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Probability And Statistics For Engineers And Scientists 3rd Edition Anthony Hayter - Solutions

Electrical capacitors can be used to deliver short high-voltage pulses. As the capacitor discharges, the time span of the resulting pulse is an important property. DS 12.11.5 contains measurements of the time spans of the discharges, in milliseconds, obtained for different capacitance values, in
DS 12.11.6 contains the circumferential strains 5 and lesion areas L of n = 12 specimens of human arteries with atherosclerosis. This disease is commonly referred to as ''hardening of the arteries"" due to the accumulation of plaque on the artery walls. The lesion area measures the amount of plaque
DS 12.11.7 contains data on the measurements of the strength of a chemical solution for different amounts of a catalyst. It is decided to perform a linear regression analysis with strength as the dependent variable and amount of catalyst as the explanatory variable.(a) Calculate the values of n
In a simple linear regression analysis with n = 20 data points, the estimates Î²^0 = 123.57, Î²^1 = -3.90. and(a) Construct a two-sided 95% prediction interval for a future response value when the input value is 40. (b) Compute the analysis of variance table and calculate the
A simple linear regression model is to be fitted to the data set of temperature and yield values for bacteria cultivation in DS 12.11.8. with temperature as the explanatory variable and yield as the dependent variable.(a) Calculate β^0 β^1, fa, and σ^2.(b) Is there sufficient evidence to
A data set has n = 20.∑ 20i=1 xi = 8.552, ∑20i=1yi, = 398.2, ∑20i=1 x2i, = 5.196. ∑20i =1 y2i = 9356 and ∑20i=1 = 216.6. Calculate ^β0, ^β1, and ^σ2. What is the fitted value when x = 0.5?
A data set has n = 30,∑30i=1, x i = -67.11. ∑30i=1 1322.7, ∑30i=1 x2l = 582.0, ∑30i=1 y2i = 60,600 and ∑30i=1 xiy1 = -3840. Calculate β^0 β^1 and σ^2. What is the fitted value when x = -2.0?
Estimating the costs of drilling oil wells is an important consideration for the oil industry. DS 12.2.1 contains the total costs and the depths of 16 offshore oil wells located in the Philippines (taken from "Identifying the major determinants of exploration drilling costs: A first approximation
A warehouse manager is interested in the possible improvements to labor efficiency if air-conditioning is installed in the warehouse. The data set given in DS 12.2.2 is collected which shows the times taken to unload a fully laden truck at various temperature levels. (a) Fit a linear regression
DS 12.2.3 is a data set concerning the aerobic fitness of a sample of 20 male subjects. An exercising individual breathes through an apparatus that measures the amount of oxygen in the inhaled air that is used by the individual. The maximum value per unit time of the utilized oxygen is then scaled
A Realtor collects the data set given in DS 12.2.4 concerning the sizes of a random selection of newly constructed houses in a certain area together with their appraised values for tax purposes. (a) Fit a linear regression model with appraised value as the dependent variable and size as the
During the installation of a large computer system, it is useful to know how long specific tasks will take, particularly programming changes. A great deal of effort is spent estimating the amount of time such tasks will take and learning how to effectively use such estimations. Having an accurate
The data set in DS 12.2.6 concerns the relationship between the temperature and resistance of vacuum transducer bobbins, which are used in the automobile industry. (a) Fit a linear regression model with resistance as the dependent variable and temperature as the explanatory variable. (b) What is
In a simple linear regression analysis with n = 18 data points, an estimate β^1 = 0.522 is obtained with s.e.(β^1) = 0.142. (a) Calculate a two-sided 99% confidence interval for the slope parameter β1. (b) Test the null hypothesis H0: β1 =0 against a two-sided alternative hypothesis.
Several samples of ceramic were made with different baking times. The densities of the samples were measured and the data set given in DS 12.3.1 was obtained. Use a simple linear regression model to investigate whether there is any evidence that the density of the ceramic is affected by the baking
In a simple linear regression analysis with n = 22 data points, an estimate β^1 = 56.33 is obtained with s.e.(β^1) = 3.78. (a) Calculate a two-sided 95% confidence interval for the slope parameter β1s. (b) Test the null hypothesis H0: β1 = 50.0 against a two-sided alternative hypothesis
Consider the data set of oil well drilling costs given in DS 12.2.1. (a) What is the standard error of β^1? (b) Construct a two-sided 95% confidence interval for the slope parameter β1. (c) Test the null hypothesis H0: β1 = 0. Interpret your answers.
Consider the data set of the times taken to unload a truck at a warehouse given in DS 12.2.2.(a) What is the standard error of β1?(b) Construct a two-sided 90% confidence interval for the slope parameter β1.(c)Test the null hypothesis H0: β1 = 0.(d) Does your analysis indicate that there is
Consider the data set of aerobic fitness measurements given in DS 12.2.3. (a) What is the standard error of β1? (b) Construct and interpret a one-sided 95% confidence interval for the slope parameter β1 that provides an upper bound. (c) Test the null hypothesis H0: β1 = 0. Is it clear that on
Consider the data set of appraised house values given in DS 12.2.4.(a) What is the standard error of β1?(b) Construct a two-sided 99% confidence interval for the slope parameter β1.(c) Test the null hypothesis H0: β1 = 0. Interpret your answers.
Consider the data set of the times taken for programming changes given in DS 12.2.5.(a) What is the standard error of β1?(b) Construct a two-sided 95% confidence interval for the slope parameter β1.(c) Why is the null hypothesis H0: β1 - 1 of particular interest? Test this null hypothesis.
Consider the data set of vacuum transducer bobbin resistances given in DS 12.2.6. (a) What is the standard error of β^1? (b) Construct a two-sided 99% confidence interval for the slope parameter β^1. (c) Test the null hypothesis H0: β^1 = 0. Is it clear that resistance increases with temperature?
A simple linear regression is performed on 20 data pairs (xj, yi. It is found that β^1 = 54.87 and s.e. (β^1) = 21.20. Is the p-value for the two-sided hypothesis test of H0: β1 = 0 (a) greater than 10%, (b) between 1% and 10%,or(c) less than 1%?
A linear regression model is fitted to the data X y 37.0 65.0 36.4 67.2 35.8 70.3 34.3 71.9 33.7 73.8 32.1 75.7 31.5 77.9 with x as the input variable and v as the output variable. Find β^0, β^0, and σ^2. Construct a 99% confidence interval for the expected value of the output
A simple linear regression analysis is performed to determine how the strength of a substance depends on its density. Fight samples of the substance are prepared, with densities 11.2. 12.5. 13.4, 14.9, 16.0, 16.6. 17.9, and 20.1, and their strengths are found. The estimates β^0 = 75.32, β^1 =
The amount of catalyst (x) and the yield (y) of a chemical experiment are analyzed using a simple linear regression model. There are 30 observations (xi,yl), and it is found that the fitted model is y = 51.98 + 3.44x Suppose that the sum of squared residuals is ∑30i=1 e2i = 329.77, and that
A linear regression model is fitted to the datawith x as the input variable and y as the output variable. Find Î²^0 , Î²^1 and Ïƒ^2. Construct a 99% prediction interval for an observation of the output variable when the input variable is equal to 20.
Calculate the missing values in the analysis of variance table for a simple linear regression analysis shown in Figure 12.36. What is the p-value? What is the coefficient of determination R2?
Consider the data set of vacuum transducer bobbin resistances given in DS 12.2.6. Compute the analysis of variance table and calculate the coefficient of determination R2. Check that the F-statistic is the square of the t-statistic for testing H0: β1 = 0, calculated earlier.
Repeat Problem 12.6.1 for the analysis of variance table shown in Figure 12.37.
A data set has n = 10, ∑10i=1 y1, = 908.8. ∑10i=1 y2i = 83,470, and σ^2 = 0.9781. Compute the analysis of variance table and calculate the coefficient of determination R2.
A data set has n = 25, ∑25i=1 = 1356.25,∑25i=1 yi, = -6225, x2i = 97.025, ∑25i=1 y2i = 10.414.600, and ∑25i=1 = -738,100. Compute the analysis of variance table and calculate the coefficient of determination R2.
Consider the data set of oil well costs given in DS 12.2.1. Compute the analysis of variance table and calculate the coefficient of determination R2. Check that the f -statistic is the square of the t-statistic for testing H0: β1 = 0, calculated earlier.
Consider the data set of the times taken to unload a truck at a warehouse given in DS 12.2.2. Compute the analysis of variance table and calculate the coefficient of determination R2. Check that the F-statistic is the square of the F-statistic for testing H0: β1 = 0, calculated earlier. What is
Consider the data set of aerobic fitness measurements given in DS 12.2.3. Compute the analysis of variance table and calculate the coefficient of determination R2. Explain the interpretation of the coefficient of determination.
Consider the data set of appraised house values given in DS 12.2.4. Compute the analysis of variance table and calculate the coefficient of determination R2. Check that the F-statistic is the square of the F-statistic for testing H0: β1 = 0, calculated earlier. What does the value of the
Consider the data set of the times taken for programming changes given in DS 12.2.5. Compute the analysis of variance table and calculate the coefficient of determination R2. Is the p-value in the analysis of variance table meaningful?
Make a plot of the data set given in DS 12.8.1. What intrinsically linear function should provide a good model for this data set? What transformation of the variables is needed? Fit a straight line to the transformed variables and write the fitted model back in terms of the original variables. What
Repeat Problem 12.8.1 for the data set given in DS 12.8.2.
A bioengineer measures the growth rate of a substance by counting the number of cells N present at various times as shown in DS 12.8.3. Fit the model N = γ0e γ |t and calculate two-sided 95c/c confidence intervals for the unknown parameters γ0) and γ1.
In an experiment to investigate the suitability of using a silicone tube to model the behavior of a human artery, the data set in DS 12.8.4 is collected, which relates the pressure differential P across the walls of the tube to the cross-sectional area A of the tube. (a) Show that the model P =
An experimenter has data on the yield and the temperature of a chemical process and wishes to fit the model yield = γ0eγ1 temperature A linear regression model is fitted to the data y = ln(yield) and x = temperature, with the results n = 25. β^0 = 2.628. β^1 = 0.341, and s.e. (β^1) = 0.025.
12.8.6 Explain how simple linear regression can be used to fit the model eγ/γ0 = γ1/x2. How would you find the parameter estimates γ^0 and γ^1?
A crack in a plastic compound affects the strength of the material. In order to investigate the relationship between the strength and the length of a crack, an experiment was conducted where cracked pieces of the plastic compound were subjected to increasing loads until the point at which the)
Suppose that variables A and B have a positive correlation. Provide a possible explanation for the claim that changes in B do not cause a change in A.
The multiple linear regression model y = β0 + β1x1 + β2x2 + β3x3 is fitted to a data set of n = 30 observations. The total sum of squares is SST = 108.9, and the error sum of squares is SSE = 12.4. (a) What is the coefficient of determination R2? (b) Write out the analysis of variance
The multiple linear regression model y = β0 + βlxl +β2x2 + β3x3 is fitted to a data set with n = 15 observations, and the parameter estimates β^0 = 65.98. β^1 = 23.65, β^2 = 82.04, and β^3, = 17.04 are obtained. (a) What is the fitted value of the expected value of the response variable
The modelY = Î²0 + Î²1x1 + Î²2x2 + Î²3x3is fitted to n = 20 data observations. Suppose thatConstruct the ANOVA table and put bounds on the p-value. What hypothesis is being tested by the p-value? What proportion of the variability of the y variable is
The multiple linear regression model y = β0 + β1x1 + β2x2 + β3x3 + β4x4 is fitted to a data set of n = 22 observations. The total sum of squares is 45.76; the error sum of squares is 23.98. (a) What is the coefficient of determination R2? (b) Write out the ANOVA table. (c) What is the estimate
The multiple linear regression model V = β0 + β1x1 + β2x2 + β3x3 + β4x4 + β5x5 + β6x6 is fitted to a data set of n = 45 observations. The total sum of squares is SST = 11.62. and the error sum of squares is SSE = 8.95. (a) What is the coefficient of determination R2? (b) Write out the
The multiple linear regression model y = β0 + β1x1 + β2x2 + β3x3 is fitted to a data set of n = 12 observations. (a) If β^12 = 132.4 and s.e.( β^1) = 27.6, construct a two-sided 95% confidence interval for (b) What is the p-value for the null hypothesis H0: β1 = 0?
The multiple linear regression model y = β0 + β1x1 + β2x2 + β3x3 is lined to a data set of n = 15 observations. (a) If β^1 = 0.954 and s.e. (β^1) = 0.616, construct a two-sided 95% confidence interval for β^1. (b) What is the p-value for the null hypothesis H0 : β1 = 0?
The multiple linear regression model y = β0 + β1x1 + β2x2 + β3x3 is fitted to a data set of n = 44 observations. The parameter estimates β1 = 11.64, p% = 132.9, and β3 = 0.775 are obtained with standard errors s.e.(β^1) = 1.03, s.e.( β^2) = 22.8, and s.e.( β^3) =0.671. Should any of the
The multiple linear regression model y = β0 + β1x1 + β2x2 is fitted to a data set with n = 20 observations, and the parameter estimates β^0 = 104.9, P\ = 12.76, and β^1 = 409.6 are obtained. (a) What is the fitted value of the expected value of the response variable when x1 = 10 and x2 =
The data set given in DS 13.2.1 concerns the sales volume of a company, the price at which the company sells its product, and the price of a competing product for each of n = 10 quarter periods. Use this data set to fit the multiple linear regression modely = β0 + β1x1 + β2x2with the response
Two polymers are ingredients in the manufacture of a synthetic fiber. The data set given in DS 13.2.2 shows the results of an experiment conducted to measure the fiber strength resulting from different concentrations of the two polymers. The polymer concentrations examined represent their standard
Consider again the problem of estimating the costs of drilling oil wells, which was originally discussed in Problem 12.2.4. The data set given in DS 13.2.3 contains the variables geology, downtime, and rig-index in addition to the variables depth and cost considered before. The variable geology is
The data set given in DS 13.2.4 extends the data set used in Problem 12.2.6 to include an individual's heart rate at rest, percentage body fat and weight, together with the variables age and V02-max considered before. Fit a multiple linear regression model to assess whether the new input variables
A categorical input variable has three levels. How can indicator variables be used to include it in a multiple linear regression model?
Consider fitting the multiple linear regression modely = Î²0 + Î²1x1, + Î²2x2to the data set in DS 13.3.1.(a) What is the 10 Ã— 1 vector of observed values of the response variable Y?(b) What is the 10 Ã— 3 design matrix X?(c) What is the 3
Consider fitting the multiple linear regression model y = B0 + fax\ + 32x2 to the data set in DS 13.3.2. a. What is the 8 × 1 vector of observed values of the response variable Y? b. What is the 8 × 3 design matrix X? c. What is the 3 × 3 matrix X'X? d. What is the 3 × 3 matrix (X'Xr1? e.What
The multiple linear regression model y = β0 + β1x1 + β2x2 + β3x3 is fitted to the data set in DS 13.3.3. Use matrix algebra to derive the parameter estimates β^0, β^1,β^2, and β^3
Consider Example 70 and the data set in figure 13.7. (a) Make a plot of the residuals c, against the lilted values y,. Make a plot of the residuals c, against the temperature values v,. Do either of these plots alert you to any problems with the regression analysis? (b) bind the standardized
Consider the modeling of oil well drilling costs described in Problem 13.2.3 and the data set in DS 13.2.3. Suppose that a model is used with cost as the dependent variable and with depth and downtime as input variables, (a) Make plots of the residuals against the tilled values and against each of
Consider the modeling of aerobic fitness described in Problem 13.2.4 and the data set in DS 13.2.4. Suppose that a model is used with V02-max as the dependent variable and with heart rate at rest and percentage body fat as input variables.(a) Make plots of the residuals against the lilted values
Consider the data set in DS 13.6.1. (a) Plot the response variable y against the input variable x and confirm that the quadratic model y = β0 + β1x + β2x2 appears to be appropriate. (b) Write out the analysis of variance table using the fact that SSE = 39.0. (c) The parameter estimates are β^0)
Use hand calculations to fit the multiple linear regression model 1y = Î²0 + Î²1x1 + Î²2x2to the data set in DS 13.6.2.(a) Write down the vector of observed values of the response variable Y and the design matrix X.(b) Calculate X'X.(c) Verify that(d) Verify that
DS 13.6.3 contains an extension of the data set presented in DS 12.11.3 concerning the power loss that occurs in the bearing of an automobile engine. In addition to the diameter of the bearing, information is provided on the clearance and the length of the bearing. What is the fitted model when the
The data set in DS 13.6.4 shows the yields of a bacteria culture obtained for different amounts of an additive x1 and for different growing temperatures x2.(a) Investigate the experimental design employed by looking at the values of the additive levels and the temperature levels used in the
The regression model y = -67.5 + 34.5x1, - 0.44x2 + 108.6x3 + 55.8x4 is obtained from n = 44 observations. The first observation is y1 = 288.9. x1 = 12.3. x2 = 143.4. x3 = -7.2, x4 = 14.4 and it has a standardized residual - 1.98 and a leverage value 0.0887. If SST = 20554, what is R2?
The multiple linear regression modely - β0 + β1x1 + β2x2is litted to a data set of n = 30 observations.(a) Suppose that β^1 = -45.2 and s.e.(β^1) = 39.5, and that β^2 = 3.55 and s.e.( β^2) = 5.92. Is it clear that both x1 and x2: should be removed from the model?(b) Suppose that β^1 = -45.2
A two-factor experiment is conducted to compare different mixes of gasoline in different types of car. The response variable measures the driving characteristics of the car based on acceleration and other properties relating to the gasoline. Larger values of the response variable relate to improved
A two-factor experiment is conducted to investigate how the tensile strength of a graphite-epoxy composite depends upon the formulation of the composite and the temperature of the composite. Four different composite formulations are considered in the experiment and they are taken to be factor A.
The data set in DS 14.1.1 shows the results of an experiment performed to test the machinery used to measure the hardness of a material. The hardness of a material is measured by pushing a tip into the material with a specified force. The depth h of the resulting indentation is then measured. A
The data set in DS 14.1.2 shows the results of an experiment performed to investigate how the deviation from specification of the width of a contact lens at its center point may depend on the type of material used to make the lens and the amount of magnification provided by the lens.(a) Construct
When a specimen of glass is placed in a solution, leaching occurs whereby some of the constituents of the glass are absorbed into the solution. DS 14.1.3 contains the results of a two-factor experiment to investigate how the glass leaching depends on the glass composition and the acidity of the
The amount of improvement of a genetic disease is measured for 18 patients who are randomly assigned to the nine experimental configurations, with two patients to each configuration, corresponding to three dosage levels of each of two active ingredients of a drug. The experimental results are given
A mechanical component can be made using three different designs and from two different material types. Six components are manufactured according to the six possible combinations of design and material, and DS 14.1.5 contains the lifetimes in hours of these components when they are employed. What
A three-factor experiment is performed to taste test different blends of a fruit juice mixture. The response variable scores how the participant liked the fruit juice based on the answers to a questionnaire. Factor A is the blend of fruit juice with a = 3 levels. Factor B is the gender of the
A three-factor experiment is conducted to investigate the yields of different brands of rice grown in controlled greenhouse conditions. The response variable is the rice yield, and factor A is rice variety with a = 3 levels. Two levels of fertilizer are considered (factor B) together with two
DS 14.2.2 contains the data set collected from a three-factor experiment to investigate how gas mileage depends on the amounts of two types of additive in the gasoline and the driving conditions.Construct the analysis of variance table. How do the three factors influence the gas mileage?
DS 14.2.3 contains the data from a three-factor experiment to investigate the distance at detection for four different radar systems of two different aircraft flying at day and at night. Construct the analysis of variance table and summarize what you learn from it.
DS 14.2.4 contains the results of a 24 experiment conducted to investigate the accuracies of two digital weighing machines. The response variable is the deviation of the weight reading from the true weight of a particular object that is used throughout the experiment. The four factors are the
DS 14.2.5 contains the results of a 24 experiment conducted to investigate the driving distances of two golfers using different clubs and balls and under different weather conditions. What do you learn from this experiment? Is there any evidence that the weather conditions have any effect on the
An ANOVA table for an experiment with four factors A. B. C and D gave the p-values shown in Figure 14.37. Classify each term as being "significant," "not-significant," or "redundant."
A three-factor experiment (3 × 2 × 2) gave the following data:When factor C is lowA lowA middleA highBlow504755403847When factor C is highA lowA middleA highB low485172B high424064Suppose that there is no three-way interaction effect, but that there is an A*C two-way
In an injection molding procedure plastic parts are produced by injecting the plastic material into a mold. Some variability is experienced in the resulting weight (or equivalently density) of the part. DS 14.3.1 contains the results of a two-factor experiment to investigate how the weight of the
Semiconductors (chips) are produced on wafers that contain 100 chips. The wafer yield is defined to be the proportion of these chips that are acceptable for use. One company has two different factory locations for producing chips and can use one of three different coatings for the wafers. The data
DS 14.3.3 contains the recovery times in days for 16 patients allocated at random to a two-factor experiment to compare four drugs and two levels of severity of the illness. Analyze the data set and summarize your conclusions.
A three-factor experiment is performed to investigate how the hardness of a metal bar depends on which furnace is used to manufacture the bar and the location of the bar in the furnace. The response variable is the hardness of the metal, and one of the factors designates which of two furnaces is
DS 14.3.5 contains the results of a 2' experiment that investigates the preparation of polymers by dispersion polymerization in an organic medium. The response variable is the mean diameter of the polymers in microns (m-6). The four factors are monomer concentration, stabilizer concentration,
In the production of low-pressure gas cylinders a lathe operator has to take the bottom and top parts of a cylinder, tit them with a footling and a collar, and then weld the halves together. An experiment was conducted to investigate the time taken lo complete this process. Two different lathe
A company that manufactures large electric motors has to pay attention to the noise levels produced by the motors. Measurements of noise levels were obtained for three different motor speeds. These measurements were made at two positions, in front of the machine and at the side of the machine, and
(a) Is it plausible that the service times are normally distributed with a mean of 70 seconds ami a standard deviation of 20 seconds'.' (b) Is it plausible that the service times are exponentially distributed with a mean of 70 seconds? (c) Consider the null hypothesis that the median service time
Use the sign test and the signed rank test to analyze the paired data set given in DS 9.2.5 concerning the radioactive carbon dating methods. Do you find any evidence of a difference between the two dating methods.?
Use the sign test and the signed rank test to analyze the paired data set of golf shots given in DS 9.2.6. Do you find any evidence of a difference between the two ball types?
(a) Is it plausible that the paving slab weights are normally distributed with a mean of 1.1 kg and a standard deviation of 0.05 kg? How about with a mean of 1.0 kg and a standard deviation of 0.05 kg? (b) Consider the null hypothesis that the median paving slab weight is 1.1kg. What statistic is
Construct the empirical cumulative distribution function for the data set of paint thicknesses given in DS 6.1.8. Draw 95% confidence bands around the empirical cumulative distribution function. Analyze the median (or mean) paint thickness using the sign lest, the signed rank test, and the t-test.
Construct the empirical cumulative distribution function for the data set of deformity angles given in DS 6.1.9. Draw 95% confidence bands around the empirical cumulative distribution function. Analyze the median (or mean) deformity angle using the sign test, the signed rank test, and the t-test.
Suppose that the data set in DS 15.1.1 consists of values that can be taken to be independent observations from a particular distribution. Consider testing whether the median of the distribution is equal to 18.0 against a two-sided alternative. (a) What is the value of the test statistic used by

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