacteria count of a new
35. The proportion of a given heat rise that has dissipated a time t after the source is cut off is of the form P = 1 −e−αt for some unknown constant α. Given the data P .07 .21 .32 .38 .40 .45 .51 t .1 .2 .3 .4 .5 .6 .7 estimate the value of α. Estimate the value of t at which half of the
34. The chlorine residual in a swimming pool at various times after being cleaned is as given:Time (hr) Chlorine Residual (pt/million)2 1.8 4 1.5 6 1.45 8 1.42 10 1.38 12 1.36 Fit a curve of the form Y ≈ ae −bx What would you predict for the chlorine residual 15 hours after a cleaning?
33. In 1957 the Dutch industrial engineer J. R. DeJong proposed the following model for the time it takes to perform a simple manual task as a function of the number of times the task has been practiced:T ≈ ts−n where T is the time, n is the number of times the task has been practiced, and t
32. It has been determined that the relation between stress (S) and the number of cycles to failure (N) for a particular type of alloy is given byEstimate A and m. A S= Nm where A and m are unknown constants. An experiment is run yielding the following data. Stress N (thousand psi) (million cycles
31. The weight and systolic blood pressure of randomly selected males in age group 25 to 30 are shown in the following table.a. Estimate the regression coefficients.b. Do the data support the claim that systolic blood pressure does not depend on an individual’s weight?c. If a large number of
30. The following are the body mass index (BMI) and the systolic blood pressure of eight randomly chosen men who do not take any blood pressure medication.BMI Systolic Blood Pressure 20.3 116 22.0 110 26.4 131 28.2 136 31.0 144 32.6 138 17.6 122 19.4 115 Give an interval that, with 95 percent

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