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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
11.48 With reference to Exercise 11.8 on page 419, construct(a) a 95% confidence interval for the average course grade of students who make a 35 on the placement test;(b) a 95% prediction interval for the course grade of a student who made a 35 on the placement test.
11.47 The following data were obtained in a study of the relationship between the weight and chest size of infants at birth.Weight (kg) Chest Size (cm)2.75 29.5 2.15 26.3 4.41 32.2 5.52 36.5 3.21 27.2 4.32 27.7 2.31 28.3 4.30 30.3 3.71 28.7(a) Calculate r.(b) Test the null hypothesis that ρ = 0
11.46 Test the hypothesis that ρ = 0 in Exercise 11.43 against the alternative that ρ = 0. Use a 0.05 level of significance.
11.45 With reference to Exercise 11.13 on page 420, assume a bivariate normal distribution for x and y.(a) Calculate r.(b) Test the null hypothesis that ρ = −0.5 against the alternative that ρ < −0.5 at the 0.025 level of significance.(c) Determine the percentage of the variation in the
11.44 With reference to Exercise 11.1 on page 418, assume that x and y are random variables with a bivariate normal distribution.(a) Calculate r.(b) Test the hypothesis that ρ = 0 against the alternative that ρ = 0 at the 0.05 level of significance.
11.43 Compute and interpret the correlation coefficient for the following grades of 6 students selected at random:Mathematics grade 70 92 80 74 65 83 English grade 74 84 63 87 78 90
11.42 A non-athletic group of adolescents from a college was examined. Researchers identified 16 students, four each with four different weights. They wanted to estimate a formula for predicting the weight of a student from the information about the weight each could lift. The weight lifted by the
11.41 Evaluating nitrogen deposition from the atmosphere is a major role of the National Atmospheric Deposition Program (NADP), a partnership of many agencies. NADP is studying atmospheric deposition and its effect on agricultural crops, forest surface waters, and other resources. Nitrogen oxides
11.40 The data for total sales (x) of a company and its expenditure towards advertisement (y) for 2014 is given as follows:x (in million units) y (in’000 dollars)170 23 190 24 220 31 235 30 245 32 300 34 235 28 245 26 300 35 320 34 320 36 320 32(a) Fit a linear regression model relating to
11.39 A regression model is desired relating temperature and the proportion of impurities passing through solid helium. Temperature is listed in degrees centigrade.The data are as follows:Temperature (◦C) Proportion of Impurities−260.5 0.425−255.7 0.224−264.6 0.453−265.0 0.475−270.0
11.38 Heat treating is often used to carburize metal parts such as gears. The thickness of the carburized layer is considered an important feature of the gear, and it contributes to the overall reliability of the part.Because of the critical nature of this feature, a lab test is performed on each
11.37 Organophosphate (OP) compounds are used as pesticides. However, it is important to study their effect on species that are exposed to them. In the laboratory study Some Effects of Organophosphate Pesticides on Wildlife Species, by the Department of Fisheries and Wildlife Conservation at
11.36 The dataset consists of variables relating to blood pressures of 15 Peruvians (n = 15) who have moved from rural, high-altitude areas to urban, loweraltitude areas. The variables in this dataset are: systolic blood pressure (Y ), weight (X1 ), height (X2 ), and pulse.Weight Height Pulse
11.35 The following data are a result of an investigation as to the effect of reaction temperature x on percent conversion of a chemical process y. (See Myers, Montgomery and Anderson-Cook, 2009.) Fit a simple linear regression, and use a lack-of-fit test to determine if the model is adequate.
11.34 Use an analysis-of-variance approach to test the hypothesis that β1 = 0 against the alternative hypothesisβ1 = 0 in Exercise 11.5 on page 418 at the 0.05 level of significance.
11.33 Suppose we have a linear equation through the origin (Exercise 11.28) μY |x = βx.(a) Estimate the regression line passing through the origin for the following data:x 0.5 1.5 3.2 4.2 5.1 6.5 y 1.3 3.4 6.7 8.0 10.0 13.2(b) Suppose it is not known whether the true regression should pass
11.32 Test for linearity of regression in Exercise 11.8 on page 419. Comment.
11.31 Test for linearity of regression in Exercise 11.3 on page 418. Use a 0.05 level of significance. Comment.
11.30 For the data in Exercise 11.29, find a 95% prediction interval at x = 25. Root MSE Dependent Moan 1.48794 R-Square 21.50000 Adj R-Sq 0.9509 0.9447 Parameter Estimates Parameter Standard Variable DF Estimate Intercept 1 44.78018 Error 1.92919 * Value Pr> It 23.21
11.28 There are important applications in which, due to known scientific constraints, the regression line must go through the origin (i.e., the intercept must be zero). In other words, the model should read Yi = β1xi + i, i= 1, 2, . . . , n, and only a simple parameter requires estimation. The
11.27 Consider the regression of mileage for certain automobiles, measured in miles per gallon (mpg) on their weight in pounds (wt). The data are from Consumer Reports (April 1997). Part of the SAS output from the procedure is shown in Figure 11.13.(a) Estimate the mileage for a vehicle weighing
11.26 With reference to Exercise 11.3 on page 418, use the value of s2 found in Exercise 11.19(a) to compute(a) a 99% confidence interval for the average amount of chemical that will dissolve in 100 grams of water at 50◦C;(b) a 99% prediction interval for the amount of chemical that will dissolve
11.25 Using the value of s2 found in Exercise 11.17(a), construct a 95% confidence interval for the amount of converted sugar corresponding to x = 1.6 in Exercise 11.5 on page 418.
11.24 Using the value of s2 found in Exercise 11.17(a), graph the regression line and the 95% confidence bands for the mean response μY |x for the data of Exercise 11.5 on page 418.
11.23 With reference to Exercise 11.6 on page 419, use the value of s2 found in Exercise 11.18(a) to compute(a) a 95% confidence interval for the mean shear resistance when x = 24.5;(b) a 95% prediction interval for a single predicted value of the shear resistance when x = 24.5.
11.22 Using the value of s2 found in Exercise 11.16(a), construct a 95% confidence interval for μY |85 in Exercise 11.2 on page 418.
11.21 Test the hypothesis that β1 = 6 in Exercise 11.9 on page 419 against the alternative that β1 < 6.Use a 0.025 level of significance.
11.20 Test the hypothesis that β0 = 10 in Exercise 11.8 on page 419 against the alternative that β0 < 10.Use a 0.05 level of significance.
11.19 With reference to Exercise 11.3 on page 418,(a) evaluate s2 ;(b) construct a 99% confidence interval for β0 ;(c) construct a 99% confidence interval for β1 .
11.18 With reference to Exercise 11.6 on page 419,(a) evaluate s2 ;(b) construct a 99% confidence interval for β0 ;(c) construct a 99% confidence interval for β1 .
11.17 With reference to Exercise 11.5 on page 418,(a) evaluate s2 ;(b) construct a 95% confidence interval for β0 ;(c) construct a 95% confidence interval for β1 .
11.16 With reference to Exercise 11.2 on page 418,(a) evaluate s2 ;(b) construct a 95% confidence interval for β0 ;(c) construct a 95% confidence interval for β1 . Chemical Oxygen Demand Reduction 60 50- 50 40- 30- 20- 10- 10 20 30 40 Solids Reduction g- 50
11.15 With reference to Exercise 11.1 on page 418,(a) evaluate s2 ;(b) test the hypothesis that β1 = 0 against the alternative that β1 = 0 at the 0.05 level of significance and interpret the resulting decision.
11.14 A professor in the school of mathematics in a university polled a dozen colleagues about the number of academic workshops they attended in the previous five years (x) and the number of papers they submitted to refereed journals (y) during the same period. The summary of the data are given as
11.13 A study of the amount of rainfall and the quantity of air pollution removed produced the following data:Daily Rainfall, Particulate Removed, x (0.01 cm) y (μg/m3 )4.3 126 4.5 121 5.9 116 5.6 118 6.1 114 5.2 118 3.8 132 2.1 141 7.5 108 (a) Find the equation of the regression line to predict
11.12 A study was done to study the effect of ambient temperature x on the electric power consumed by a chemical plant y. Other factors were held constant, and the data were collected from an experimental pilot plant.y (BTU) x (◦F) y (BTU) x (◦F)250 27 265 31 285 45 298 60 320 72 267 34 295 58
11.11 The thrust of an engine (y) is a function of exhaust temperature (x) in ◦F when other important variables are held constant. Consider the following data.y x y x 4300 1760 4010 1665 4650 1652 3810 1550 3200 1485 4500 1700 3150 1390 3008 1270 4950 1820(a) Plot the data.(b) Fit a simple linear
11.10 The following data are the selling prices z of a certain make and model of used motorcycle w years old. Fit a curve of the form μz |w = γδw by means of the nonlinear sample regression equation ˆz = cdw .[Hint: Write lnˆz = lnc + (lnd)w = b0 + b1w.]w (years) z (dollars) w (years) z
11.9 A study was made by a retail merchant to determine the relation between weekly advertising expenditures and sales.Advertising Costs ($) Sales ($)40 385 20 400 25 395 20 365 30 475 50 440 40 490 20 420 50 560 40 525 25 480 50 510(a) Plot a scatter diagram.(b) Find the equation of the regression
11.8 A mathematics placement test is given to all entering freshmen at a small college. A student who receives a grade below 35 is denied admission to the regular mathematics course and placed in a remedial class.The placement test scores and the final grades for 20 students who took the regular
11.7 The following is a portion of a classic data set called the “pilot plot data” in Fitting Equations to Data by Daniel and Wood, published in 1971. The response y is the acid content of material produced by titration, whereas the regressor x is the organic acid content produced by extraction
11.6 In a certain type of metal test specimen, the normal stress on a specimen is known to be functionally related to the shear resistance. The following is a set of coded experimental data on the two variables:Normal Stress, x Shear Resistance, y 26.8 26.5 25.4 27.3 28.9 24.2 23.6 27.1 27.7 23.6
11.5 A study was made on the amount of converted sugar in a certain process at various temperatures. The data were coded and recorded as follows:Temperature, x Converted Sugar, y 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 8.1 7.8 8.5 9.8 9.5 8.9 8.6 10.2 9.3 9.2 10.5(a) Estimate the linear
11.4 The following data were collected from 10 individuals to determine the relationship between age and systolic blood pressure for the purpose of a medical study.Systolic blood Age (x) pressure (in mmHg)38 140 46 200 25 122 18 120 32 138 36 130 52 142 44 145 28 128 20 114(a) Find the equation of
11.3 The amounts of a chemical compound y that dissolved in 100 grams of water at various temperatures x were recorded as follows:x (◦C) y (grams)0 15 30 45 60 75 812 25 31 44 48 610 21 33 39 51 814 24 28 42 44(a) Find the equation of the regression line.(b) Graph the line on a scatter
11.2 The grades of a class of 9 students on a midterm report (x) and on the final examination (y) are as follows:x 77 50 71 72 81 94 96 99 67 y 82 66 78 34 47 85 99 99 68(a) Estimate the linear regression line.(b) Estimate the final examination grade of a student who received a grade of 85 on the
11.1 A study was conducted at Virginia Tech to determine if certain static arm-strength measures have an influence on the “dynamic lift” characteristics of an individual. Twenty-five individuals were subjected to strength tests and then were asked to perform a weightlifting test in which weight
10.111 z-Value for Testing p1−p2 = d0: To test the null hypothesis H0 that p1 −p2 = d0, where d0 = 0, we base our decision on z =ˆp1 − ˆp2 − d0ˆp1 ˆq1/n1 + ˆp2 ˆq2/n2, which is a value of a random variable whose distribution approximates the standard normal distribution as long as
10.110 Studies show that the concentration of PCBs is much higher in malignant breast tissue than in normal breast tissue. If a study of 50 women with breast cancer reveals an average PCB concentration of 22.8 × 10−4 gram, with a standard deviation of 4.8 × 10−4 gram, is the mean
10.109 The following data show the numbers of defects in 100,000 lines of code in a particular type of software program developed in the United States and Japan. Is there enough evidence to claim that there is a significant difference between the programs developed in the two countries? Test on
10.108 In a study conducted by the Water Resources Research Center and analyzed by the Laboratory for Interdisciplinary Statistical Consulting at Virginia Tech, two different wastewater treatment plants are compared. Plant A is located where the median household income is below $22,000 a year, and
10.107 In a study conducted by the Department of Mechanical Engineering and analyzed by the Laboratory for Interdisciplinary Statistical Consulting at Virginia Tech, steel rods supplied by two different companies were compared. Ten sample springs were made out of the steel rods supplied by each
10.106 A study was conducted at the Department of Human Nutrition, Foods and Exercise at Virginia Tech to determine if 8 weeks of training truly reduces the cholesterol levels of the participants. A treatment group consisting of 15 people was given lectures twice a week on how to reduce cholesterol
10.105 A study was conducted at the Virginia-Maryland Regional College of Veterinary Medicine Equine Center to determine if the performance of a certain type of surgery on young horses had any effect on certain kinds of blood cell types in the animal. Fluid samples were taken from each of six foals
10.104 A study was made to determine whether there is a difference between the proportions of parents in the states of Maryland (MD), Virginia (VA), Georgia(GA), and Alabama (AL) who favor placing Bibles in the elementary schools. The responses of 100 parents selected at random in each of these
10.103 In a random sample of 90 students recruited from a college, 36, 42, and 39 students were from departments A, B, and C, respectively. Can we conclude at a 0.05 level of confidence that the placement of students from all three departments was in equal proportion?
10.102 State the null and alternative hypotheses to be used in testing the following claims, and determine generally where the critical region is located:(a) At most, 20% of next year’s wheat crop will be exported to the Soviet Union.(b) On the average, American homemakers drink 3 cups of coffee
10.101 In a study analyzed by the Laboratory for Interdisciplinary Statistical Analysis at Virginia Tech, a group of subjects was asked to complete a certain task on the computer. The response measured was the time to completion. The purpose of the experiment was to test a set of facilitation tools
10.100 Consider the situation of Exercise 10.54 on page 380. Oxygen consumption in mL/kg/min, was also measured.Subject With CO Without CO 1 26.46 25.41 2 17.46 22.53 3 16.32 16.32 4 20.19 27.48 5 19.84 24.97 6 20.65 21.77 7 28.21 28.17 8 33.94 32.02 9 29.32 28.96 It is conjectured that oxygen
10.99 A study was conducted to determine whether the proportion of female professionals in the Indian software industry from rural areas was more than that of their male counterparts. Of the 420 female professionals selected at random, 174 were from the rural areas, whereas, out of the 380 male
10.98 In a socio-economic survey conducted among a random sample of 1200 villagers from a coastal village A, 744 earn their living by fishing. In a nearby village B, 672 out of 1200 earn living by fishing. Can we conclude that at 0.05 level of significance, the proportion of the villagers in
10.97 State the null and alternative hypotheses to be used in testing the following claims and determine generally where the critical region is located:(a) The mean snowfall at Lake George during the month of February is 21.8 centimeters.(b) No more than 20% of the faculty at the local university
10.96 In a study to estimate the proportion of wives who regularly watch soap operas, it is found that 52 of 200 wives in Denver, 31 of 150 wives in Phoenix, and 37 of 150 wives in Rochester watch at least one soap opera. Use a 0.05 level of significance to test the hypothesis that there is no
10.95 A survey was conducted in two Virginia cities to determine voter sentiment about two gubernatorial candidates in an upcoming election. Five hundred voters were randomly selected from each city and the following data were recorded:City Voter Sentiment Richmond Norfolk Favor A Favor B Undecided
10.94 A survey was conducted in Indiana, Kentucky, and Ohio to determine the attitude of voters concerning school busing. A poll of 200 voters from each of these states yielded the following results:Voter Attitude Do Not State Support Support Undecided Indiana 82 97 21 Kentucky 107 66 27 Ohio 93 74
10.93 To determine current attitudes about prayer in public schools, a survey was conducted in four Virginia counties. The following table gives the attitudes of 200 parents from Craig County, 150 parents from Giles County, 100 parents from Franklin County, and 100 parents from Montgomery
10.92 A college infirmary conducted an experiment to determine the degree of relief provided by three cough remedies. Each cough remedy was tried on 50 students and the following data recorded:Cough Remedy NyQuil Robitussin Triaminic No relief 11 13 9 Some relief 32 28 27 Total relief 7 9 14 Test
10.91 The following responses concerning the standard of living at the time of an independent opinion poll of 1000 households versus one year earlier seem to be in agreement with the results of a study published in Across the Board (June 1981):Standard of Living Somewhat Not as Period Better Same
10.90 According to a Johns Hopkins University study published in the American Journal of Public Health, widows live longer than widowers. Consider the following survival data collected on 100 widows and 100 widowers following the death of a spouse:Years Lived Widow Widower Less than 5 25 39 5 to 10
10.89 A criminologist conducted a survey to determine whether the incidence of certain types of crime varied from one part of a large city to another. The particular crimes of interest were assault, burglary, larceny, and homicide. The following table shows the numbers of crimes committed in four
10.88 A random sample of 200 married men, all retired, was classified according to education and number of children:Number of Children Education 0–1 2–3 Over 3 Elementary 14 37 32 Secondary 19 42 17 College 12 17 10 Test the hypothesis, at the 0.05 level of significance, that the size of a
10.87 A study selects 60 boys and 40 girls from a school and classifies them according to their ability in mathematics.Gender Classification Boys Girls Below average 18 10 Average 32 18 Above average 10 12 Use a 0.05 level of significance and test the hypothesis that gender and ability in
10.86 In an experiment to study the dependence of hypertension on smoking habits, the following data were taken on 180 individuals:Non- Moderate Heavy smokers Smokers Smokers Hypertension 21 36 30 No hypertension 48 26 19 Test the hypothesis that the presence or absence of hypertension is
10.85 For Exercise 1.19 on page 51, test the goodness of fit between the observed class frequencies and the corresponding expected frequencies of a normal distribution with μ = 1.8 and σ = 0.4, using a 0.01 level of significance.
10.84 For Exercise 1.18 on page 51, test the goodness of fit between the observed class frequencies and the corresponding expected frequencies of a normal distribution with μ = 65 and σ = 21, using a 0.05 level of significance.
10.83 A coin is thrown until a head occurs and the number X of tosses recorded. After repeating the experiment 256 times, we obtained the following results:x 1 2 3 4 5 6 7 8 f 136 60 34 12 9 1 3 1 Test the hypothesis, at the 0.05 level of significance, that the observed distribution of X may be
10.82 Three marbles are selected from an urn containing 5 red marbles and 3 green marbles. After the number X of red marbles is recorded, the marbles are replaced in the urn and the experiment repeated 112 times. The results obtained are as follows:x 0 1 2 3 f 1 31 55 25 Test the hypothesis, at the
10.81 A die is tossed 240 times and shows the following results:x 1 2 3 4 5 6 f 32 38 46 44 34 46 Verify whether the die is unbiased. Use a 0.05 level of significance.
10.80 The grades in a statistics course for a particular semester were as follows:Grade A B C D F f 14 18 32 20 16 Test the hypothesis, at the 0.05 level of significance, that the distribution of grades is uniform.
10.79 The ratio of the students attending a mathematics exhibition from high school, higher secondary classes, undergraduate classes, and graduate classes is 3:5:6:2. Out of the 512 mathematics students who attended the exhibition, 98 were from high school, 162 were from the higher secondary
10.78 Hydrocarbon emissions from cars are known to have decreased dramatically during the 1980s. A study was conducted to compare the hydrocarbon emissions at idling speed, in parts per million (ppm), for automobiles from 1980 and 1990. Twenty cars of each model year were randomly selected, and
10.77 An experiment was conducted to compare the alcohol content of soy sauce on two different production lines. Production was monitored eight times a day.The data are shown here.Production line 1:0.48 0.39 0.42 0.52 0.40 0.48 0.52 0.52 Production line 2:0.38 0.37 0.39 0.41 0.38 0.39 0.40 0.39
10.76 Two types of instruments for measuring the amount of sulfur monoxide in the atmosphere are being compared in an air-pollution experiment. Researchers wish to determine whether the two types of instruments yield measurements having the same variability. The readings in the following table were
10.73 A study is conducted to compare the lengths of time required by men and women to assemble a certain product. Past experience indicates that the distribution of times for both men and women is approximately normal but the variance of the times for women is less than that for men. A random
10.72 Large-Sample Test of σ2 = σ2 0: When n ≥30, we can test the null hypothesis that σ2 = σ2 0, orσ = σ0 , by computing z = s − σ0σ0 /√2n which is a value of a random variable whose sampling distribution is approximately the standard normal distribution.(a) With reference to Example
10.71 A soft-drink dispensing machine is said to be out of control if the variance of the contents exceeds 1.15 deciliters. If a random sample of 25 drinks from this machine has a variance of 2.03 deciliters, does this indicate at the 0.05 level of significance that the machine is out of control?
10.70 Past data indicate that the amount of money contributed by the working residents of a large city to a volunteer rescue squad is a normal random variable with a standard deviation of $1.40. It has been suggested that the contributions to the rescue squad from just the employees of the
10.69 Aflotoxins produced by mold on peanut crops in Virginia must be monitored. A sample of 64 batches of peanuts reveals levels of 24.17 ppm, on average, with a variance of 4.25 ppm. Test the hypothesis thatσ2 = 4.2 ppm against the alternative that σ2 = 4.2 ppm. Use a P-value in your
10.68 Past experience indicates that the time required for high school seniors to complete a standardized test is a normal random variable with a standard deviation of 6 minutes. Test the hypothesis that σ = 6 against the alternative thatσ < 6 if a random sample of the test times of 20 high
10.67 A study conducted by an agriculture department reveals that the water content in tender coconuts is normally distributed, with a variance of 0.075 liter.Test the hypothesis that σ2 = 0.075 against the alternative that σ2 = 0.075 using the water content found in a random sample of 10 tender
10.66 Group Project: The class should be divided into pairs of students for this project. Suppose it is conjectured that at least 25% of students at your university exercise for more than two hours a week. Collect data from a random sample of 50 students. Ask each student if he or she works out for
10.65 It is believed that the proportion of boys getting placed through campus recruitment is higher than that of girls. Among the 365 boys and 324 girls in the final semester of an engineering college, 180 and 142 got placed in various companies through the campus recruitment drive, respectively.
10.64 In a study on the fertility of married women conducted by Martin O’Connell and Carolyn C. Rogers for the Census Bureau in 1979, two groups of childless wives aged 25 to 29 were selected at random, and each was asked if she eventually planned to have a child.One group was selected from among
10.63 In a study to estimate the proportion of smokers among the residents in a certain city and its suburbs, it is found that 24 of 104 urban residents are smokers, while 34 of 108 suburban residents are smokers.Is there a significant difference between the proportions of smokers among urban and
10.62 In a controlled laboratory experiment, scientists at the University of Minnesota discovered that 25% of a certain strain of rats subjected to a 20% coffee bean diet and then force-fed a powerful cancer-causing chemical later developed cancerous tumors. Would we have reason to believe that the
10.61 In a winter of an epidemic flu, the parents of 2000 babies were surveyed by researchers at a wellknown pharmaceutical company to determine if the company’s new medicine was effective after two days.Among 120 babies who had the flu and were given the medicine, 29 were cured within two days.
10.60 On an enquiry related to the frequent reporting of mouth cancer cases, it is estimated that more than 60% of the patients chewed tobacco. Does this seem to be a valid estimate if, in a random sample of 120 people, 84 were found to be tobacco users? Use a 0.05 level of significance in your
10.59 Two-thirds of the employees of a town prefer using the public transport system for their travel. Do we have reason to believe this claim if, in a random sample of 900 employees, 582 use public transport? Use a P-value in your conclusion.
10.58 It is believed that at least 90% of the residents of a certain city and its suburbs did not support the idea of constructing a nuclear power plant near the city.What conclusion would you draw if only 9 in a sample 180 residents favored the idea? Use a 0.05 level of significance.
10.57 A new radar device is being considered for a certain missile defense system. The system is checked by experimenting with aircraft in which a kill or a no kill is simulated. If, in 300 trials, 250 kills occur, accept or reject, at the 0.04 level of significance, the claim that the probability
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