Questions and Answers of Applied Statistics and Probability

An article in Electronic Packaging and Production (2002, Vol. 42) considered the effect of X-ray inspection of integrated circuits. The rads (radiation dose) were studied as a function of current (in
Table E12-3 provides the highway gasoline mileage test results for 2005 model year vehicles from DaimlerChrysler. The full table of data (available on the book€™s Web site) contains the
The data from a patient satisfaction survey in a hospital are in Table E12-1.The regressor variables are the patient€™s age, an illness severity index (higher values indicate greater
You have fit a multiple linear regression model and the (X'X)ˆ’1matrix is:(a) How many regressor variables are in this model?(b) If the error sum of squares is 307 and there are 15
Hsuie, Ma, and Tsai (‘‘Separation and Characterizations of Thermotropic Copolyesters of p-Hydroxybenzoic Acid, Sebacic Acid, and Hydroquinone,’’ (1995, Vol. 56) studied the effect of the
Can the percentage of the workforce who are engineers in each U.S. state be predicted by the amount of money spent in on higher education (as a percent of gross domestic product), on venture capital
A class of 63 students has two hourly exams and a final exam. How well do the two hourly exams predict performance on the final?The following are some quantities of interest:(a) Calculate the least
Exercise 11.1 described a regression model between percent of body fat (%BF) as measured by immersion and BMI from a study on 250 male subjects. The researchers also measured 13 physical
Table E11-21 shows the population and the average count of wood storks sighted per sample period for South Carolina from 1991 to 2004. Fit a regression line with population as the response and the
The data in Table E11-20 related diamond carats to purchase prices. It appeared in Singapore€™s Business Times, February 18, 2000.(a) Graph the data. What is the relation between carat and
Cesium atoms cooled by laser light could be used to build inexpensive atomic clocks. In a study reported in IEEE Transactions on Instrumentation and Measurement (2001, Vol. 50, pp.
Consider the propellant data is Exercise 11-13. Assume that strength less than 2100 psi is considered a failure. Relate propellant age to the probability of failure with a logistic regression
The World Health Organization defines obesity in adults as having a body mass index (BMI) higher than 30. Of the 250 men in the study mentioned in Exercise 11-1, 23 are by this definition obese. How
The vapor pressure of water at various temperatures is in Table E11-11:(a) Draw a scatter diagram of these data. What type of relationship seems appropriate in relating y to x?(b) Fit a simple linear
Consider the (x, y) data in Table E11-10. Calculate the correlation coefficient. Graph the data and comment on the relationship between x and y. Explain why the correlation coefficient does not
Refer to the NFL quarterback ratings data in Exercise 11-5.(a) Estimate the correlation coefficient between the ratings and the average yards per attempt.(b) Test the hypothesis H0: Ï = 0
The weight and systolic blood pressure of 26 randomly selected males in the age group 25 to 30 are shown in the Table E11-8. Assume that weight and blood pressure are jointly normally
Repeat Exercise 11-60 using an error term with a standard deviation of 4. What impact has increasing the error standard deviation had on the values of R2?(a) Fit the linear regression model by least
Refer to the data in Exercise 11-15 on oxygen demand. Find a 99% confidence interval on each of the following:(a) Î²1 (b) Î²0(c) Find a 95% confidence interval on mean BOD
Refer to the data in Exercise 11-14 on the microstructure of zirconia. Find a 95% confidence interval on each of the following:(a) Slope(b) Intercept(c) Mean length when x = 1500(d) Find a 95%
Show that by expanding etXin a power series and taking expectations term by term we may write the momentgenerating function asThus, the coefficient of tr / r! in this expansion is
A study was performed to investigate new automobile purchases. A sample of 20 families was selected. Each family was surveyed to determine the age of their oldest vehicle and their total family
Consider the simple linear regression model Y = β0 + β1x + ε, with E(e) = 0, V(e) = σ2, and the errors e uncorrelated.(a) Show that E(σ̂) = E(MSE) σ2. (b) Show that E(MSR) = σ2 + β12
Consider the following data. Suppose that the relation - ship between Y and x is hypothesized to be Y = (Î²0+ Î²1x + e)ˆ’ . 1 Fit an appropriate model to the data.
An article in the IEEE Transactions on Instrumentation and Measurement [€œDirect, Fast, and Accurate Measurement of VTand K of MOS Transistor Using VT-Sift Circuit€ (1991, Vol.
The market research department of a soft drink manufacturer is investigating the effectiveness of a price discount coupon on the purchase of a two-liter beverage product. A sample of 5500 customers
The compressive strength of an alloy fastener used in aircraft construction is being studied. Ten loads were selected over the range 2500€“4300 psi, and a number of fasteners were tested at
A study was conducted attempting to relate home ownership to family income. Twenty households were selected and family income was estimated along with information concerning home ownership (y = 1
Determine if the following models are intrinsically linear. If yes, determine the appropriate transformation to generate the linear model.(a) Y = β0 xβ1 ε(b) Y = 3 + 5x/x + ε(c) Y = β0 βx1
The monthly absolute estimate of global (land and ocean combined) temperature indexes (degrees C) in 2000 and 2001 (www.ncdc.noaa.gov/oa/climate/) are:(a) Graph the data and fit a regression line to
In an article in IEEE Transactions on Instrumentation and Measurement (2001, Vol. 50, pp. 986€“990), researchers reported on a study of the effects of reducing current draw in a magnetic
Suppose that data are obtained from 20 pairs of (x, y) and the sample correlation coefficient is 0.75.(a) Test the hypothesis that H0: ρ = 0 against H1: ρ> 0 with α = 0.05. Calculate the
Suppose that data are obtained from 20 pairs of (x, y) and the sample correlation coefficient is 0.8.(a) Test the hypothesis that H0: ρ = 0 against H1: ρ ≠ 0 with α = 0.05. Calculate the
Studentized Residuals. Show that the variance of the ith residual isThe ith studentized residual is defined as(a) Explain why ri has unit standard deviation.(b) Do the standardized residuals have
An article in the Journal of the American Statistical Association [€œMarkov Chain Monte Carlo Methods for Computing Bayes Factors: A Comparative Review€ (2001, Vol. 96, pp.
Refer to the compressive strength data in Exercise 11-3. Use the summary statistics provided to calculate R2 and provide a practical interpretation of this quantity.Exercise 11-3An article in
Consider the simple linear regression model y = 10 + 30x + ε where the random error term is normally and independently distributed with mean zero and standard deviation 1. Use software to generate a
Refer to the NFL quarterback ratings data in Exercise 11-5. Find a 95% confidence interval on each of the following:(a) Slope (b) Intercept(c) Mean rating when the average yards per attempt is
Using the regression from Exercise 11-2,(a) Find a 95% confidence interval for the slope.(b) Find a 95% confidence interval for the mean weight for a man 25 years old.(c) Find a 95% prediction
Using the regression from Exercise 11-1,(a) Find a 95% confidence interval for the slope.(b) Find a 95% confidence interval for the mean percent body fat for a man with a BMI of 25.(c) Find a 95%
Consider the data in Exercise 11-15 on y = oxygen demand and x = time.(a) Test for significance of regression using α = 0.01. Find the P-value for this test. What conclusions can you draw?(b)
An article in The Journal of Clinical Endocrinology and Metabolism [€œSimultaneous and Continuous 24-Hour Plasma and Cerebrospinal Fluid Leptin Measurements: Dissociation of Concentrations
Consider the data in Exercise 11-16 on y = deflection and x = stress level.(a) Test for significance of regression using α = 0.01. What is the P-value for this test? State the conclusions that
Consider the data from Exercise 11-9 on y = green liquor Na2S concentration and x = production in a paper mill.(a) Test for significance of regression using α = 0.05. Find the P-value for this
Consider the National Football League data in Exercise 11-5.(a) Test for significance of regression using α = 0.01. Find the P-value for this test. What conclusions can you draw?(b) Estimate the
Consider the following computer output.The regression equation isAnalysis of Variance(a) Fill in the missing information. You may use bounds for the P-values.(b) Can you conclude that the model
Consider the following computer output. The regression equation isAnalysis of Variance(a) Fill in the missing information. You may use bounds for the P-values.(b) Can you conclude that the model
Consider the simple linear regression model y = 10 + 25x + ε where the random error term is normally and independently distributed with mean zero and standard deviation 2. Use software to generate a
Suppose that in Exercise 11-24 weight is measured in kg instead of lbs.(a) How will the estimates of the slope and intercept change?(b) Estimate the error standard deviation.(c) Estimate the standard
Recall the regression of weight on age from Exercise 11-2.(a) Estimate the error standard deviation.(b) Estimate the standard deviation of the slope.(c) What is the value of the t-statistic for the
Recall the regression of percent body fat on BMI from Exercise 11-1.(a) Estimate the error standard deviation.(b) Estimate the standard deviation of the slope.(c) What is the value of the t-statistic
Suppose that we wish to fit a regression model for which the true regression line passes through the point (0, 0). The appropriate model is Y = Î²x + Îµ. Assume that we have n
Consider the regression model developed in Exercise 11-8. Suppose that engine displacement is measured in cubic centimeters instead of cubic inches.(a) Write the new regression model.(b) What change
In an article in Statistics and Computing [€œAn Iterative Monte Carlo Method for Nonconjugate Bayesian Analysis€ (1991, pp. 119€“128)], Carlin and Gelfand investigated
An article in Wood Science and Technology [€œCreep in Chipboard, Part 3: Initial Assessment of the Influence of Moisture Content and Level of Stressing on Rate of Creep and Time to
An article in the Journal of the Environmental Engineering Division [€œLeast Squares Estimates of BOD Parameters€ (1980, Vol. 106, pp. 1197€“1202)] took a sample from
An article in the Journal of the American Ceramic Society [€œRapid Hot-Pressing of Ultrafi ne PSZ Powders€ (1991, Vol. 74, pp. 1547€“1553)] considered the
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
Table E11-3 presents the highway gasoline mileage performance and engine displacement for DaimlerChrysler vehicles for model year 2005 (U.S. Environmental Protection Agency).(a) Fit a simple linear
An article in Technometrics by S. C. Narula and J. F. Wellington [€œPrediction, Linear Regression, and a Minimum Sum of Relative Errors€ (1977, Vol. 19)] presents data on the
See Table E11-1 for data on the ratings of quarterbacks for the 2008 National Football League season (The Sports Network). It is suspected that the rating (y) is related to the average number of
An article in Concrete Research [“Near Surface Characteristics of Concrete: Intrinsic Permeability” (1989, Vol. 41)] presented data on compressive strength x and intrinsic permeability y of
On average, do people gain weight as they age? Using data from the same study as in Exercise 11-1, we provide some summary statistics for both age and weight.(a) Calculate the least squares estimates
Diabetes and obesity are serious health concerns in the United States and much of the developed world. Measuring the amount of body fat a person carries is one way to monitor weight control progress,
Diabetes and obesity are serious health concerns in the United States and much of the developed world. Measuring the amount of body fat a person carries is one way to monitor weight control progress,
Derive an expression for β for the test of the equality of the variances of two normal distributions. Assume that the two-sided alternative is specified.
In some situations involving proportions, you are interested in the ratio θ = p1 / p2 rather than the difference p1 – p2. Let θ = p1 / p2. It can be shown that ln(θ) has an approximate normal
Construct a data set for which the paired t-test statistic is very large, indicating that when this analysis is used, the two population means are different, but t0 for the two-sample t-test is very
Suppose that you wish to test H0: μ = μ0 versus H1: μ ≠ μ0, where the population is normal with known σ. Let 0 < ε < α, and define the critical region so that you will reject H0 if z0
Suppose that you wish to test the hypothesis H0: μ1 = μ2 versus H1: μ1 ≠ μ2, where both variances σ21 and σ22 are known. A total of n1 + n2 = N observations can be taken. How should
Suppose that you wish to test H0: l1 = l2 versus H1: μ1 ≠ μ2, where σ21 and σ22 are known. The total sample size N is to be determined, and the allocation of observations to the two populations
Three different pesticides can be used to control pest infestation of grapes. It is suspected that pesticide 3 is more effective than the other two. In a particular vineyard, three different
Consider the unlisted telephone number data in Exercise 10-101. Find 95% CIs on the difference in the proportions of unlisted telephone numbers for Phoenix and Scottsdale residents using both
Consider the seat belt usage data in Exercise 10-102. Find 95% CIs on the difference in the proportions of seat belt usage for drivers of foreign and domestic cars using both procedures described in
Consider the grinding force data in Exercise 10-40. Is there any indication that the variances of grinding force are different at the two different levels of temperature?Exercise 10-40An article in
Consider the foam thickness data from Exercise 10-39. Is there any indication that the variances of foam thickness are different at the two different levels of temperature?Exercise 10-39An article in
Consider the situation described in Exercise 10-87.(a) Redefine the parameters of interest to be the proportion of lenses that are unsatisfactory following tumble polishing with polishing fluids 1 or
Two machines are used to fill plastic bottles with dishwashing detergent. The standard deviations of fill volume are known to be σ1 = 0.10 fluid ounces and σ2 = 0.15 fluid ounces for the two
Consider the following computer output.(a) Is this a one-sided or a two-sided test?(b) Fill in the missing values. You may use bounds for the P-value.(c) What are your conclusions if Î±
Consider the following computer output.(a) Fill in the missing values. You may use bounds for the P-value.(b) Is this a two-sided test or a one-sided test?(c) What are your conclusions if
Consider the highway speed limit data introduced in Exercise 10-88. Find a 99% CI on the difference in the two proportions using the alternate CI procedure described in this section. Compare the
Rework the election data reported in Exercise 10-85 using the alternate CI procedure described in this section. Compare the lengths of the CI from Exercise 10-85 with this one. Discuss the possible
The New England Journal of Medicine reported an experiment to judge the efficacy of surgery on men diagnosed with prostate cancer. The randomly assigned half of 695 (347) men in the study had
Air pollution has been linked to lower birthweight in babies. In a study reported in the Journal of the American Medical Association, researchers examined the proportion of low-weight babies born to
A random sample of 500 adult residents of Maricopa County indicated that 385 were in favor of increasing the highway speed limit to 75 mph, and another sample of 400 adult residents of Pima County
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,
In the 2004 presidential election, exit polls from the critical state of Ohio provided the following results: For respondents with college degrees, 53% voted for Bush and 46% voted for Kerry. There
An article in Knee Surgery, Sports Traumatology, Arthroscopy (2005, Vol. 13, pp. 273–279) considered arthroscopic meniscal repair with an absorbable screw. Results showed that for tears greater
Consider the following computer output.(a) Is this one-sided or a two-sided test?(b) Fill in the missing values.(c) Can the null hypothesis be rejected if = 0.10? What if = 0.05? Test and CI for Two
Consider the following computer output.(a) Is this a one-sided or a two-sided test?(b) Fill in the missing values.(c) Can the null hypothesis be rejected?(d) Construct an approximate 90% CI for the
Is there evidence to suggest that the standard deviations of the algae concentrations in the two types of rivers (flow rates) in Exercise 10-36 differ? Use α = 0.05.
Consider the swimming data in Exercise 10-37. Is there evidence to suggest that the standard deviations of the heats differ for slower swimmers in heat five and the faster swimmers in heat seven?
Consider the etch rate data in Exercise 10-23.(a) Test the hypothesis H0: Ïƒ21 = Ïƒ22 against H1: Ïƒ21 ‰ Ïƒ22 using Î± = 0.05, and draw
Consider the film speed data in Exercise 10-26.(a) Test H0: σ21 = σ22 versus H1: σ21 ≠ σ22 using α = 0.02.(b) Suppose that one population standard deviation is 50% larger than the other. Is
Consider the weight of paper data from Technometrics in Exercise 10-32. Is there evidence that the variance of the weight measurement differs for the sheets of paper? Use Î± = 0.05.
Reconsider the coefficient of restitution data in Exercise 10-34. Do the data suggest that the standard deviation is the same for both brands of drivers (use Î± = 0.05)? Explain how to
Reconsider the overall distance data for golf balls in Exercise 10-33. Is there evidence to support the claim that the standard deviation of overall distance is the same for both brands of balls (use
Exercise 10-28 presented measurements of plastic coating thickness at two different application temperatures. Test H0: σ21 = σ22 against H1 : σ21 ≠ σ22 using α = 0.01.Exercise 10-28The
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 α = 0.05 in reaching your conclusion.Exercise
Consider the gear impact strength data in Exercise 10-24. Is there sufficient evidence to conclude that the variance of impact strength is different for the two suppliers? Use α = 0.05.Exercise

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