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Applied Statistics And Probability For Engineers 6th Edition Douglas C. Montgomery, George C. Runger - Solutions

In Example 9-6 we described how the “spring-like effect” in a golf club could be determined by measuring the coefficient of restitution (the ratio of the outbound velocity to the inbound velocity of a golf ball fired at the club head). Twelve randomly selected drivers produced by two club
Consider the shear strength experiment described in Example 10-9. Construct a 95% confidence interval on the difference in mean shear strength for the two methods. Is the result you obtained consistent with the findings in Example 10-9? Explain why.
Reconsider the shear strength experiment described in Example 10-9. Do each of the individual shear strengths have to be normally distributed for the paired t-test to be appropriate, or is it only the difference in shear strengths that must be normal? Use a normal probability plot to investigate
Consider the parking data in Example 10-10. Use the paired t-test to investigate the claim that the two types of cars have different levels of difficulty to parallel park use a = 0.10. Compare your results with the confidence interval constructed in Example 10-10 and comment on why they are the
Reconsider the parking data in Example 10-10. Investigate the assumption that the differences in parking times are normally distributed.
The manager of a fleet of automobiles is testing two brands of radial tires and assigns one tire of each brand at random to the two rear wheels of eight cars and runs the cars until the tires wear out. The data (in kilometers) follow. Find a 99% confidence interval on the difference in mean life
A computer scientist is investigating the usefulness of two different design languages in improving programming tasks. Twelve expert programmers, familiar with both languages, are asked to code a standard function in both languages, and the time (in minutes) is recorded. The data follow:(a) Find a
Fifteen adult males between the ages of 35 and 50 participated in a study to evaluate the effect of diet and exercise on blood cholesterol levels. The total cholesterol was measured in each subject initially and then three months after participating in an aerobic exercise program and switching to a
An article in the Journal of Aircraft (Vol. 23, 1986, pp. 859€“864) describes a new equivalent plate analysis method formulation that is capable of modeling aircraft structures such as cranked wing boxes and that produces results similar to the more computationally intensive finite
Ten individuals have participated in a diet-modification program to stimulate weight loss. Their weight both before and after participation in the program is shown in the following list is there evidence to support the claim that this particular diet-modification program is effective in producing a
Two different analytical tests can be used to determine the impurity level in steel alloys. Eight specimens are tested using both procedures, and the results are shown in the following tabulation. Is there sufficient evidence to conclude that both tests give the same mean impurity level, using a =
Consider the weight-loss data in Exercise 10-41. Is there evidence to support the claim that this particular diet modification program will result in a mean weight loss of at least 10 pounds? Use a = 0.05.
Consider the weight-loss experiment in Exercise 10-41. Suppose that, if the diet-modification program results in mean weight loss of at least 10 pounds, it is important to detect this with probability of at least 0.90. Was the use of 10 subjects an adequate sample size? If not, how many subjects
For an F distribution, find the following:(a) f0.25,5,10 (b) f0.10,24,9(c) f0.05,8,15 (d) f0.75,5,10(e) f0.90,24,9 (f ) f0.95,8,15
For an F distribution, find the following:(a) f0.25,7,15 (b) f0.10,10,12(c) f0.01,20,10 (d) f0.75,7,15(e) f0.90,10,12 (f) f0.99,20,10
Two chemical companies can supply a raw material. The concentration of a particular element in this material is important. The mean concentration for both suppliers is the same, but we suspect that the variability in concentration may differ between the two companies. The standard deviation of
Consider the etch rate data in Exercise 10-21. Test the hypothesis H0: σ2/1 = σ2/2 against H1: σ2/1  σ2/2 using a = 0.05, and draw conclusions.
Consider the etch rate data in Exercise 10-21. Suppose that if one population variance is twice as large as the other, we want to detect this with probability at least 0.90 (using a = 0.05). Are the sample sizes n1 = n2 = 10 adequate?
Consider the diameter data in Exercise 10-17. Construct the following: (a) A 90% two-sided confidence interval on σ1/ σ 2 (b) A 95% two-sided confidence interval on σ1/ σ2. Comment on the comparison of the width of this interval with the width of the interval in part (a). (c)
Consider the foam data in Exercise 10-18. Construct the following: (a) A 90% two-sided confidence interval on σ2/1 σ2/2 (b) A 95% two-sided confidence interval on σ2/1 σ2/2. Comment on the comparison of the width of this interval with the width of the interval in part
Consider the film speed data in Exercise 10-24. Test H0: σ2/1 = σ2/2 versus using a = 0.02.
Consider the gear impact strength data in Exercise 10-22. Is there sufficient evidence to conclude that the variance of impact strength is different for the two suppliers? Use a = 0.05.
Consider the melting point data in Exercise 10-25. Do the sample data support a claim that both alloys have the same variance of melting point? Use a = 0.05 in reaching your conclusion.
Exercise 10-28 presented measurements of plastic coating thickness at two different application temperatures. Test H0: σ2/1 = σ2/2 against H1: σ2/1  σ2/2 using a = 0.01.
A study was performed to determine whether men and women differ in their repeatability in assembling components on printed circuit boards. Random samples of 25 men and 21 women were selected, and each subject assembled the units. The two sample standard deviations of assembly time were smen = 0.98
Reconsider the assembly repeatability experiment described in Exercise 10-56. Find a 98% confidence interval on the ratio of the two variances. Provide an interpretation of the interval.
Reconsider the film speed experiment in Exercise 10-24. Suppose that one population standard deviation is 50% larger than the other. Is the sample size n1 = n2 = 8 adequate to detect this difference with high probability? Use a = 0.01 in answering this question.
Reconsider the overall distance data for golf balls in Exercise 10-31. Is there evidence to support the claim that the standard deviation of overall distance is the same for both brands of balls (use a = 0.05)? Explain how this question can be answered with a 95% confidence interval on.
Reconsider the coefficient of restitution data in Exercise 10-32. Do the data suggest that the standard deviation is the same for both brands of drivers (use a = 0.05)? Explain how to answer this question with a confidence interval on σ1/ σ2.
Two different types of injection-molding machines are used to form plastic parts. A part is considered defective if it has excessive shrinkage or is discolored. Two random samples, each of size 300, are selected, and 15 defective parts are found in the sample from machine 1 while 8 defective parts
Two different types of polishing solution are being evaluated for possible use in a tumble-polish operation for manufacturing interocular lenses used in the human eye following cataract surgery. Three hundred lenses were tumble polished using the first polishing solution, and of this number 253 had
Consider the situation described in Exercise 10-61. Suppose that p1 = 0.05 and p2 = 0.01.(a) With the sample sizes given here, what is the power of the test for this two-sided alternate?(b) Determine the sample size needed to detect this difference with a probability of at least 0.9. Use a = 0.05.
Consider the situation described in Exercise 10-61. Suppose that p1 = 0.05 and p2 = 0.02.(a) With the sample sizes given here, what is the power of the test for this two-sided alternate?(b) Determine the sample size needed to detect this difference with a probability of at least 0.9. Use a =0.05.
A random sample of 500 adult residents of Maricopa County found that 385 were in favor of increasing the highway speed limit to 75 mph, while another sample of 400 adult residents of Pima County found that 267 were in favor of the increased speed limit. Do these data indicate that there is a
Construct a 95% confidence interval on the difference in the two fractions defective for Exercise 10-61.
Construct a 95% confidence interval on the difference in the two proportions for Exercise 10-65. Provide a practical interpretation of this interval.
A procurement specialist has purchased 25 resistors from vendor 1 and 35 resistors from vendor 2. Each resistor€™s resistance is measured with the following results:(a) What distributional assumption is needed to test the claim that the variance of resistance of product from vendor 1 is
An article in the Journal of Materials Engineering (1989, Vol. 11, No. 4, pp. 275–282) reported the results of an experiment to determine failure mechanisms for plasma sprayed thermal barrier coatings. The failure stress for one particular coating (NiCrAlZr) under two different test conditions is
Consider Supplemental Exercise 10-69. (a) Construct a 95% confidence interval on the ratio of the variances σ1/ σ2, of failure stress under the two different test conditions. (b) Use your answer in part (b) to determine whether there is a significant difference in variances of the two
A liquid dietary product implies in its advertising that use of the product for one month results in an average weight loss of at least 3 pounds. Eight subjects use the product for one month, and the resulting weight loss data are reported below. Use hypothesis-testing procedures to answer the
The breaking strength of yarn supplied by two manufacturers is being investigated. We know from experience with the manufacturers’ processes that σ1 = 5 psi and σ2 = 4 psi. A random sample of 20 test specimens from each manufacturer results in psi and psi, respectively. (a) Using a 90%
The Salk polio vaccine experiment in 1954 focused on the effectiveness of the vaccine in combating paralytic polio. Because it was felt that without a control group of children there would be no sound basis for evaluating the efficacy of the Salk vaccine, the vaccine was administered to one group,
Consider Supplemental Exercise 10-72. Suppose that prior to collecting the data, you decide that you want the error in estimating σ1 = σ2 by x1 = x2 to be less than 1.5 psi. Specify the sample size for the following percentage confidence: (a) 90% (b) 98% (c) Comment on the effect of
A random sample of 1500 residential telephones in Phoenix in 1990 found that 387 of the numbers were unlisted. x2 x1 = 88 = 91A random sample in the same year of 1200 telephones in Scottsdale found that 310 were unlisted.(a) Find a 95% confidence interval on the difference in the two proportions
In a random sample of 200 Phoenix residents who drive a domestic car, 165 reported wearing their seat belt regularly, while another sample of 250 Phoenix residents who drive a foreign car revealed 198 who regularly wore their seat belt. (a) Perform a hypothesis-testing procedure to determine if
Consider the previous exercise, which summarized data collected from drivers about their seat belt usage.(a) Do you think there is a reason not to believe these data? Explain your answer.(b) Is it reasonable to use the hypothesis-testing results from the previous problem to draw an inference about
Consider the situation described in Exercise 10-62.(a) Redefine the parameters of interest to be the proportion of lenses that are unsatisfactory following tumble polishing with polishing fluids 1 or 2. Test the hypothesis that the two polishing solutions give different results using a = 0.01.(b)
Consider the situation of Exercise 10-62, and recall that the hypotheses of interest are H0: p1 = p2 versus H1: p1 ( p2.We wish to use a = 0.01. Suppose that if p1 = 0.9 and p2 = 0.6, we wish to detect this with a high probability, say, at least 0.9. What sample sizes are required to meet this
A manufacturer of a new pain relief tablet would like to demonstrate that its product works twice as fast as the competitor€™s product. Specifically, the manufacturer would like to testWhere μ1 is the mean absorption time of the competitive product and μ2 is the mean absorption time
Suppose that we are testing H0: μ1 = μ2 versus H1: μ1  μ2 and we plan to use equal sample sizes from the two populations. Both populations are assumed to be normal with unknown but equal variances. If we use a = 0.05 and if the true mean μ1 = μ2 + what sample
Consider the fire-fighting foam expanding agents investigated in Exercise 10-18, in which five observations of each agent were recorded. Suppose that, if agent 1 produces a mean expansion that differs from the mean expansion of agent 1 by 1.5, we would like to reject the null hypothesis with
A fuel-economy study was conducted for two German automobiles, Mercedes and Volkswagen. One vehicle of each brand was selected, and the mileage performance was observed for 10 tanks of fuel in each car. The data are as follows (in miles per gallon):(a) Construct a normal probability plot of each of
Reconsider the fuel-economy study in Supplemental Exercise 10-83. Rework part (d) of this problem using an appropriate hypothesis-testing procedure. Did you get the same answer as you did originally? Why?
An experiment was conducted to compare the filling capability of packaging equipment at two different wineries. Ten bottles of pinot noir from Ridgecrest Vineyards were randomly selected and measured, along with 10 bottles of pinot noir from Valley View Vineyards. The data are as follows (fill
Consider Supplemental Exercise 10-85. Suppose that the true difference in mean fill volume is as much as 2 fluid ounces; did the sample sizes of 10 from each vineyard provide good detection capability when a = 0.05? Explain your answer.
A Rockwell hardness-testing machine presses a tip into a test coupon and uses the depth of the resulting depression to indicate hardness. Two different tips are being compared to determine whether they provide the same Rockwell C-scale hardness readings. Nine coupons are tested, with both tips
Two different gauges can be used to measure the depth of bath material in a Hall cell used in smelting aluminum. Each gauge is used once in 15 cells by the same operator.(a) State any assumptions necessary to test the claim that both gauges produce the same mean bath depth readings. Check those
An article in the Journal of the Environmental Engineering Division (“Distribution of Toxic Substances in Rivers,” 1982, Vol. 108, pp. 639–649) investigates the concentration of several hydrophobic organic substances in the Wolf River in Tennessee. Measurements on hexachlorobenzene (HCB) in
An article in Concrete Research (“Near Surface Characteristics of Concrete: Intrinsic Permeability,” Vol. 41, 1989), presented data on compressive strength x and intrinsic permeability y of various concrete mixes and cures. Summary quantities are n = 14, Σyi = 572, g = 23,530, Σxi =
Regression methods were used to analyze the data from a study investigating the relationship between roadway surface temperature (x) and pavement deflection (y). Summary quantities were n = 20, Σyi = 12.75, Σy2/i 8.86, Σxi = 1478, Σx2/i 143,215.8, and Σxiyi = 1083.67.
Consider the regression model developed in Exercise 11-2.(a) Suppose that temperature is measured in oC rather than oF. Write the new regression model that result.(b) What change in expected pavement deflection is associated with a 1oC change in surface temperature?
Montgomery, Peck, and Vining (2001) present data concerning the performance of the 28 National Football League teams in 1976. It is suspected that the number of games won (y) is related to the number of yards gained rushing by an opponent (x). The data are shown in the following table.(a) Calculate
An article in Techno metrics by S. C. Narula and J. F. Wellington (€œPrediction, Linear Regression, and a Minimum Sum of Relative Errors,€ Vol. 19, 1977) presents data on the selling price and annual taxes for 24 houses. The data are shown in the following table.(a) Assuming
The number of pounds of steam used per month by a chemical plant is thought to be related to the average ambient temperature (inoF) for that month. The past year€™s usage and temperature are shown in the following table:(a) Assuming that a simple linear regression model is appropriate,
The data shown in the following table are highway gasoline mileage performance and engine displacement for a sample of 20 cars.(a) Fit a simple linear model relating highway miles per gallon (y) to engine displacement (x) using least squares.(b) Find an estimate of the mean highway gasoline mileage
An article in the Tappi Journal (March, 1986) presented data on green liquor Na2S concentration (in grams per liter) and paper machine production (in tons per day) Te data read from a graph) are shown as follows:(a) Fit a simple linear regression model with y =green liquor Na2S concentration and x
An article in the Journal of Sound and Vibration (Vol. 151, 1991, pp. 383€“394) described a study investigating the relationship between noise exposure and hypertension. The following data are representative of those reported in the article.(a) Draw a scatter diagram of y (blood pressure
An article in Wear (Vol. 152, 1992, pp. 171€“181) presents data on the fretting wear of mild steel and oil viscosity. Representative data follow, with x = oil viscosity and y = wear ear volume (10 €“4 cubic millimeters).(a) Construct a scatter plot of the data. Does a simple
An article in the Journal of Environmental Engineering (Vol. 115, No. 3, 1989, pp. 608€“619) reported the results of a study on the occurrence of sodium and chloride in surface streams in central Rhode Island. The following data are chloride concentration y (in milligrams per liter) and
A rocket motor is manufactured by bonding together two types of propellants, an igniter and a sustainer. The shear strength of the bond y is thought to be a linear function of the age of the propellant x when the motor is cast. Twenty observations are shown in the table on the next page.(a) Draw a
Show that in a simple linear regression model the point (x, y) lies exactly on the least squares regression line.
Consider the simple linear regression model Y =B0 + B1x + e suppose that the analyst wants to use z = x – x as the regress or variable.(a) Using the data in Exercise 11-12, construct one scatter plot of the (xi, yi) points and then another of the (zi = xi – x, y) points. Use the two plots to
Suppose we wish to fit the model y*I = B*0 + B*1 (xi – x) +E, where y*i = yi, y (i = 1, 2, .., p , n). Find the least squares estimates of B*0 and B*1. How do they relate to B0 and B1?
Suppose we wish to fit a regression model for which the true regression line passes through the point (0, 0). The appropriate model is Y = Bx = E. Assume that we have n pairs of data (x1, y1), (x2, y2), p , (xn, yn). Find the least squares estimate of B.
Using the results of Exercise 11-16, fit the model Y = Bx + E to the chloride concentration-roadway area data in Exercise 11-11. Plot the fitted model on a scatter diagram of the data and comment on the appropriateness of the model.
Consider the data from Exercise 11-1 on x = compressive strength and y = intrinsic permeability of concrete. (a) Test for significance of regression using a = 0.05. Find the P-value for this test. Can you conclude that the model specifies a useful linear relationship between these two variables?
Consider the data from Exercise 11-2 on x = roadway surface temperature and y = pavement deflection.(a) Test for significance of regression using a = 0.05. Find the P-value for this test. What conclusions can you draw?(b) Estimate the standard errors of the slope and intercept.
Consider the National Football League data in Exercise 11-4.(a) Test for significance of regression using a = 0.01. Find the P-value for this test. What conclusions can you draw?(b) Estimate the standard errors of the slope and intercept.(c) Test (using a = 0.01) H0: B1 = - 0.01 versus H1: B1 =
Consider the data from Exercise 11-5 on y = sales price and x = taxes paid.(a) Test H0: B1 = 0 using the t-test; use a = 0.05.(b) Test H0: B1 = 0 using the analysis of variance with a = 0.05. Discuss the relationship of this test to the test from part (a).(c) Estimate the standard errors of the
Consider the data from Exercise 11-6 on y = steam usage and x = average temperature.(a) Test for significance of regression using a = 0.01. What is the P-value for this test? State the conclusions that result from this test.(b) Estimate the standard errors of the slope and intercept.(c) Test the
Exercise 11-7 gave 20 observations on y = highway gasoline mileage and x = engine displacement.(a) Test for significance of regression using a = 0.01. Find the P-value for this test. What conclusions can you reach?(b) Estimate the standard errors of the slope and intercept.(c) Test H0: B1 = –
Exercise 11-8 gave 13 observations on y = green liquor Na2S concentration and x = production in a paper mill.(a) Test for significance of regression using a = 0.05. Find the P-value for this test. (b) Estimate the standard errors of the slope and intercept.(c) Test H0: B0 = 0 versus H1: B0 = 0
Exercise 11-9 presented data on y = blood pressure rise and x = sound pressure level.(a) Test for significance of regression using a = 0.05. What is the P-value for this test?(b) Estimate the standard errors of the slope and intercept.(c) Test H0: B0 = 0 versus H1: B0 = 0 using a = 0.05. Find the
Exercise 11-11 presented data on y = chloride concentration in surface streams and x = roadway area.(a) Test the hypothesis H0: B1 = 0 versus H1: B1 = 0 using the analysis of variance procedure with a = 0.01.(b) Find the P-value for the test in part (a).(c) Estimate the standard errors of and(d)
Refer to Exercise 11-12, which gives 20 observations on y = shear strength of a propellant and x = propellant age.(a) Test for significance of regression with a = 0.01. Find the P-value for this test.(b) Estimate the standard errors of and(c) Test H0: B1 = - 30 versus H1: B1 = -30 using a = 0.01.
Suppose that each value of xi is multiplied by a positive constant a, and each value of yi is multiplied by another positive constant b. Show that the t-statistic for testing H0: B1 = 0 versus H1: B1 = 0 is unchanged in value.
Consider the no-intercept model Y = Bx + E with the E’s NID (0, σ2). The estimate of σ2 is s2 Σ = (yi – Bxi) 2 / (n – 1) and v(B) = σ2 / Σ= 1x2/i (a) Devise a test statistic for H0: B = 0 versus H1: B  0. (b) Apply the test in (a) to the model from
The type II error probability for the t-test for H0: B1 = B1, 0 can be computed in a similar manner to the t-tests of Chapter 9. If the true value of B1 is B1, the value =|B1, 0 – B1”/(σ√(n – 1)/Sxx is calculated and used as the horizontal scale factor on the operating
Refer to the data in Exercise 11-1 on y = intrinsic permeability of concrete and x = compressive strength. Find a 95% confidence interval on each of the following: (a) Slope (b) Intercept(c) Mean permeability when x = 2.5(d) Find a 95% prediction interval on permeability when x = 2.5. Explain why
Exercise 11-2 presented data on roadway surface temperature x and pavement deflection y. Find a 99% confidence interval on each of the following:(a) Slope (b) Intercept(c) Mean deflection when temperature x = 85oF (d) Find a 99% prediction interval on pavement deflection when the temperature is 90oF
Exercise 11-4 presented data on the number of games won by NFL teams in 1976. Find a 95% confidence interval on each of the following:(a) Slope (b) Intercept(c) Mean number of games won when opponents rushing yardage is limited to x = 1800(d) Find a 95% prediction interval on the number of games
Refer to the data on y = house selling price and x = taxes paid in Exercise 11-5. Find a 95% confidence interval on each of the following:(a) B1 (b) B0(c) Mean selling price when the taxes paid are x = 7.50(d) Compute the 95% prediction interval for selling price when the taxes paid are x = 7.50.
Exercise 11-6 presented data on y = steam usage and x = monthly average temperature.(a) Find a 99% confidence interval for B1.(b) Find a 99% confidence interval for B0.(c) Find a 95% confidence interval on mean steam usage when the average temperature is 55oF.(d) Find a 95% prediction interval on
Exercise 11-7 presented gasoline mileage performance for 20 cars, along with information about the engine displacement. Find a 95% confidence interval on each of the following:(a) Slope (b) Intercept(c) Mean highway gasoline mileage when the engine displacement is x = 150 in3(d) Construct a 95%
Consider the data in Exercise 11-8 on y = green liquor Na2S concentration and x = production in a paper mill. Find a 99% confidence interval on each of the following:(a) B1 (b) B0(c) Mean Na2S concentration when production x = 910 tons/day(d) Find a 99% prediction interval on Na2S concentration
Exercise 11-9 presented data on y = blood pressure rise and x = sound pressure level. Find a 95% confidence interval on each of the following:(a) B1 (b) B0(c) Mean blood pressure rise when the sound pressure level is 85 decibels(d) Find a 95% prediction interval on blood pressure rise when the
Refer to the data in Exercise 11-10 on y = wear volume of mild steel and x = oil viscosity. Find a 95% confidence interval on each of the following: (a) Intercept (b) Slope(c) Mean wear when oil viscosity x = 30
Exercise 11-11 presented data on chloride concentration y and roadway area x on watersheds in central Rhode Island. Find a 99% confidence interval on each of the following:(a) B1 (b) B0(c) Mean chloride concentration when roadway area x = 1.0%(d) Find a 99% prediction interval on chloride
Refer to the data in Exercise 11-12 on rocket motor shear strength y and propellant age x. Find a 95% confidence interval on each of the following:(a) Slope B1 (b) Intercept B0(c) Mean shear strength when age x = 20 weeks(d) Find a 95% prediction interval on shear strength when age x = 20 weeks.
Refer to the NFL team performance data in Exercise 11-4.(a) Calculate R2 for this model and provide a practical interpretation of this quantity.(b) Prepare a normal probability plot of the residuals from the least squares model. Does the normality assumption seem to be satisfied?(c) Plot the

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