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applied statistics and probability for engineers

Questions and Answers of Applied Statistics And Probability For Engineers

=+40 rows of 25 trees each. To test the sugar content of fruit from a sample of 30 trees, researcher A suggests randomly selecting five rows and then randomly selecting six trees from each sampled
=+9. Citrus trees are usually grown in orderly arrangements of rows to facilitate automated farming and harvesting practices. Suppose a group of 1000 trees is laid out in
=+b. Suppose the inspector decides that only 15 items must be tested. Describe a method by which a valid random sample of 15 from the lot can be formed from the 20 items already selected.
=+a. Before evaluating the 20 items, the inspector decides that a sample of size 30 should be used instead. If the inspector obtains a second random sample of size 10 from the remaining 980 items,
=+Suppose an inspector selects a random sample of 20 items from a lot of 1000 items.
=+8. Small manufactured goods are often gathered into large batches, called lots, for purposes of handling and shipping. Random sampling is commonly used to evaluate the quality of items in a given
=+7. Sometimes it is difficult or impossible to determine the population size before selecting a random sample.Describe how you would go about selecting a random sample of trees from a 1-square-mile
=+6. Devise a procedure for selecting a random sample of words from a dictionary. Explain why your procedure guarantees that, for any n, each collection of n words has an equally likely chance of
=+its product specifications listing. Here, would ISO ppm more properly be considered an operational definition or a benchmark?
=+measuring print speed. The standard, known as“ISO ppm,” allows a consumer to make “applesto-apples” comparisons of real-world print speeds under standard conditions. It is now common for
=+5. Print speed (often measured in pages per minute, ppm) is an important property to consider when buying a printer. However, printer manufacturers measure this property in different ways, making
=+4. To test the accuracy of a new numerical algorithm, a programmer uses the algorithm to produce the first 200 digits of the number . The programmer checks the accuracy of 200 digits by comparing
=+3. Give an operational definition for measuring the daytime temperature in a city. In your definition, take into account factors such as time of day and location.
=+2. Give an operational definition for measuring the fuel efficiency of a car. In your definition, take into account factors such as the driving speed, octane rating, distance driven, tire
=+1. What is the primary difference between an operational definition and a benchmark?
=+b. What proportion of observed variation in energy content can be explained by the approximate relationship between energy content and the four predictors?
=+a. Predict the value of energy content when plastics is 17.03, paper is 23.46, garbage is 32.45, and water is 53.23. Also determine the corresponding residual.
=++ 7.64 paper + 4.30 garbage - 37.4 water Predictor Coef StDev T P Constant 2244.9 177.9 12.62 0.000 plastics 28.925 2.824 10.24 0.000 paper 7.644 2.314 3.30 0.003 garbage 4.297 1.916 2.24 0.034
=+ Using Minitab to fit a regression function with the four aforementioned variables as predictors of energy content resulted in the following output:The regression equation is enercont = 2245 +
=+proximate analysis variable x4 5 % moisture by weight for waste specimens obtained from a certain region.Obs Plastics Paper Garbage Water Energy Content 1 18.69 15.65 45.01 58.21 947 2 19.43 23.51
=+56. Efficient design of certain types of municipal waste incinerators requires that information about energy content of the waste be available. The authors of the article “Modeling the Energy
=+amplitude from cycles to failure. Fit an appropriate curve, investigate the quality of the fit, and predict amplitude when cycles to failure 5 5000.Obs Cycfail Strampl Obs Cycfail Strampl 1 1326
=+55. Failures in aircraft gas turbine engines due to high cycle fatigue is a pervasive problem. The article“Effect of Crystal Orientation on Fatigue Failure of Single Crystal Nickel Base Turbine
=+c. Given the slope coefficients from the regression, summarize the relationship between vane shear strength and fluid content by mass as pore fluid viscosity changes from 0%, to 25%, and to 50%.
=+b. Determine the least squares regression line for each pair. For each, determine the corresponding coefficient of determination.
=+a. Create the scatterplots for the pairs (x, y=), (x, y==), and (x, y===). Does each scatterplot suggest that a linear relationship holds for the respective variables?
=+data below corresponds to a graph from the article:x 35.0 37.5 40.0 42.5 45.0 47.5 y9 75.0 63.0 57.0 45.0 28.5 38.0 y0 52.0 41.5 38.0 35.0 20.0 16.0 y- 33.5 24.5 22.0 19.0 13.0 10.0
=+54. Ground motions resulting from an earthquake can be heavily influenced by the dynamic properties of the soils overlying bedrock. The authors of “Influence of Pore Fluid Viscosity on the
=+b. Let x951/x and y9 5 ln(y). Fit a straight line to the (x9, y9) data, use it as a basis for predicting viscosity when temperature is 720, and calculate a quantitative assessment of the extent to
=+a. Would a straight line fit to this data give accurate predictions of viscosity?
=+53. The accompanying data resulted from an investigation of the relationship between temperature (x, in°F) and viscosity (y, in poise) for specimens of bitumen removed from tar sand deposits:x:
=+c. The values of SSTo and SSResid were 6110.2 and 123.4, respectively. Can a substantial percentage of the observed variation in strength be attributed to the postulated approximate relationship
=+b. The observed value of pull strength was 24.35 when wire length was 9 and die height was 100.What value of pull strength would you have predicted under these circumstances, and what is the value
=+a. Interpret the coefficients of x1 and x2 in the given equation.
=+52. An experiment carried out to investigate the relationship between y 5 wire bond pull strength in a semiconductor product and the two predictors x1 5 wire length and x2 5 die height resulted
=+c. What happens if the best-fit equation is used to predict stress when strain is .03? Note: The largest strain value in the sample was .017.
=+b. The observed values of stress were 91, 97, 108, 111, 114, 110, 112, 102, 98, and 91. Using the best-fit quadratic gave corresponding predicted values of 94.16, 98.87, 102.93, 109.07, 111.16,
=+a. One observation in the sample was made when strain was .005, and the resulting value of stress was 111. What value of stress would you have predicted in this situation, and what is the value of
=+for each of n 5 10 values of strain. A scatterplot of the resulting data suggested a quadratic relationship between the two variables. Employing the principle of least squares gave yn 5 88.791 1
=+51. The relationship between x 5 strain (in./in.) and y 5 stress (ksi) for an experimental alloy tension member was investigated by making an observation on stress
=+50. Refer to Exercise 49. Consider predicting speed from stride rate, so that the response variable y is speed. Suppose that the values of speed in the sample are expressed in meters/second. How
=+c. Calculate and interpret the coefficient of determination for the regression of stride rate on speed of part (a) and for the regression of speed on stride rate of part (b). How are these two
=+b. Calculate the equation of the least squares line that you would use to predict speed from stride rate.
=+a. Calculate the equation of the least squares line that you would use to predict stride rate from speed.
=+49. An investigation was carried out to study the relationship between speed (ft/sec) and stride rate (number of steps taken/sec) among female marathon runners. Resulting summary quantities
=+b. The residuals, listed in the same order as the x values, are 21.03 20.92 21.35 20.78 20.68 20.11 0.21 20.59 0.13 0.45 0.06 0.62 0.94 0.80 20.14 0.93 0.04 0.36 1.92 0.78 0.35 0.67 1.02 1.09 0.66
=+function of elapsed time (hr) under specified conditions. The following data was read from a graph in the article: n 5 33; x 5 .17, .33, .50, .67, . . . , 5.50;y 5 .50, 1.25, 1.50, 2.75, 3.50,
=+48. As the air temperature drops, river water becomes supercooled and ice crystals form. Such ice can significantly affect the hydraulics of a river. The article“Laboratory Study of Anchor Ice
=+c. The values of SSTo and SSResid were 1060.0 and 390.0, respectively. Calculate and interpret the coefficient of determination.
=+Why do these residuals have different signs?
=+b. The largest strength value when temperature was 120 was 40 and the smallest was 29. What value of strength would you have predicted for this temperature, and what are the values of the
=+a. Interpret the value of the slope.
=+47. An investigation of the relationship between the temperature (°F) at which a material is treated and the strength of the material involved an experiment in which four different strength
=+b. Find a transformation that produces an approximate linear relationship between the transformed values. Then fit a line to the transformed data and use it to obtain an equation that describes
=+a. Would you fit a straight line to the data and use it as a basis for predicting y 5 stress range from x 5 number of cycles? Why or why not?
=+Stress: 121 71 108 99 77 Cycles: 1.257 11.250 2.240 4.030 6.650 Stress: 70 79 56 89 75 Cycles: 6.970 6.430 19.140 3.950 9.000 Stress: 95 90 110 77 64 Cycles: 2.290 4.470 2.150 10.490 19.260 Stress:
=+46. Orthotropic steel bridge decks with closed ribs have been widely used in suspension bridges, cablestayed bridges, and urban elevated expressways due to their overall light weights, ease of
=+45. Refer to Exercise 42. Compute the covariance between x and y and then the value of the population correlation coefficient. Do these two variables appear to be strongly related? Explain.
=+c. For what proportion of these machines will the number of cosmetic flaws exceed the number of major defects?
=+b. What proportion of these machines will have no major defects or cosmetic flaws? What proportion will have at least one defect or flaw?
=+a. What is the joint mass function of these two variables?
=+44. Let x denote the number of major defects for a particular piece of machinery and y be the number of cosmetic flaws on this same piece. Suppose that x and y are independent variables with
=+e. If a bus occupies three vehicle spaces and a car occupies just one, what is the mean value of the number of vehicle spaces occupied during a signal cycle? Hint: Let h(x, y) 5 x 1 3y.
=+d. What is the mean value of the number of cars per signal cycle?
=+c. In what proportion of cycles will the number of cars be the same as the number of buses?
=+b. In what proportion of cycles will there be at most one vehicle of each type?
=+a. In what proportion of cycles will there be exactly one car and one bus?
=+43. The joint distribution of the number of cars (x) and the number of buses (y) per signal cycle at a particular left turn lane is displayed in the accompanying table:y f(x, y) 0 1 2 0 .025 .015
=+c. What is the marginal mass function of x? What is the marginal mass function of y?
=+b. What proportion of customers have both deductible amounts less than $500?
=+a. What proportion of customers have $500 deductible amounts for both types of policies?
=+amount and y denote the automobile deductible amount for a customer who has both types of policies. The joint mass function of x and y is as follows:y f(x, y) 0 250 500 x 200 .20 .10 .20 500 .05
=+42. A large insurance agency provides services to a number of customers who have purchased both a homeowner’s policy and an automobile policy. For each type of policy, a deductible amount must be
=+. Be sure to specify all function coefficients. For each fit, also include the coefficient of multiple determination and interpret its value.
=+ Use a statistical computer package to fit(a) a 1 b1x1 1 b2x2 1 b3x3, (b) a 1 b1x1 1 b2x2 1 b3x3 1 b4x1x2 1 b5x1x3 1 b6x2x3, and (c) a 1 b1x1 1 b2x2 1 b3x3 1 b4x1x2 1 b5x1x3 1 b6x2x31 b7x1 2 1 b8x2
=+41. A new surface finishing method has been developed for nanofinishing flat and three-dimensional workpiece surfaces. The authors of “Parametric Analysis of an Improved Ball End
=+. Be sure to specify all function coefficients. For each function, also include the coefficient of multiple determination and interpret its value.
=+shear (Vmax, in kN) is influenced by x15transversereinforcement yield stress (MPa) and x25concrete cylinder compressive strength (MPa).y: 314.9 359.0 300.7 271.3 266.9 x1: 469 469 469 400 400 x2:
=+40. The collapse of reinforced concrete buildings during earthquakes can result in significant loss of property and life. Often such collapses are caused by concrete column axial failure. The
=+d. Note the difference in magnitude of the residuals you just computed for the two regressions. Explain how it is reasonable for one of these to have a smaller residual magnitude given the
=+c. For the fit with predictors x1, x2, and x3 as well as quadratic and interaction predictors, what is the predicted value of -carotene when lineolic acid 5 40, kerosene 5 20, and antioxidant 5
=+b. For the fit using a 1 b1x1 1 b2x2 1 b3x3, what is the predicted value of -carotene when lineolic acid 5 40, kerosene 5 20, and antioxidant 5 5? What is the corresponding residual?
=+a. What is the coefficient of multiple determination for each fitted function?
=+R–Square Coeff Var Root MSE beta Mean 0.986568 15.08576 0.043778 0.290195 Parameter Estimate Standard Error tValue Pr >|t|Intercept -2.368673650 0.25095313 -9.44
=+A request to the SAS package to fit a function with predictors x1, x2, and x3 as well as quadratic and interaction predictors yielded the following output:Dependent Variable: beta Source DF Sum of
=+Model 3 0.02352595 0.00784198 0.09 0.9648 Error 16 1.40326270 0.08770392 C. Total 19 1.42678865 R-Square Coeff Var Root MSE beta Mean 0.016489 102.0515 0.296148 0.290195 Parameter Estimate Standard
=+ A request to the SAS package to fit a 1 b1x1 1 b2x2 1 b3x3 yielded the following output:Dependent Variable: beta Source DF Sum of Squares Mean Square FValue Pr. F
=+three predictors amount of lineolic acid, amount of kerosene, and amount of antioxidant (all g/dm3).Obs Linoleic Kerosene Antiox Betacaro 1 30.00 30.00 10.00 0.7000 2 30.00 30.00 10.00 0.6300 3
=+39. Use of sucrose as a carbon source for the production of chemicals is uneconomical. Beet molasses is a readily available and lower-priced substitute. The article “Optimization of the
=+ Use a statistical computer package to fit y 5 a 1 b1 x1 1 b2x2 using the least squares method. Be sure to specify all function coefficients. Also include the coefficient of multiple determination
=+38. An investigation of a die-casting process resulted in the accompanying data on x1 5 furnace temperature, x2 5 die close time, and y 5 temperature difference on the die surface (“A
=+c. Since b2 5 29,836, is it legitimate to conclude that if x2 increases by 1 unit while the values of the other predictors remain fixed, deposition would increase by 29,836 units? Explain your
=+b. Predict the value of deposition when x15 20,000 and x2 5 .001.
=+a. Interpret the value of the coefficient of multiple determination.
=+(Intercept) -33.46 1.490e+01 -2.246 0.0413 1 2.055e-03 2.945e-04 6.977 6.48e-06 2 29836 1.365e+04 2.185 0.0464 Residual standard error: 44.28 on 14 degrees of freedom Multiple R–squared: 0.9234,
=+a 1 b1x1 1 b2x2 from the R software:obs 1 2 flth 1 92017 .0026900 278.78 2 51830 .0030000 124.53 3 17236 .0000196 22.65 4 15776 .0000360 28.68 5 33462 .0004960 32.66 6 243500 .0038900 604.70 7
=+two rather complicated predictors x1 (mg-sec/m3)and x2 (mg/m2), defined in terms of PAH air concentrations for various species, total time, and total amount of precipitation. Here is data on the
=+37. Snowpacks contain a wide spectrum of pollutants that may represent environmental hazards. The article “Atmospheric PAH Deposition: Deposition Velocities and Washout Ratios” (J. of
=+b. What proportion of observed variation in surface roughness can be explained by the approximate relationship between surface roughness and the four predictors?
=+a. Predict the value of surface roughness when amplitude is 10, depth of cut is .5, feed rate is .25, and cutting speed is 50.

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