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statistics principles and methods

Statistics The Exploration And Analysis Of Data 6th Edition John M Scheb, Jay Devore, Roxy Peck - Solutions

=+a. Suppose you knew a person with the following characteristics: a 25-year-old, white female with a college degree(16 years of education), who has a $32,000-per-year job, is from the upper middle class and considers herself left of center, but who is neither a materialist nor a postmaterialist.
=+x6 5 ideology (4 5 conservative, 3 5 right of center, 2 5 middle of the road, 1 5 left of center, and 0 5 liberal)x7 5 social class (4 5 upper, 3 5 upper middle, 2 5 middle, 1 5 lower middle, 0 5 lower)x8 5 postmaterialist (1 if postmaterialist, 0 otherwise)x9 5 materialist (1 if materialist, 0
=+model proposed by the authors was where the variables are defined as follows y 5 ecology score (higher values indicate a greater concern for ecology)x1 5 age times 10 x2 5 income (in thousands of dollars)x3 5 gender (1 5 male, 0 5 female)x4 5 race (1 5 white, 0 5 nonwhite)x5 5 education (in
=+14.4 According to “Assessing the Validity of the PostMaterialism Index” (American Political Science Review[1999]: 649–664), one may be able to predict an individual’s level of support for ecology based on demographic and ideological characteristics. The multiple regression
=+What multiple regression model is suggested by the statement? Did you include an interaction term in the model?Why or why not?
=+Regression analyses indicated that academic adjustment and race made independent contributions to academic achievement, as measured by current GPA.Suppose
=+f. If repeated observations on rating are made on different individuals, all of whom have the values of x1, x2, and x3 specified in Part (e), in the long run approximately e between 13.5 kg and peared in the article “Di-College Women” (Journal of College Student Development [1998]: 364):
=+e. What is the mean value of rating of acceptable load when extent of left lateral bending is 25 cm, dynamic hand grip endurance is 200 sec, and trunk extension ratio is 10 N/kg?
=+b. What are the values of the population regression coefficients?c. Interpret the value of b1.d. Interpret the value of b3.
=+a. What is the population regression function?
=+14.2 A proble in a sa Streng in the gonom model to relate the dependent variable y 5 individual’s rating of acceptable load (kg)to k 5 3 independent (predictor) variables:x1 5 extent of left lateral bending (cm)x2 5 dynamic hand grip endurance (sec)x3 5 trunk extension ratio (N/kg)Suppose that
=+b. For what values of r will se be much smaller than sy?
=+a. For what value of r is se as large as sy? What is the equation of the least-squares line in this case?se < 121 2 r 22sy
=+13.75 Some straightforward but slightly tedious algebra shows that from which it follows that se 5 B n 2 1 n 2 2 121 2 r 22sy Unless n is quite small, (n 2 1)/(n 2 2) ø 1, so
=+f. A research report included the following summary quantities obtained from a simple linear regression analysis:
=+e. A student reported that a data set consisting of n 5 6 observations yielded residuals 2, 0, 5, 3, 0, and 1 from the least-squares line.
=+d. SSResid is always positive.
=+c. Does it make sense to test hypotheses about b?
=+b. The simple linear regression model states that y 5 a 1bx.
=+a. What is the difference between e1, e2, . . . , en and the n residuals?
=+13.74 Give a brief answer, comment, or explanation for each of the following.
=+you think the simple linear regression model is appropriate here? Explain. What would you expect to see in a plot of the standardized residuals versus x?
=+13.73 The accompanying figure is from the article “Root and Shoot Competition Intensity Along a Soil Depth Gradient” (Ecology [1995]: 673–682). It shows the relationship between above-ground biomass and soil depth within the experimental plots. The relationship is described by the linear
=+e. Compute and interpret a 95% interval estimate for the true average iron concentration of core samples taken at 70 m.
=+d. Use the estimated regression equation to construct a 95% prediction interval for the iron concentration of a single core sample taken at a depth of 50 m.
=+c. Calculate the slope and intercept of the estimated regression line relating y 5 iron concentration and x 5 depth.
=+b. Using a .05 significance level, do the data strongly suggest a correlation between depth and iron concentration?
=+a. Using a .05 significance level, test appropriate hypotheses to determine whether a correlation exists between depth and zinc concentration.
=+13.72 ● The article “Statistical Comparison of Heavy Metal Concentrations in Various Louisiana Sediments”(Environmental Monitoring and Assessment [1984]: 163–170) gave the accompanying data on depth (m), zinc concentration (ppm), and iron concentration (%) for 17 core samples.Core Depth
=+resulting output gave the P-value 5 .0076 for the model utility test. Does the percentage raise appear to be linearly related to productivity? Explain.
=+a recent period might not have been based strictly on objective performance criteria. A sample of n 5 20 employees was selected, and the values of x, a quantitative measure of productivity, and y, the percentage salary increase, were determined for each one. A computer package was used to fit
=+13.71 The employee relations manager of a large company was concerned that raises given to employees during
=+d. Based on the plot in Part (c), do you think that a linear model is appropriate for describing the relationship between y and x? Explain.
=+c. Using the estimated regression line of Part (a), compute the residuals and construct a plot of the residuals versus x (that is, of the (x, residual) pairs).
=+b. Do the data indicate a linear relationship between y and x? Test using a .10 significance level.
=+a. Use the data to calculate the estimated regression line.
=+13.70 ● The article “Improving Fermentation Productivity with Reverse Osmosis” (Food Technology [1984]: 92–96)gave the following data (read from a scatterplot) on y 5 glucose concentration (g/L) and x 5 fermentation time(days) for a blend of malt liquor.x 1 2 3 4 5 6 7 8 y 74 54 52 51 52
=+happen to the slope if this observation is included in the computations?
=+13.69 The accompanying scatterplot, based on 34 sediment samples with x 5 sediment depth (cm) and y 5 oil and grease content (mg/kg), appeared in the article “Mined Land Reclamation Using Polluted Urban Navigable Waterway Sediments” (Journal of Environmental Quality[1984]: 415– 422).
=+, and so on. The summary quantities provide no way of distinguishing among the four data sets.Based on a scatterplot for each set, comment on the appropriateness or inappropriateness of fitting the simple linear regression model in each case.
=+identical, so all quantities computed from these will be identical for the four sets: the estimated regression line, SSResid, se, r 2
=+American Statistician [1973]: 17–21).Data Set 1–3 1 2 3 4 4 Variable x y y y x y 10.0 8.04 9.14 7.46 8.0 6.58 8.0 6.95 8.14 6.77 8.0 5.76 13.0 7.58 8.74 12.74 8.0 7.71 9.0 8.81 8.77 7.11 8.0 8.84 11.0 8.33 9.26 7.81 8.0 8.47 14.0 9.96 8.10 8.84 8.0 7.04 6.0 7.24 6.13 6.08 8.0 5.25 4.0 4.26
=+13.68 ● Consider the following four (x, y) data sets: the first three have the same x values, so these values are listed only once (from “Graphs in Statistical Analysis”
=+if the slopes of the true regression lines for the two different frog populations are equal. (Summary statistics are given in the table.)Leptodactylus ocellatus x 3.8 4.0 4.9 7.1 8.1 8.5 8.9 9.1 9.8 y 1.0 1.2 1.7 2.0 2.7 2.5 2.4 2.9 3.2 Bufa marinus x 3.8 4.3 6.2 6.3 7.8 8.5 9.0 10.0 y 1.6 1.7
=+The given data are a subset of the data in the article“Diet and Foraging Model of Bufa marinus and Leptodactylus ocellatus” (Journal of Herpetology [1984]: 138–146). The independent variable x is body length (cm) and the dependent variable y is mouth width (cm), with n 5 9 observations for
=+When H0 is true, this statistic has a t distribution based on(n 1 m 2 4) df.
=+where SSResid and SSResid9 are the residual sums of squares for the first and second samples, respectively.With Sxx and denoting the quantity for the first and second samples, respectively, the test statistic is
=+same for both populations. Then this common variance can be estimated by
=+second samples, respectively. The investigator may then wish to test the null hypothesis H0: b 2 5 0 (that is, b 5 ) against an appropriate alternative hypothesis. Suppose that s 2, the variance about the population line, is the
=+13.67 ● In some studies, an investigator has n (x, y) pairs sampled from one population and m (x, y) pairs from a second population. Let b and denote the slopes of the first and second population lines, respectively, and let b and b9 denote the estimated slopes calculated from the first and
=+c. Compute a 95% confidence interval fora. Does the result indicate that a 5 0 is plausible? Explain.
=+a test at level of significance .05 to see whether the y intercept of the true regression line differs from zero.
=+b. Compute the estimated standard deviation sa. Carry out
=+from a thermocouple placed on a traversing vehicle. Selected data are given (read from a scatterplot in the article).x 22 21 0 1 2 3 4 y 23.9 22.1 22.0 21.2 0.0 1.9 0.6 x 5 6 7 y 2.1 1.2 3.0 Estimate the true regression line.t 5 a 2 hypothesized value sa a 6 1t critical value 2sa sa1bx*
=+a. The article “Comparison of Winter-Nocturnal Geostationary Satellite Infrared-Surface Temperature with Shelter-Height Temperature in Florida” (Remote Sensing of the Environment [1983]: 313–327) used the simple linear regression model to relate surface temperature as measured by a
=+x 5 0, since a 1 b(0) 5a. This implies that sa the estimated standard deviation of the statistica, results from substituting x* 5 0 in the formula for . The desired confidence interval is then and a test statistic is
=+13.66 ● Occasionally an investigator may wish to compute a confidence interval fora, the y intercept of the true regression line, or test hypotheses abouta. The estimated y intercept is simply the height of the estimated line when
=+b. Estimate true average percentage area covered by pores for all 50-year-olds in the population in a way that conveys information about the precision of estimation.
=+a. Suppose that the researchers had believed a priori that the average decrease in percentage area associated with a 1-year age increase was .5%. Do the data contradict this prior belief? State and test the appropriate hypotheses using a .10 significance level.
=+article “Morphometry of Nerve Fiber Bundle Pores in the Optic Nerve Head of the Human” (Experimental Eye Research [1988]: 559–568) presented the accompanying data on x 5 age and y 5 percentage of the cribriform area of the lamina scleralis occupied by pores.x 22 25 27 39 42 43 44 46 46 y
=+13.65 ● Reduced visual performance with increasing age has been a much-studied phenomenon in recent years.This decline is due partly to changes in optical properties of the eye itself and partly to neural degeneration throughout the visual system. As one aspect of this problem, th
=+g. Give an estimate of true average peak photovoltage when percentage of light absorption is 20, and do so in a way that conveys information about precision.
=+f. Give an estimate of the average change in peak photovoltage associated with a 1% increase in light absorption.Your estimate should convey information about the precision of estimation.
=+e. The authors claimed that there is a useful linear relationship between the two variables. Do you agree? Carry out a formal test.
=+d. Predict peak photovoltage when percent absorption is 19.1, and compute the value of the corresponding residual.
=+c. How much of the observed variation in peak photovoltage can be explained by the model relationship?
=+b. Assuming that the simple linear regression model is appropriate, obtain the equation of the estimated regression line.
=+a. Construct a scatterplot of the data. What does it suggest?
=+13.64 ● The article “Photocharge Effects in Dye Sensitized Ag[Br,I] Emulsions at Millisecond Range Exposures” (Photographic Science and Engineering [1981]:138–144) gave the accompanying data on x 5 % light absorption and y 5 peak photovoltage.x 4.0 8.7 12.7 19.1 21.4 24.6 28.9 29.8
=+c. Would you use the simple linear regression model to predict trail length when hardness is 10.0? Explain your reasoning.
=+b. Using se 5 2.35, 5 4.5, and 5 250, predict trail length when soil hardness is 6.0 in a way that conveys information about the reliability and precision of the prediction.
=+a. Does the relationship between soil hardness and trail length appear to be linear, with shorter trails associated with harder soil (as the article asserted)? Carry out an appropriate test of hypotheses.
=+13.63 A sample of n 5 61 penguin burrows was selected, and values of both y 5 trail length (m) and x 5 soil hardness (force required to penetrate the substrate to a depth of 12 cm with a certain gauge, in kg) were determined for each one (“Effects of Substrate on the Distribution of Magellanic
=+d. How would the estimate when length is 250 mm compare to the estimate of Part (c)? Answer without actually calculating the new estimate.
=+c. Estimate average maximum size when length is 325 mm in a way that conveys information about the precision of estimation.
=+b. Does it appear that the average change in maximum size associated with a 1-mm increase in length is less than.8 mm? State and test the appropriate hypotheses.
=+a. Does there appear to be a useful linear relationship between length and size?
=+The regression equation is Size = –89.1 = 0.729length 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 SS MS F p Regression 1 36736 36736 232.87 0.000 Error 9 1420 158 Total
=+13.62 Exercise 5.46 presented data on x 5 squawfish length and y 5 maximum size of salmonid consumed, both in mm. Use the accompanying MINITAB output along with the values 5 343.27 and Sxx 5 69,112.18 to answer the following questions.
=+d. MINITAB also reported that 5 1.029. Why is this larger than ?
=+c. MINITAB reported that 5 .689. Predict ski time for a single biathlete whose treadmill time is 10 min, and do so in a way that conveys information about the precision of prediction.
=+b. Estimate the average change in ski time associated with a 1-minute increase in treadmill time, and do so in a way that conveys information about the precision of estimation.
=+a. Carry out a test at significance level .01 to decide whether the simple linear regression model is useful.
=+13.61 Exercise 13.8 gave data on x 5 treadmill run time to exhaustion and y 5 20-km ski time for a sample of 11 biathletes. Use the accompanying MINITAB output to answer the following questions.The regression equation is ski = –88.8 – 2.33tread Predictor Coef Stdev t-ratio p Constant 88.796
=+b. Predict flood damage resulting from a claim made when depth of flooding is 3.5 ft, and do so in a way that conveys information about the precision of the prediction.
=+a. Do the data suggest the existence of a positive linear relationship (one in which an increase in y tends to be associated with an increase in x)? Test using a .05 significance level.
=+13.60 Data on x 5 depth of flooding and y 5 flood damage were given in Exercise 5.75. Summary quantities are
=+c. Would the simple linear regression model give accurate predictions? Why or why not?yˆ
=+b. Is the sign of r consistent with your intuition? Explain.(Higher scale values correspond to more developed sense of humor and greater extent of depression.)
=+a. The investigators reported that P-value , .05. Do you agree?
=+13.59 A random sample of n 5 347 students was selected, and each one was asked to complete several questionnaires, from which a Coping Humor Scale value x and a Depression Scale value y were determined (“Depression and Sense of Humor” (Psychological Reports [1994]:1473–1474). The resulting
=+the coefficient of determination was .470. Does this suggest that there is a useful linear relationship between the two variables? Carry out an appropriate test.
=+c. The sample size was n 5 58, and the reported value of
=+b. What value of biomass concentration would you predict when elapsed time is 40 days?
=+a. The estimated regression equation was given as 5 106.3 2 .640x. What is the estimate of average change in biomass concentration associated with a 1-day increase in elapsed time?
=+One such study, reported in “The Ecology of Plants, Large Mammalian Herbivores, and Drought in Yellowstone National Park” (Ecology [1992]: 2043–2058), proposed using the simple linear regression model to relate y 5 green biomass concentration (g/cm3) to x 5 elapsed time since snowmelt
=+13.58 The effects of grazing animals on grasslands have been the focus of numerous investigations by ecologists.
=+8. After consulting with your partner, write a paragraph explaining why it is a good idea to include a model utility test (H0: b 5 0) as part of a regression analysis.
=+7. Would you recommend using the least-squares regression line as a way of predicting heights for women at this university? Explain.
=+why the conclusion from this test is consistent with your explanation in Step 4.

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