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Statistics The Exploration And Analysis Of Data 6th Edition John M Scheb, Jay Devore, Roxy Peck - Solutions
=+a. One observation was (25, 70). What is the corresponding residual?
=+5.49 The paper “Effects of Canine Parvovirus (CPV) on Gray Wolves in Minnesota” (Journal of Wildlife Management [1995]: 565–570) summarized a regression of y 5 percentage of pups in a capture on x 5 percentage of CPV prevalence among adults and pups. The equation of the least-squares line,
=+5.48 A study was carried out to investigate the relationship between the hardness of molded plastic (y, in Brinell units) and the amount of time elapsed since termination of the molding process (x, in hours). Summary quantities include n 5 15, SSResid 5 1235.470, and SSTo 5 25,321.368.
=+c. What effect does the deletion in Part (b) have on the value of r 2? Can you explain why this is so?
=+b. Delete the observation with the largest x value from the sample and recalculate the equation of the least-squares line. Does this observation greatly affect the equation of the line?
=+5.47 ● The paper “Crop Improvement for Tropical and Subtropical Australia: Designing Plants for Difficult Climates” (Field Crops Research [1991]: 113–139) gave the following data on x 5 crop duration (in days) for soybeans and y 5 crop yield (in tons per hectare):x 92 92 96 100 102 y 1.7
=+b. What proportion of observed variation in y can be attributed to the approximate linear relationship between the two variables?
=+a. What maximum salmonid size would you predict for a squawfish whose length is 375 mm, and what is the residual corresponding to the observation (375, 165)?
=+Use the accompanying output from MINITAB to answer the following questions.The regression equation is size 89.1 + 0.729 length Predictor Coef Stdev t ratio p Constant 89.09 16.83 5.29 0.000 length 0.72907 0.04778 15.26 0.000 s = 12.56 R-sq = 96.3% R-sq(adj) = 95.9%Analysis of Variance Source DF
=+5.46 ● The paper “Feeding of Predaceous Fishes on Out-Migrating Juvenile Salmonids in John Day Reservoir, Columbia River” (Transactions of the American Fisheries Society [1991]: 405–420) gave the following data on y 5 maximum size of salmonids consumed by a northern squawfish (the most
=+c. How effectively do you think the least-squares line summarizes the relationship between x and y? Explain your reasoning.
=+b. Interpret the value of se.
=+a. One observation in the sample was (9900, 893). What average SAT score would you predict for this district, and what is the corresponding residual?
=+226 C hapt e r 5 ■ Summarizing Bivariate Data proportion of the variability in telomere length? Justify your answer.5.45 The article “Cost-Effectiveness in Public Education”(Chance [1995]: 38–41) reported that for a regression of y 5 average SAT score on x 5 expenditure per pupil, based
=+c. Does the linear relationship between perceived stress and telomere length account for a large or small 6Bold exercises answered in back ● Data set available online but not required ▼ Video solution available
=+b. What is the value of r 2?
=+a. Compute the equation of the least-squares line.
=+5.44 ● The paper “Accelerated Telomere Shortening in Response to Life Stress” (Proceedings of the National Academy of Sciences [2004]: 17312–17315) described a study that examined whether stress accelerates aging at a cellular level. The accompanying data on a measure of perceived stress
=+d. Compute the residuals, and construct a residual plot.Are there any features of the plot that indicate that a line is not an appropriate description of the relationship between x and y? Explain.
=+c. Compute an estimate of the average runoff volume when rainfall volume is 80.
=+b. Calculate the slope and intercept of the least-squares line.
=+5.43 ● ▼ The article “Characterization of Highway Runoff in Austin, Texas, Area” (Journal of Environmental Engineering [1998]: 131–137) gave a scatterplot, along with the least-squares line for x 5 rainfall volume (in cubic meters) and y 5 runoff volume (in cubic meters), for a
=+a. Interpret the value of r 2.b. Find and interpret the value of se(the sample size was n 5 14).
=+5.42 Exercise 5.22 gave the least-squares regression line for predicting y 5 clutch size from x 5 snout-vent length(“Reproductive Biology of the Aquatic Salamander Amphiuma tridactylum in Louisiana,” Journal of Herpetology[1999]: 100–105). The paper also reported r 2 5 .7664 and SSTo 5
=+b. The article stated that SAT II was the best predictor of first-year college grades. Do you think that predictions based on a least-squares line with y 5 first-year college GPA and x 5 SAT II score would have been very accurate? Explain why or why not.
=+15.4 percent, and SAT I was last at 13.3 percent.”a. If the data from this study were used to fit a leastsquares line with y 5 first-year college GPA and x 5 high school GPA, what would the value of r 2have been?
=+5.41 The article “Examined Life: What Stanley H. Kaplan Taught Us About the SAT” (The New Yorker [December 17, 2001]: 86–92) included a summary of findings regarding the use of SAT I scores, SAT II scores, and high school grade point average (GPA) to predict first-year college GPA. The
=+a y a x a xy 5 1114.5 2 5 57,939 2 5 62.600235 n 5 12 a x 5 22.027 a y 5 793a. Is the observation for Hospital 11 an influential observation? Justify your answer.b. Is the observation for Hospital 11 an outlier? Explain.c. Is the observation for Hospital 5 an influential observation? Justify
=+The least-squares regression line with y 5 inpatient costto-charge ratio and x 5 outpatient cost-to-charge ratio is yˆ 5 21.1 1 1.29x.45 55 65 75 Inpatient 100 90 80 70 60 50 Outpatient
=+5.40 ● Cost-to-charge ratio (the percentage of the amount billed that represents the actual cost) for inpatient and outpatient services at 11 Oregon hospitals is shown in the following table (Oregon Department of Health Services, 2002). A scatterplot of the data is also shown.Cost-to-Charge
=+Summary quantities area. Obtain the equation of the least-squares line.b. Construct a residual plot, and comment on any interesting features.
=+5.39 ● The following data on degree of exposure to 242Cm alpha particles (x) and the percentage of exposed cells without aberrations (y) appeared in the paper “Chromosome Aberrations Induced in Human Lymphocytes by D-T Neutrons” (Radiation Research [1984]: 561–573):x 0.106 0.193 0.511
=+d. The observation for the West, (40.0, 899), has an x value that is far removed from the other x values in the sample. Is this observation influential in determining the values of the slope and/or intercept of the least-squares line? Justify your answer.
=+c. Construct a residual plot. Are there any unusual features of the plot?
=+Does the value of r indicate that the linear relationship between pollution and medical cost is strong, moderate, or weak? Explain.
=+The equation of the least-squares regression line for this data set is , where y 5 medical cost and x 5 pollution.
=+5.38 ● Data on pollution and cost of medical care for elderly people were given in Exercise 5.17 and are also shown here. The following data give a measure of pollution (micrograms of particulate matter per cubic meter of air) and the cost of medical care per person over age 65 for six
=+c. Compute the residuals, and construct a residual plot.Are there any unusual features in the plot?yˆ
=+5.37 ● The following data on x 5 soil depth (in centimeters) and y 5 percentage of montmorillonite in the soil were taken from a scatterplot in the paper “Ancient Maya Drained Field Agriculture: Its Possible Application Today in the New River Floodplain, Belize, C.A.” (Agricultural
=+b. Compute the 10 residuals, and construct a residual plot.Are there any features of the residual plot that indicate that the relationship between year and number of transplants performed would be better described by a curve rather than a line? Explain.
=+a. Construct a scatterplot of these data, and then find the equation of the least-squares regression line that describes the relationship between y 5 number of transplants performed and x 5 year. Describe how the number of transplants performed has changed over time from 1990 to 1999.
=+5.36 ● The following table gives the number of organ transplants performed in the United States each year from 1990 to 1999 (The Organ Procurement and Transplantation Network, 2003):Number of Transplants Year (in thousands)1 (1990) 15.0 2 15.7 3 16.1 4 17.6 5 18.3 6 19.4 7 20.0 8 20.3 9 21.4
=+ Verify your conjectures by using the given formulas for b anda. (Hint: Replace y with cy, and see what happens—and remember, this conversion will affect .)
=+b. More generally, suppose that each y value in a data set consisting of n (x, y) pairs is multiplied by a conversion factor c (which changes the units of measurement for y).What effect does this have on the slope b (i.e., how does the new value of b compare to the value before conversion), on
=+a. Because 1 lb 5 0.4536 kg, strength observations can be re-expressed in kilograms through multiplication by this conversion factor: new y 5 0.4536(old y). What is the equation of the least-squares line when y is expressed in kilograms?
=+5.35 ● The accompanying data resulted from an experiment in which weld diameter x and shear strength y (in pounds) were determined for five different spot welds on steel. A scatterplot shows a pronounced linear pattern.With and , the least-squares line is .x 200.1 210.1 220.1 230.1 240.0 y
=+5.34 Explain why the slope b of the least-squares line always has the same sign (positive or negative) as does the sample correlation coefficient r.
=+b. If a particular person whose sales experience is 1.5 standard deviations below the average experience is predicted to have an annual sales value that is 1 standard deviation below the average annual sales, what is the value of r?
=+a. Suppose that the sample correlation coefficient is r 5.75 and that the average annual sales is . If a y 5 100 24.54 1 0.1123x yˆ 5 a y a x 2 5 412.81 a xy 5 6933.48 2 5 117,123.85 a x 5 1368.1 a y 5 80.9 particular salesperson is 2 standard deviations above the mean in terms of experience,
=+5.33 The sales manager of a large company selected a random sample of n 5 10 salespeople and determined for each one the values of x 5 years of sales experience and y 5 annual sales (in thousands of dollars). A scatterplot of the resulting (x, y) pairs showed a marked linear pattern.
=+5.32 Explain why it can be dangerous to use the leastsquares line to obtain predictions for x values that are substantially larger or smaller than those contained in the sample.
=+e. Is vital capacity completely determined by chest circumference? Explain.
=+d. What vital capacity would you predict for a Tibetan native whose chest circumference is 85 cm?
=+c. On average, roughly what change in vital capacity is associated with a 1-cm increase in chest circumference?with a 10-cm increase?
=+b. The summary quantities are Verify that the equation of the least-squares line is, and draw this line on your scatterplot.
=+a. Construct a scatterplot. What does it suggest about the nature of the relationship between x and y?
=+5.31 ● The paper “Increased Vital and Total Lung Capacities in Tibetan Compared to Han Residents of Lhasa”(American Journal of Physical Anthropology [1991]:341–351) included a scatterplot of vital capacity (y)versus chest circumference (x) for a sample of 16 Tibetan natives, from which
=+d. Does this approximate linear relationship appear to hold for shell heights as small as 1 cm? Explain.
=+c. What breaking strength would you predict when shell height is 2 cm?
=+b. When shell height increases by 1 cm, by how much does breaking strength tend to change?
=+a. What are the slope and the intercept of this line?
=+5.30 The paper “Postmortem Changes in Strength of Gastropod Shells” (Paleobiology [1992]: 367–377) included scatterplots of data on x 5 shell height (in centimeters)and y 5 breaking strength (in newtons) for a sample of n 5 38 hermit crab shells. The least-squares line was
=+c. Would you recommend using the least-squares equation from Part (a) to predict runoff sediment concentration for gradually sloped plots? If so, explain why it would be appropriate to do so. If not, provide an alternative way to make such predictions.
=+b. What would you predict runoff sediment concentration to be for a steeply sloped plot with 18% bare ground?
=+Concentration 100 250 300 600 Bare ground (%) 20 25 20 30 Concentration 500 500 900 800 Bare ground (%) 35 40 35 Concentration 1100 1200 1000a. Using the data for steeply sloped plots, find the equation of the least-squares line for predicting y runoff sediment concentration using x 5
=+plots with varying amounts of grazing damage, measured by the percentage of bare ground in the plot, are given for gradually sloped plots and for steeply sloped plots.Gradually Sloped Plots Bare ground (%) 5 10 15 25 Concentration 50 200 250 500 Bare ground (%) 30 40 Concentration 600 500 Steeply
=+5.29 ● Representative data read from a plot that appeared in the paper “Effect of Cattle Treading on Erosion from Sale Price Size Land-to-(millions (thousands Building Property of dollars) of sq. ft.) Ratio 1 10.6 2166 2.0 2 2.6 751 3.5 3 30.5 2422 3.6 4 1.8 224 4.7 5 20.0 3917 1.7 6 8.0 2866
=+d. Based on your choice in Part (c), find the equation of the least-squares regression line you would use for predicting y 5 sale price.
=+c. If you wanted to predict sale price and you could use either size or land-to-building ratio as the basis for making predictions, which would you use? Explain.
=+b. Calculate and interpret the value of the correlation coefficient between sale price and land-to-building ratio.
=+5.28 ● The following data on sale price, size, and land-tobuilding ratio for 10 large industrial properties appeared in the paper “Using Multiple Regression Analysis in Real Estate Appraisal” (Appraisal Journal [2002]: 424–430):a. Calculate and interpret the value of the correlation
=+The newspaper article “FDA OKs Use of Home Defibrillators” (San Luis Obispo Tribune, November 13, 2002)reported that “every minute spent waiting for paramedics to arrive with a defibrillator lowers the chance of survival by 10 percent.” Is this statement consistent with the given
=+5.26 The data given in Example 5.5 on x 5 call-to-shock time (in minutes) and y 5 survival rate (percent) were used to compute the equation of the least-squares line, which was
=+d. Explain why it would not be reasonable to use the least-squares line to predict strength when carbonation depth is 100 mm.
=+c. What would you predict for strength when carbonation depth is 25 mm?
=+a. Construct a scatterplot. Does the relationship between carbonation depth and strength appear to be linear?
=+5.25 ● Representative data on x 5 carbonation depth (in millimeters) and y 5 strength (in megapascals) for a sample of concrete core specimens taken from a particular building were read from a plot in the article “The Carbonation of Concrete Structures in the Tropical Environment of
=+c. What first-year GPA would you predict for a student with a 4.0 high school GPA?
=+b. Interpret the value ofb, the slope of the least-squares line, in the context of this problem.
=+a. Find the equation of the least-squares regression line.
=+5.24 Data on high school GPA (x) and first-year college GPA (y) collected from a southeastern public research university can be summarized as follows (“First-Year Academic Success: A Prediction Combining Cognitive and Psychosocial Variables for Caucasian and African American Students,”
=+c. Nevada, a western state not included in the data set, had a 1996 fourth-grade math proficiency of 14%. What yˆ 5 2147 1 6.175x would you predict for Nevada’s 2000 eighth-grade math proficiency percentage? How does your prediction compare to the actual eighth-grade value of 20 for Nevada?
=+b. Find the equation of the least-squares line that summarizes the relationship between x 5 1996 fourth-grade math proficiency percentage and y 5 2000 eighth-grade math proficiency percentage.
=+a. Construct a scatterplot, and comment on any interesting features.
=+5.23 ● Percentages of public school students in fourth grade in 1996 and in eighth grade in 2000 who were at or above the proficient level in mathematics were given in the article “Mixed Progress in Math” (USA Today, August 3, 2001) for eight western states:4th grade 8th grade State (1996)
=+b. Would you be reluctant to predict the clutch size whensnout-vent length is 22 cm? Explain.
=+Interpret the slope in the context of this problem.
=+What is the value of the slope of the least-squares line?
=+a. What is the value of the y intercept of the least-squares line?
=+5.22 In the article “Reproductive Biology of the Aquatic Salamander Amphiuma tridactylum in Louisiana” (Journal of Herpetology [1999]: 100–105), 14 female salamanders were studied. Using regression, the researchers predicted y 5 clutch size (number of salamander eggs) from x 5 snout-vent
=+e. Explain why it would not be a good idea to use the least-squares line to predict the volume of grey matter for a child whose head circumference z score was 3.0.
=+d. Predict the volume of cerebral grey matter for a child whose head circumference z score at age 12 months was 1.8.
=+c. Find the equation of the least-squares line.
=+b. What is the value of the correlation coefficient?
=+a. Construct a scatterplot for these data.
=+5.21 ● ▼ The accompanying data on x 5 head circumference z score (a comparison score with peers of the same age—a positive score suggests a larger size than for peers)at age 6 to 14 months and y 5 volume of cerebral grey matter (in ml) at age 2 to 5 years were read from a graph in the
=+b. y 5 a measure of patient satisfaction with hospital care(higher values indicate higher satisfaction)c. y 5 a measure of patient quality of care.
=+a. y 5 a measure of nurse’s job satisfaction (higher values indicate higher satisfaction)
=+5.20 The relationship between hospital patient-to-nurse ratio and various characteristics of job satisfaction and patient care has been the focus of a number of research studies. Suppose x 5 patient-to-nurse ratio is the predictor variable. For each of the following potential dependent variables,
=+b. Would the least-squares line for predicting number of servings of fruits and vegetables using number of hours spent watching TV as a predictor have a positive or negative slope? Explain.
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