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statistics for engineers and scientists

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

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 the
The following is the result of the last 30 games played by an athletic team, with W signifying a win and L a loss.W W W LW W LW W LW LW W LW W W W LW LW W W LW LW L Are these data consistent with pure randomness?
Running the text disk program on the data of Example 12.4c yields Figure 12.4, which is quite close to the exact value of .1225. Running the program using the data of Example 12.4d yields Figure 12.5, which is again quite close to the exact value of .0364. ■
Suppose that in testing whether 2 production methods yield identical results, 9 items are produced using the first method and 13 using the second. If, among all 22 items, the sum of the ranks of the 9 items produced by method 1 is 72, what conclusions would you draw?
Suppose we are interested in determining whether a certain population has an underlying probability distribution that is symmetric about 0. If a sample of size 20 from this population results in a signed rank test statistic of value 142, what conclusion can we draw at the 10 percent level of
A financial institution has decided to open an office in a certain community if it can be established that the median annual income of families in the community is greater than $90,000. To obtain information, a random sample of 80 families was chosen, and the family incomes determined. If 28 of
If a sample of size 200 contains 120 values that are less than m0 and 80 values that are greater, what is the p-value of the test of the hypothesis that the median is equal to m0?
Data are said to be from a lognormal distribution with parameters μ and σ if the natural logarithms of the data are normally distributed with mean μ and standard deviation σ. Use the Kolmogorov–Smirnov test with significance level.05 to decide whether the following lifetimes (in days) of a
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.Test the hypothesis that whether it rains tomorrow is independent of the color of
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.Test the hypothesis that the percentages of the surgical operations that lead to lawsuits are the same for
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
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.Use the above to test the hypothesis that having a cellular phone in your car and being involved in an
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.Is this change significant at the .05 level? Defective Acceptable Superior Before 25 After 9 218 22 103 14
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 Moderate Smoker Heavy Smoker
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. Outcome at the End of 1 Year Birthweight
Repeat Problem 16 with all of the data values doubled—that is, with these data:20 80 30 270
The following data relate the mother’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. Birthweight Maternal Age Less Than 2,500 Grams More Than 2,500 Grams 20 years or less Greater than 20 10 15 40 135
A random sample of 500 families was classified by region and income (in units of$1,000). The following data resulted.Determine the p-value of the test that a family’s income and region are independent. Income South North 0-10 42 53 10-20 55 90 20-30 47 88 >30 36 89
In Problem 4, test the hypothesis that the daily number of failures has a Poisson distribution.
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 that
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 be(a) 1,000;(b) 5,000;(c) 10,000.
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
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 1,100 earthquakes.Test, at the 5 percent
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
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 0 012 Number of Hours with
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:Test whether the signals are uniformly distributed over the 24-hour period. Frequency of Neutrino Radiation from Outer Space Hour Starting
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
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
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 0123456789 0 5 22 23 32 22 19 13 6 4 10
Determine the birth and death dates of 100 famous individuals and, using the four-category approach of Example 11.2a, test the hypothesis that the death month is not affected by the birth month. Use the chi-square approximation.
To ascertain whether a certain die was fair, 1,000 rolls of the die were recorded, with the following results.Test the hypothesis that the die is fair (that is, that pi = 16 , i = 1, . . . ,6) at the 5 percent level of significance. Use the chi-square approximation. Outcome Number of Occurrences 12
According to the Mendelian theory of genetics, a certain garden pea plant should produce either white, pink, or red flowers, with respective probabilities 14, 12, 14.To test this theory, a sample of 564 peas was studied with the result that 141 produced white, 291 produced pink, and 132 produced
Suppose we want to test the hypothesis that a given population distribution is exponential with mean 100; that is, F (x) = 1 − e−x/100. If the (ordered) values from a sample of size 10 from this distribution are 66, 72, 81, 94, 112, 116, 124, 140, 145, 155 what conclusion can be drawn?
A recent study reported that 500 female office workers were randomly chosen and questioned in each of four different countries. One of the questions related to whether these women often received verbal or sexual abuse on the job. The following data resulted.Based on these data, is it plausible that
A randomly chosen group of 20,000 nonsmokers and one of 10,000 smokers were followed over a 10-year period. The following data relate the numbers of them that developed lung cancer during that period.Test the hypothesis that smoking and lung cancer are independent. Use the 1 percent level of
A company operates four machines on three separate shirts daily. The following contingency table presents the data during a 6-month time period, concerning the machine breakdowns that resulted.Suppose we are interested in determining whether a machine’s breakdown probability during a particular
A sample of 300 people was randomly chosen, and the sampled individuals were classified as to their gender and political affiliation, Democrat, Republican, or Independent. The following table, called a contingency table, displays the resulting data.Thus, for instance, the contingency table
Suppose the weekly number of accidents over a 30-week period is as follows:Test the hypothesis that the number of accidents in a week has a Poisson distribution. 883 13 8 0 0 1 3402 8 0 2 0 19 3 334 47 4012 12 5 45 1 2
Consider an experiment having six possible outcomes whose probabilities are hypothesized to be .1, .1, .05, .4, .2, and .15. This is to be tested by performing 40 independent replications of the experiment. If the resultant number of times that each of the six outcomes occurs is 3, 3, 5, 18, 4, 7,
A contractor who purchases a large number of fluorescent lightbulbs has been told by the manufacturer that these bulbs are not of uniform quality but rather have been produced in such a way that each bulb produced will, independently, either be of quality level A, B, C, D, or E, with respective
In recent years, a correlation between mental and physical well-being has increasingly become accepted. An analysis of birthdays and death days of famous people could be used as further evidence in the study of this correlation. To use these data, we are supposing that being able to look forward to
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 groups
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 the
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
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 hypothesis
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 the
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
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 that
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 in
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.
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
Astudy has been made on pyrethrumflowers 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 for 1
If xij = ai + bj , show that mn m =a+m i=1 j=1 i=1 j=1
If xij = i + j2, determine (a) (6) =1
For data xij , i = 1, . . . ,m, j = 1, . . . , n, show that x. m =xilm = xjln i=1 j=1
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
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 final
Five servings each of three different brands of processed meat were tested for fat content. The following data (in fat percentage per gram) resulted.(a) Does the fat content differ depending on the brand?(b) Find confidence intervals for all quantities μi − μj that, with 95 percent confidence,
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 the
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 on a
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 patients
The following data relate to the ages at death of a certain species of rats that were fed 1 of 3 types of dies. 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 the
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.Test the hypothesis that the polymer
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:Test, at the 5 percent level of significance, the hypothesis that the two diets
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.Do the data indicate that the procedures yield equivalent results? Method 1 2 3 4 76.42 80.41 74.20
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.Do the ovens appear to operate at the
Explain why we cannot efficiently test the hypothesis H0 : μ1 = μ2 = · · · = μm by running t -tests on all of them2pairs of samples.
We want to know what type of filter should be used over the screen of a cathoderayoscilloscope 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 target. A
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 measured
The following data from Example 10.4a give the scores obtained when four different reading tests were given to each of five students. Use it to estimate the parameters of the model. Student Examination 1 2 3 4 5 Row Totals Xi. 1 75 73 60 70 86 364 72.8 2 78 71 64 72 90 375 75 3 80 69 62 70 85 366 4
A college administrator claims that there is no difference in first-year grade point averages for students entering the college from any of three different city high schools. The following data give the first-year grade point averages of 12 randomly chosen students, 4 from each of the three high
An auto rental firm is using 15 identical motors that are adjusted to run at a fixed speed to test 3 different brands of gasoline. Each brand of gasoline is assigned to exactly 5 of the motors. Each motor runs on 10 gallons of gasoline until it is out of fuelThe following represents the total
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.(a) Explain how the relationship between a second heart attack and one’s anger score can be analyzed via a
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 refers
For the logistics regression model, find the value x such that p(x) = .5.
A recently completed study attempted to relate job satisfaction to income (in 1,000s) 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
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
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 95
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 5.1 5.4 5.9 6.6 7.5 8.6 9.9 11.4 13.1
Explain why, for the same data, a prediction interval for a future response always contains the corresponding confidence interval for the mean response.
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 in
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)
(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 x2 x3 y 7.1 .68 4 41.53 9.9 .64 1 63.75 3.6 .58 1 16.38 9.3
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
Fit a multiple linear regression equation to the following data set. x1 x2 x3 1 11 16 2345678 a 10 9 9 4 432 x4 y 275 183 140 8 1 1 82 7 2 1 97 6 1 -1 122 5 4 -2 146 4 9 -3 246 9 3 16 -4 359 10 2 25 -5 482
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 (x103 mi) Discharge (ft/sec) Sediment Yield (Millions of tons/yr) 8 65 1.8 19 625 6.4 31 1,450 3.3 16 2,400
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. Peak x1 (m) (ft/ft) x2 Discharge
The following data set refers to Example 9.8c.(a) Estimate the number of accidents in a month when the number of cars using the highway is 3,500. (b) Use the model √Y = α + βx + e and redo part (a). Number of Cars (Daily) 2,000 Number of Accidents (Monthly) 15 2,300 27 2,500 20 2,600 21 2,800
The following data relate the proportions of coal miners who exhibit symptoms of pneumoconiosis to the number of years of working in coal mines.\Estimate the probability that a coal miner who has worked for 42 years will have pneumoconiosis. Years Working Proportion Having Penumoconiosis 5 10 0
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.
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.(a) Analyze as a simple linear regression model.(b) Plot the standardized residuals.(c) Do the results of part (b) indicate any flaw in the model?(d) If
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
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.(a) Draw a scatter diagram.(b) Fit a curve of the form Y = α + βx + γx2 + e to the data. 8 00 Distance 1 Amount
The following data represent the bacterial count of five individuals at different times after being inoculated by a vaccine consisting of the bacteria.(a) Fit a curve.(b) Estimate the bacteria count of a new patient after 8 days. Days Since Inoculation Bacterial Count 36989 121,000 134,000 7
The proportion of a given heat rise that has dissipated a time t after the source is cut off is of the form for some unknown constant α. Given the dataestimate the value of α. Estimate the value of t at which half of the heat rise is dissipated. P = 1-eat
The chlorine residual in a swimming pool at various times after being cleaned is as given:Fit a curve of the form Y ≈ ae−bx What would you predict for the chlorine residual 15 hours after a cleaning? Time (hr) 246 Chlorine Residual (pt/million) 1.8 1.5 1.45 8 1.42 10 1.38 12 1.36
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 and s
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 bywhere A andmare unknown constants. An experiment is run yielding the following data.Estimate A and m. S = A Nm
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 males

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