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applied statistics and probability for engineers

Applied Statistics For Engineers And Scientists 3rd Edition Jay L. Devore, Nicholas R. Farnum, Jimmy A. Doi - Solutions

=+Once the simple linear regression model has been judged useful by the model utility test discussed in Section 11.2, the estimated model can be used as the basis for further inferences. Let x denote a particular value of the independent or predictor variable x.In this section, we show how to
=+the same rate for a period of more than 150 hours.Here is data on x 5 leaching time (h), yfw 5 nitrate extraction percentage (freshwater), and ysw 5 nitrate extraction percentage (seawater):11.3 Inferences Based on the Estimated Regression Line
=+Using Seawater” (Hydrometallurgy, 2013: 100–105)evaluated the recovery of nitrate ions from discarded salts using freshwater and seawater leaching agents.Tests were performed in salt columns irrigated at
=+20. Mineral mining is one of the most important economic activities in Chile. Mineral products are frequently found in saline systems composed largely of natural nitrates. Freshwater is often used as a leaching agent for the extraction of nitrate, but the Chilean mining regions have scarce
=+d. When Minitab was used to fit the simple linear regression model to the raw data, the observation (6.0, 2.50) was flagged as possibly having a substantial impact on the fit. Eliminate this observation from the sample and recalculate the estimate of part (a). Based on this, does the
=+c. Would it be sensible to use the simple linear regression model as a basis for predicting % nausea when dose 5 5.0? Explain your reasoning.
=+b. Does it appear that there is a useful linear relationship between these two variables?
=+a. Assuming that the simple linear regression model is valid for relating these two variables(this is supported by the raw data), calculate and interpret an estimate of the slope parameter that conveys information about the precision and reliability of estimation.
=+similar motion at sea) and y 5 reported nausea (%).Relevant summary quantities are n 5 17 ^xi 5 221.1 ^yi 5 193^x 2i 5 3056.69 ^xi yi 5 2759.6^y 2i 5 2975 Values of dose in the sample ranged from 6.0 to 17.6.
=+19. How does lateral acceleration—side forces experienced in turns that are largely under driver control—affect nausea as perceived by bus passengers? The article “Motion Sickness in Public Road Transport: The Effect of Driver, Route, and Vehicle” (Ergonomics, 1999: 1646–1664)
=+b. A scatterplot of the n 5 32 observations on y 5 odor concentration (OU/m3) and x 5 CH4 concentration (ppm) also suggested the plausibility of a positive linear relationship. The coefficient of determination for the simple linear regression of y on x was 0.33. State and test the relevant
=+a. A scatterplot of the n 5 32 observations on y 5 odor concentration (OU/m3) and x 5 H2S concentration (ppb) suggested the plausibility of a positive linear relationship. The coefficient of determination for the simple linear regression of y on x was .58. State and test the relevant hypotheses
=+a Canadian swine farm for a period of one year. One study objective was to identify possible relationships, if any, between odor and presence of other gases such as ammonia (NH3), hydrogen sulfide (H2S), carbon dioxide (CO2), and methane (CH4). Identifying such relationships would be helpful in
=+v18. In what was surely an unpleasant data collection experience, the article “Annual Variations of Odor Concentrations and Emissions from Swine Gestation, Farrowing, and Nursery Buildings” (J. of the Air and Waste Mgmnt., 2011: 1361–1368) reported on monthly odor concentrations and
=+ Does there appear to be a positive linear relationship between these two variables in the sampled population? State and test the relevant hypotheses.
=+17. A sample of n 5 13 steel specimens was selected, and the values of x 5 nickel content and y 5 percentage austentite were determined, resulting in^(xi 2 x)2 5 1.183 ^(yi 2 y)2 5 .05080^(xi 2 x)(yi 2 y) 5 .2073
=+16. The value of the sample correlation coefficient is.722 for the n 5 14 observations on average anterior maximum inclination angle (AMIA) in both the clockwise (Cl) and counterclockwise (Co) directions given in Exercise 10 (Section 3.2) of Chapter 3. Carry out a test at significance level
=+which amounts to multiplying each y value by the same conversion factorc. How does this change affect the value of the t-ratio for testing model utility? Explain your reasoning.
=+15. Suppose that the unit of measurement for y 5 wear loss in Example 11.5 is changed from mm3 to in3,
=+d. Suppose it had previously been believed that when air void increased by 1 percent, the associated true average change in dielectric constant would be at least 2.05. Does the sample data contradict this belief? State and test the relevant hypotheses.
=+c. Use the output to calculate a confidence interval with a confidence level of 95% for the slope of the population regression line and interpret the resulting interval.
=+b. Determine and interpret the value of r 2for this regression. What is the corresponding value of r?Note that the sign of r can be determined based on the output.
=+a. What are the values of SSRegr, SSResid, and SSTo?
=+for 18 asphalt mixture samples having 5% asphalt content. The following R output is from a simple linear regression of y on x:Estimate Std.Error tvalue Pr(>|t|)(Intercept) 4.858691 0.059768 81.293 F)AirVoid 1 0.071422 0.071422 56.635 1.214e-06 Residuals 16 0.020178 0.001261
=+14. Exercise 20 (Section 3.3) of Chapter 3 presented data on y 5 dielectric constant and x 5 air void (%)
=+b. If the roles of the two variables were reversed, so that the amount of oil recovered from wheat straw was the independent variable, what would be the value of the t-ratio for testing model utility? (Answer without actually carrying out another regression analysis, and explain your reasoning.)
=+a. Does the simple linear regression model appear to specify a useful relationship between these two variables? State the relevant hypotheses, and carry out a test in two different ways.
=+13. Exercise 22 (Section 3.3) of Chapter 3 gave SAS output from a regression of amount of oil recovered from wheat straw on amount of oil added.
=+12. Use the computer output given in Exercise 9 of the previous section to decide whether the simple linear regression model specifies a useful relationship between flux and inverse foil thickness.
=+11. In the same way that bysb is the t-ratio for testing H0: 5 0, the t-ratio aysa is appropriate for testing H0: 5 0, where sa is the estimated standard deviation of the statistic a and the test is again based on n 2 2 df. The null hypothesis says that the vertical intercept of the
=+the accompanying Minitab output to decide whether there is a useful linear relationship between rainfall and runoff, and then calculate a confidence interval for the true average change in runoff volume associated with a 1-m3 increase in rainfall volume.The regression equation is runoff =
=+10. Exercise 4 of Section 11.1 gave data on x 5 rainfall volume and y 5 runoff volume (both in m3). Use
=+d. Verify that the sum of the residuals is zero and that squaring and summing the residuals results in the value of SSResid given in the output.
=+c. Predict the value of flux that would result from a single observation made when inverse foil thickness is 45.
=+b. Calculate a point estimate of true average flux when inverse foil thickness is 23.5.
=+a. Interpret the estimated slope and the coefficient of determination.
=+the substantial linear pattern was used as a basis for an important conclusion about material behavior.This is the Minitab output from fitting the simple linear regression model to the data.The regression equation is flux = –0.398 + 0.260 invthick Predictor Coef Stdev t-ratio p Constant
=+9. The authors of the article “Long-Term Effects of Cathodic Protection on Prestressed Concrete Structures” (Corrosion, 1997: 891–908) presented a scatterplot of y 5 steady-state permeation flux(Aycm2) versus x 5 inverse foil thickness (cm21);
=+ Calculate estimates of the parameters for the model in part (a), and then obtain a point prediction of dynamic shear modulus when temperature is 35°F.
=+b. Summary quantities calculated from the data are n 5 7 ^ xi 5 211.4 ^ y i 5 40.64^ xi 2 5 8449.68 ^ (y i)2 5 282.58^ xi y i 5 917.48
=+a. What probabilistic model for relating y =dynamic shear modulus to x 5 testing temperature is implied by the simple linear regression relationship between x and y9?
=+8. Exercise 30 in Section 3.4 gave data on x 5 testing temperature and y 5 dynamic shear modulus for a particular asphalt binder type. A scatterplot of x and y 5 log(y) shows a substantial linear pattern, suggesting that these variables are related by the simple linear regression model.
=+c. Given the slope coefficients from the regression, summarize the relationship between critical rating and pile length as timber damage changes from 0%, to 20%, and to 40%.
=+b. For each pair, calculate point estimates of the slope and intercept of the respective population regression line and determine the corresponding coefficients of determination.
=+a. Create the scatterplots for the pairs (x, y ), (x, y ), and (x, y ). Does each scatterplot suggest that a simple linear regression model holds for the respective variables?
=+7. Timber piles are often used to buttress multiplespan simply supported (MSSS) bridges that are commonly found in rural areas. The authors of “Bridge Timber Piles Load Rating under Eccentric Loading Conditions” (J. Bridge Engr., 2012: 700–710) examined the effect of various geometric
=+b. Calculate a point estimate of mean diffusivity when temperature is 1.5. How does this point estimate compare to a point prediction of the diffusivity value that would result from making one more observation when temperature is 1.5?c. Estimate the error standard deviation .d. Calculate and
=+a. Assuming that the variables are related by the simple linear regression model, determine the equation of the estimated regression line.
=+6. A study reported in the article “The Effects of Water Vapor Concentration on the Rate of Combustion of an Artificial Graphite in Humid Air Flow”(Combustion and Flame, 1983: 107–118) gave data on x 5 temperature of a nitrogen–oxygen mixture (1000s of °F) under specified conditions and
=+c. Calculate a point estimate of the true average bond capacity when lateral pressure is .45f cu.
=+be heavily influenced by the specimen’s fcu, bond capacity was expressed as the ratio of bond strength(MPa) to 1f cu.Pressure:Ratio:0 0.123 00.100 00.101.1 0.172.1 0.133.1 0.107.2 0.217 Pressure:Ratio:.2 0.172.2 0.151.3 0.263.3 0.227.3 0.252.4 0.310.4 0.365 Pressure:Ratio:.4 0.239.5 0.365.5
=+levels of lateral pressure on 21 concrete cube specimens, each with an embedded 16-mm plain steel round bar, and measured the corresponding bond capacity. Due to differing concrete cube strengths(fcu, in MPa), the applied lateral pressure was equivalent to a fixed proportion of the specimen’s
=+5. The bond behavior of reinforcing bars is an important determinant of strength and stability. The article “Experimental Study on the Bond Behavior of Reinforcing Bars Embedded in Concrete Subjected to Lateral Pressure” (J. of Materials in Civil Engr., 2012: 125–133) reported the results
=+e. What proportion of the observed variation in runoff volume can be attributed to the simple linear regression relationship between runoff and rainfall?
=+d. Calculate a point estimate of the error standard deviation .
=+c. Calculate a point estimate of the true average runoff volume when rainfall volume is 50.
=+b. Calculate point estimates of the slope and intercept of the population regression line.
=+a. Does a scatterplot of the data support the use of the simple linear regression model?
=+4. The article “Characterization of Highway Runoff in Austin, Texas, Area” (J. of Envir. Engr., 1998:131–137) gave a scatterplot, along with the least squares line, of x 5 rainfall volume (m3) and y 5 runoff volume (m3) for a particular location. The accompanying values were read from the
=+b. If the coefficients in the simple linear regression model are 5 20.607 and 5 25200.762, what would you predict for the value of vapor pressure when temperature is 300?
=+a. What is the implied probabilistic relationship between V and T?
=+3. Let V be the vapor pressure of water (mm Hg) at a specific temperature T (°K). The Clausius–Clapeyron equation from physical chemistry suggests that y 5 ln(V) is related to x 5 1/ T according to the simple linear regression model.
=+c. What is P(2.4 , y , 2.6 when x 5 250)? If an investigator makes five independent experimental runs, each for a temperature of 250°F, what is the probability that all five observed reaction times are between 2.4 and 2.6?
=+b. What is the true average reaction time when temperature is 200°F? When temperature is 250°F?
=+a. What is the true average change in reaction time associated with a 1°F increase in temperature? A 10°F increase in temperature?
=+2. In a certain chemical process the reaction time y (hr)is known to be related according to the simple linear regression model to the temperature x (°F)in the chamber in which the reaction takes place.The model equation is y 5 5.00 2 .01x 1e, with 5 .075.
=+d. Suppose that 5 .025 and consider making repeated observations on flow rate when the pressure drop is 10 in. What is the long-run proportion of observed flow rates that will exceed .835 [that is, what is P(y . .835 when x 5 10)]?
=+vc. What is the expected (i.e., true average) flow rate when the pressure drop is 10 in.? When the pressure drop is 15 in.?
=+b. What change in flow rate can be expected when pressure drop increases from 10 in. to 15 in.?
=+1. The flow rate y (m3/min) in a device used for airquality measurement depends on the pressure drop x (in. of water) across the device’s filter. Suppose that for x values between 5 and 20, the two variables are related according to the simple linear regression model with true regression line
=+ Assuming that second- and higher-order interactions are negligible, conduct tests (at 5 .01) for the presence of main effects.
=+castings. The following data was obtained:Test run Obs Test run Obs a 70.4 acd 66.6 b 72.1 ace 67.5 c 70.4 ade 64.0 d 67.4 bcd 66.8 e 68.0 bce 70.3 abc 73.8 bde 67.9 abd 67.0 cde 65.9 abe 67.8 abcde 68.0
=+53. A half-fraction of a 25 experiment is used to study the effects of heating time (A), quenching time (B), drawing time (C), position of heating coils (D), and measurement position (E) on the hardness of steel
=+b. Create the ANOVA table for this experiment.Which factors appear to have an effect on surface uniformity? (Use 5 .01).
=+ experiment was conducted in an effort to increase the coating uniformity. In the following table, higher values of the response variable are associated with higher surface uniformity:Surface uniformity Repli- RepliSpray Belt cation cation Run volume speed Brand 1 2 1 2 2 2 40 36 2 1 2 2 25 28
=+52. ln an automated chemical coating process, the speed with which objects on a conveyor belt are passed through a chemical spray (belt speed), the amount of chemical sprayed (spray volume), and the brand of chemical used (brand) are factors that may affect the uniformity of the coating applied.A
=+e. Suppose that additional experiments show that the AB and BC interactions are not significant.If the objective of the study is to maximize lignin removal, what setting of each factor do you recommend?
=+d. Draw an effects plot for the important effects identified in part (c).
=+c. Suppose that additional experimentation shows that only those effects whose magnitudes exceed 40 are important. Which factors or interactions have a significant effect on lignin removal?
=+The response variable lignin removal (g/kg) was studied using a full 23 design with no replication:Run Time Pressure Temp Lignin 1 21 21 21 30 2 1 21 21 110 3 21 1 21 241 4 1 1 21 192 5 21 21 1 116 6 1 21 1 201 7 21 1 1 230 8 1 1 1 191
=+51. Shea tree oxidation experiments were conducted to determine which of three factors (reaction time, air pressure, reaction temp.) affect various aspects in converting the woody biomass into a renewable biofuel.Optimal enzymatic conversion of the Shea tree into ethanol occurs when the
=+f. Is the disparity in magnitudes of the standard deviations a possible cause for concern in this experiment?
=+of employees to five per checkpoint? Note: X-ray machines and metal detectors each require one operator.
=+e. What is the best way to staff a security checkpoint if management wants to limit the number Processing time Ticket X-ray Metal Number of Standard Test checkers machines detectors replicates Mean deviation 1 2 2 2 67 39.10 1.29 2 3 2 1 67 46.50 4.30 3 2 2 1 67 50.56 5.41 4 3 2 2 67 35.07 1.05
=+d. Which settings (high or low) of the factors in part (c) lead to minimizing processing time?
=+c. Using the SSE from part (b), determine which effects are significant (at 5 .05).
=+b. Pool the standard deviations of the replicated runs to find a value for SSE.
=+of Airport Security Checkpoints Under Increased Threat Conditions,” J. of Transp. Engr.,1996: 264–269). Each of the possible combinations of these factors was studied by using eight separate random samples of 67 travelers. The processing times(in seconds) are summarized in the table below.
=+50. Even under the increased levels of security sought by current airport security practices, airports try to assure rapid processing of individuals through security checkouts. In an experiment designed to find combinations of factors that will minimize travelers’processing times at security
=+b. Use the appropriate F ratios to show that none of the two-factor interactions are significant at 5 .05.
=+a. Construct an ANOVA table for this experiment including only main effects and two-factor interactions (as did the authors of the cited article).
=+49. The article cited in Exercise 21 also reported on another experiment in which the authors investigated whether the percent by weight of nickel in the alloy layer is affected by niobium powder paste thickness (A, at three levels), scanning speed (B, at three levels), and laser power (C, at
=+different laundry treatments (factor A), three different types of pen (factor B), and six different fabrics(factor C) were used in the experiment. Three observations were obtained for each combination of the factor levels. Perform an analysis of variance using 5 .01 for all tests, and state
=+variable “degree of removal of marks” (larger values of this variable are associated with more complete removal of marks): SSA 5 39.171, SSB 5 .665, SSC 5 21.508, SS(AB) 5 1.432, SS(AC) 5 15.953, SS(BC) 5 1.382, SS(ABC) 5 9.016, and SSE 5 115.820. Four
=+48. The article “An Assessment of the Effects of Treatment, Time, and Heat on the Removal of Erasable Pen Marks” (J. Testing and Eval., 1991: 394–397)reports the following sums of squares for the response
=+the cladding layer is affected by these same factors.Each factor had three levels and there was one observation at each factor combination. Here is the ANOVA table from the article, which only considered main effects and two-factor interactions:SOURCE DF SS MS? ? ? 162.38? 0.080570 ? ?? ?
=+47. Exercise 20 described an experiment involving three processing parameters: laser power (A), scanning velocity (B), and powder flow rate (C). Another experiment considered how depth penetration of
=+46. The authors of the article cited in Exercise 15 also performed an experiment to see whether the maximum peak to valley profile height (Rmax) is affected by the abrasive size (A), abrasive quantity (B), and quill gap (C); the experiment involved three sizes, three quantities, and three gaps,
=+SSA 5 30,763.0, SSB 5 34,185.6, SSE 5 97,436.8, and SST 5 205,966.6.a. Construct an ANOVA table for this experiment.b. Using 5 .05, can you conclude that there is a significant interaction between the two factors?c. Test, at 5 .05, the hypothesis that factor A has no effect on compressive
=+different curing times were used in combination with four different mixes, with three replicate observations obtained for each of the 12 factor–level combinations. The resulting sums of squares were

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