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Probability & Statistics For Engineers & Scientists 7th Edition Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying - Solutions
Using the data from Exercise 12.28, test, at level 0.05 (a) Ho: β1 = 0 versus H1: β1 ≠ 0; (b) Ho: β2 = 0 versus H1: β2 ≠ 0. (c) Do you have any reason to believe that the model in Exercise 12.28 should be changed? Why or win not?
Using the data Exercise 12.16,(a) Estimate σ2 using the multiple regression of y on x1, x2, and x3.(b) Compute a 95% prediction interval for the observed device gain for three regressor at x1 = 15.0, x2 = 220.0, and x3 = 6.0.
Compute and interpret the coefficient of multiple determinations for the variables of Exercise 12.3.
Test whether the regression explained by the model in Exercise 12.3 is significant at the 0.01 level of significance.
Test whether the regression explained by the model in Exercise 12.9 is significant at the 0.01 level of significance.
For the model of Exercise 12.11 test the hypothesis H0: β1 = β2 = 0, H1: Β1 and β2 are not both zero.
Repeat Exercise 12.26 using an F-statistic.
A small experiment was conducted to fit a multiple regression equation relating the yield y to temperature x1, reaction time x2, and concentration of one of the reactants 23. Two levels of each variable were chosen and measurements corresponding to the coded independent variables were recorded as
Consider the electric power data of Exercise 12.9. Test H0: β1 = 02 = 0 making use of R (β1, β2β3, β4). Give a P-value, and draw conclusions.
Consider the data for Exercise 12.36. Compute the followingsComment.
Consider the data of Exercise 11.63. Fit a regression model using weight and drive ratio as explanatory variables. Compare this model with the SLR (simple linear regression) model using weight alone. Use R2, R2adj, and any t- (or F)-statistics you may need to compare the SLR with the multiple
Consider Example 12.3. Figure displays a SAS printout of an analysis of the model containing variables x1, x2, and x3. Focus on the confidence interval of the mean response μ y at the (x1, x2, x3) locations representing the 13 data points. Consider an item in the printout indicated by C.V.
Consider Example 12.4 on page 451. Compare the two competing models Use R2adj in your comparison in addition to testing Ho: ?11 = ?22 = ?12 = 0. In addition, use C.V. discussed in Exercise 12.40.
In Example 12.8 a case is made for eliminating x\, powder temperature, from the model since the P-value based on the F-test is 0.2154 while P-values for x2 and x3 are near zero.(a) Reduce the model by eliminating xi, thereby producing a full and restricted (or reduced) model and compare them on the
Consider the data of Exercise 12.15. Can the response, wear, be explained adequately by a single variable (either viscosity or load) in an SLR rather than with the full two-variable regression? Justify your answer thoroughly through tests of hypotheses as well as the comparison of three competing
For the data set given in Exercise 12.16, can the response be explained adequately by any two regressor variables? Discuss.
A study was done to assess the cost effectiveness of driving a four door sedan instead of a van or an SUV (sports utility vehicle). The continuous variables are odometer reading and octane of the gasoline used, the response variable is miles per gallon. The data are presented here.(a) Fit a linear
A study was done to determine whether gender of the credit card holder was an import ant factor in generating profit for a certain credit card company. The variables considered were income, the number of family members, and gender of the card holder. The data are as follows:(a) Fit a linear
Consider the "hang time" punting data given in Example 12.13, using only the variables x2 and x3.(a) Verify the regression equation shown.(b) Predict punter hang time for a punter with LLS = 180 pounds and Power = 260 foot-pounds.(c) Construct a 95% confidence interval for the mean hang time of a
For the data of Exercise 12.11 use the techniques of(a) Forward selection with a 0.05 level of significance to choose a linear regression model;(b) Backward elimination with a 0.05 level of significance to choose a linear regression model;(c) Stepwise regression with a 0.05 level of significance to
Use the techniques of backward elimination with ? = 0.05 to choose a prediction equation for the data of Table 12.8.
For the punter data in Example 12.13, an additional response, "punting distance," was also recorded. The following are average distance values for each of the 13 punters:(a) Using the distance data rather than the hang times, estimate a multiple linear regression model of the type μ Y׀x1, x2, x3,
The following is a set of data for y, the amount of money (thousands of dollars) contributed to the alumni association at Virginia Tech by the Class of 1960, and x, the number of years following graduation: (a) Fit a regression model of the type ? Y?x = 00 + 01X. (b) Fit a quadratic model of the
For the model of Exercise 12.50(a) test the hypothesis Ho: β4 = 0, H1: β4 ≠ 0. Use a P-value in your conclusion.
For the quadratic model of Exercise 12.51(b), give estimates of the variances and covariance’s of the estimates of 0 and β11.
In an effort to model executive compensation for the year 1979, 33 firms were selected, and data were gathered on compensation, sales, profits, and employment. Consider the model (a) Fit the regression with the model above. (b) Is a model with a subset of the variables preferable to the full model?
Rayon whiteness is an important factor for scientists dealing in fabric quality. Whiteness is affected by pulp quality and other processing variables. Some of the variables include acid bath temperature, ?C (x1); cascade acid concentration, % (x2); water temperature, ?C (x3); sulfide concentration,
A client from the Department of Mechanical Engineering approached the Consulting Center at Virginia Polytechnic Institute and State University for help in analyzing an experiment dealing with gas turbine engines. Voltage output of engines was measured at various combinations of blade speed and
The pull strength of a wire bond is an important characteristic. The table below gives information on pull strength y, die height x1, post height x2, loop height x3, wire length x4, bond width on the die x5, and bond width on the post x6. [Data from Myers and Montgomery (2002)] (a) Fit a regression
For exercise 12.57, test H0: β1 = β6 = 0. Give P-values and comment.
In exercise 12.28, we have the following data concerning wear of a bearing: (a) The following model may be considered to describe this data: y i = 00 + ?1xl, + ?2x2 + ?12x1,?x2i + ? i, for i = 1, 2... 6. The %X2 is an "interaction"' term. Fit this model and estimate the parameters. (b) Use the
In the Department of Fisheries and Wildlife at Virginia Polytechnic Institute and State University, an experiment was conducted to study the effect of stream characteristics on fish biomass. The regressor variables are as follows: average depth (of 50 cells) (x1): area of in-stream cover (i.e.,
Show that, in a multiple linear regression dataset,
A small experiment is conducted to fit a multiple regression equation relating the yield y to temperature x1, reaction time x2, and concentration of one of the reactants x3. Two levels of each variable were chosen and measurements corresponding to the coded independent variables were recorded as
In a chemical engineering experiment dealing with heat transfer in a shallow fluidized bed, data are collected on the following four regressor variables: fluidizing gas flow rate, lb/hr (x1); supernatant gas flow rate, lb/hr (x2); supernatant gas inlet nozzle opening, millimeters (as); supernatant
In exercise physiology, an objective measure of aerobic fitness is the oxygen consumption in volume per unit body weight per unit time. Thirty-one individuals were used in an experiment in order to be able to model oxygen consumption against: age in years (x1); weight in kilograms (x2); time to run
Consider the data of Review Exercise 12.62. Suppose it is of interest to add some "interaction" terms. Namely, consider the model(a) Do we still have orthogonality? Comment(b) With the fitted model in part (a), can you find prediction intervals and confidence intervals on the mean response? Why or
A carbon dioxide (CO2) flooding technique is used to extract crude oil. The CO2 floods oil pockets and displaces the crude oil. In the experiment, flow tubes are dipped into sample oil pockets containing a known amount of oil. Using three different values of flow pressure and three different values
An article in the Journal of Pharmaceutical Sciences (Vol. 80, 1991) presents data on the mole fraction solubility of a solute at a constant temperature. Also measured are the dispersion x1, and dipolar and hydrogen bonding solubility parameters x2, and x3. A portion of the data is shown in the
A study was conducted to determine whether lifestyle changes could replace medication in reducing blood pressure among hypertensives. The factors considered were a healthy diet with an exercise program, the typical dosage of medication for hypertension, and no intervention. The pretreatment body
Show that in choosing the so-called best subset model from a series of candidate models, if the model is chosen that has the smallest s2, this is equivalent to choosing the model with the smallest R2adj.
From a set of streptonignic dose-response data, an experimenter desires to develop a relationship between the proportions of lymphoblasts sampled that contain aberrations and the dosage of streptonignic. Five dosage levels were applied to the rabbits used for the experiment. The data are See Myers,
In an experiment to ascertain the effect of load, x, in lb/inches2 on the probability of failure of specimens of a certain fabric type, an experiment was conducted in which numbers of specimens were exposed to load ranging from 5 lb/in.2 to 90 lb/in2. The numbers of ''failures" were observed. The
Given a standard normal distribution, find the area under the curve which lies(a) To the left of a = 1.43;(b) To the right of z = -0.89:(c) Between z = —2.16 and z = — 0.65;(d) To the left of; = -1.39;(e) To the: right of z = 1.90:(f) Between z = -0.48 and z = 1.74.
Find the value of z if the area under a standard normal curve(a) To the right, of z is 0.3622;(b) To the left of a is 0.1131;(c) Between 0 and z, with z > 0, is 0.4838;(d) Between —z and z, with z > 0, is 0.9500.
Given a standard normal distribution, value of k such that(a) P(Z < k) = 0.0427;(b) P(Z > /,•) = 0.2946;(c) P(-0.93 < Z < k) = 0.7235.
Given a normal distribution with p = 30 and a — 6, find(a) The normal-curve area to the right of x = 17;(b) The normal-curve area to the left of x = 22;(c) The normal-curve area between x = 32 and x = 41;(d) The value of a: that has 80% of the normal-curve area to the left;(e) The two values of x
Given the normally distributed variable X with mean 18 and standard deviation 2.5, find(a) P(X < 15):(b) the value of k such that P(X < k) = 0.2236;(c) the value of k such that P(X > k) = 0.1814;(d) P(17 < X < 21).
According to Chebyshev's theorem, the probability that any random variable assumes a value within 3 standard deviations of the mean is at least 8/9. If it is known that the probability distribution of a random variable X is normal with mean p, and variance a2, what is the exact value of P(μ —
A research scientist reports that mice will live an average of 40 months when their diets are sharply restricted and then enriched with vitamins and proteins. Assuming that the lifetimes of such mice are normally distributed with a standard deviation of 6.3 months, find the probability that a given
The loaves of rye bread distributed to local stores by a certain bakery have an average length of 30 centimeters and a standard deviation of 2 centimeters. Assuming that the lengths are normally distributed, what percentage of the loaves are(a) Longer than 31.7 centimeters?(b) Between 29.3 and 33.5
A soft-drink machine is regulated so that it discharges an average of 200 milliliters per cup. If the amount of drink is normally distributed with a standard deviation equal to 15 milliliters,(a) What fraction of the cups will contain more than 224 milliliters?(b) What is the probability that a cup
The finished inside diameter of a piston ring is normally distributed with a mean of 10 centimeters and a standard deviation of 0.03 centimeter.(a) What proportion of rings will have inside diameters exceeding 10.075 centimeters?(b) What is the probability that a piston ring will have an inside
A lawyer commutes daily from his suburban home to his midtown office. The average time for a one-way trip is 24 minutes, with a standard deviation of 3.8 minutes. Assume the distribution of trip times to be normally distributed.(a) What is the probability that a trip will take at least 1/2 hour?(b)
In the November 1990 issue of Chemical Engineering Progress, a study discussed the percent purity of oxygen from a certain supplier. Assume that the mean was 99.61 with a standard deviation of 0.08. Assume that the distribution of percent purity was approximately normal.(a) What percentage of the
The average life of a certain type of small motor is 10 years with a standard deviation of 2 years. The manufacturer replaces free all motors that fail while under guarantee. If he is willing to replace only 3% of the motors that fail, how long a guarantee should he offer? Assume that the lifetime
The heights of 1000 students are normally distributed with a mean of 174.5 centimeters and a standard deviation of 6.9 centimeters. Assuming that the heights are recorded to the nearest half-centimeter, how many of these students you would expect to have heights(a) Less than 160.0 centimeters?(b)
A company pays its employees an average wage of $15.90 an hour with a standard deviation of $1.50. If the wages are approximately normally distributed and paid to the nearest cent,(a) What percentage of the workers receive wages between S13.75 and S16.22 an hour inclusive?(b) The highest 5% of the
The weights of a large number of miniature poodles are approximately normally distributed with a mean of 8 kilograms and a standard deviation of 0.9 kilogram. If measurements arc recorded to the nearest tenth of a kilogram, find the fraction or these poodles with weights(a) Over 9.5 kilograms:(b)
The tensile strength of a certain metal component is normally distributed with a mean 10,000 kilograms per square centimeter and a standard deviation of 100 kilograms per square centimeter. Measurements are recorded to the nearest 50 kilograms per square centimeter.(a) What proportion of these
If a set of observations is normally distributed, what percent of these differ from the mean by?(a) More than 1.3c?(b) Less than 0.52cr?
The IQs of 600 applicants of a certain college are approximately normally distributed with a mean of 115 and a standard deviation of 12. If the college requires an IQ of at least 95, how many of these students will be rejected on this basis regardless of their other qualifications?
Given a continuous uniform distribution, show that
The daily amount of coffee, in liters, dispensed by a machine located in an airport, lobby is a random variable X having a continuous uniform distribution with A = 7 and B = 10. Find the probability that on a given day the amount, of coffee dispensed by this machine will be(a) At most 8.8
A bus arrives every 10 minutes at a bus stop. It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution.(a) What is the probability that the individual waits more than 7 minutes?(b) What is the probability that the individual waits
Evaluate P ( l < X < 4) for a binomial variable with n = 15 and p = 0.2 by using(a) Table Al in the Appendix;(b) The normal-curve approximation.
A coin is tossed 400 times. Use the normal-curve approximation to find the probability of obtaining(a) Between 185 and 210 heads inclusive;(b) Exactly 205 heads;(c) Less than 176 or more than 227 heads.
A process for manufacturing an electronic component is 1% defective. A quality control plan is to select 100 items from the process, and if none are defective, the process continues. Use the normal approximation to the binomial to find(a) The probability that the process continues for the sampling
A process yields 10% defective items. If 100 items are randomly selected from the process, what is the probability that the number of defectives(a) Exceeds 13?(b) Is less than 8?
The probability that a patient recovers from a delicate heart operation is 0.9. Of the next 100 patients having this operation, what is the probability that(a) Between 84 and 95 inclusive survive?(b) Fewer than 86 survive?
Researchers at George Washington University and the National Institutes of Health claim that approximately 75% of the people believe "tranquilizers work very well to make a person more calm and relaxed." Of the next 80 people interviewed, what is the probability that(a) At least 50 are of this
If 20% of the residents in a U.S. city prefer a white telephone over any other color available, what is the probability that among the next 1000 telephones installed in that city(a) Between 170 and 185 inclusive will be white?(b) At least 210 but not more than 225 will be white?
A drug manufacturer claims that a certain drug cures a blood disease, on the average, 80% of the time. To check the claim, government testers used the drug on a sample of 100 individuals and decided to accept the claim if 75 or more were cured.(a) What is the probability that, the claim will be
One-sixth of the male freshmen entering a large state school are out-of-state students. If the students are assigned at random to the dormitories, 180 to a building, what is the probability that in a given dormitory at least one-fifth of the students are from out of state?
A pharmaceutical company knows that approximately 5% of its birth-control pills have an ingredient that is below the minimum strength, thus rendering the pill ineffective. What is the probability that fewer than 10 in a sample of 200 pills will be ineffective?
Statistics released by the National Highway Traffic Safety Administration and the National Safety Council show that on an average weekend night, 1 out of every 10 drivers on the road is drunk. If 400 drivers are randomly checked next Saturday night, what is the probability that the number of drunk
A pair of dice is rolled 180 times. What is the probability that a total of 7 occurs(a) At least 25 times?(b) Between 33 and 41 times inclusive?(c) Exactly 30 times?
A company produces component parts for an engine. Parts specifications suggest that 95% of items meet specifications. The parts are shipped to customers in lots of 100.(a) What is the probability that more than 2 items will be defective in a given lot?(b) What is the probability that more than 10
A commonly used practice of airline companies is to sell more tickets than actual seats to a particular flight because customers who buy tickets do not always show up for the flight. Suppose that the percentage of no-shows at flight time is 2%. For a particular flight with 197 seats, a total of 200
The serum cholesterol level X in 14-year-old boys has approximately a normal distribution with mean 170 and standard deviation 30.(a) Find the probability that the serum cholesterol level of a randomly chosen 14-year-old boy exceeds 230.(b) In a middle school there are 300 14-year-old boys. Find
A telemarketing company has a special letter opening machine that opens and removes the contents of an envelope. If the envelope is fed improperly into the machine, the contents of the envelope may not be removed or may be damaged. In this case we say that, the machine has "failed."(a) If the
If a random variable X has the gamma distribution with α = 2 and 0 = 1, find P(1.8 < X < 2.4).
In a certain city, the daily consumption of water (in millions of liters) follows approximately a gamma distribution with Q = 2 and β = 3. If the daily capacity of that city is 9 million liters of water, what is the probability that on any given day the water supply is inadequate?
Use the gamma function with y = √2x to show that P(l/2) = √π.
Suppose that the time, in hours, taken to repair a heat pump is a random variable X having a gamma distribution with parameters α = 2 and 3 = 1/2. What is the probability that the next service call will require(a) At most 1 hour to repair the heat pump?(b) At least 2 hours to repair the heat pump?
(a) Find the mean and variance of the daily water consumption in Exercise 6.40.(b) According to Chebyshev's theorem, there is a probability of at least 3/4 that the water consumption on any given day will fall within what interval?
In a certain city, the daily consumption of electric power, in millions of kilowatt-hours, is a random variable X having a gamma distribution with mean μ = 6 and variance α2 = 12.(a) Find the values of a and 0.(b) Find the probability that on any given day the daily power consumption will exceed
The length of time for one individual to be served at a cafeteria is a random variable having an exponential distribution with a mean of 4 minutes. What is the probability that a person is served in less than 3 minutes on at least 4 of the next 6 days?
The life, in years, of a certain type of electrical switch has an exponential distribution with an average life 0 = 2. If 100 of these switches are installed in different systems, what is the probability that at most 30 fail during the first year?
Suppose that the service life, in years, of a hearing aid battery is a random variable having a Weibull distribution with α = 1/2 and β = 2.(a) How long can such a battery be expected to last?(b) What is the probability that such a battery will be operating after 2 years?
Derive the mean and variance of the Weibull distribution.
The lives of a certain automobile seal have the Weibull distribution with failure rate Z(t) = 1/√t. Find the probability that such a seal is still intact after 4 years.
The continuous random variable X has the beta distribution with parameters α and 3 if its density function is given by where α > 0 and β > 0. If the proportion of a brand of television set requiring service during the first year of operation is a random variable having a beta distribution with
In a biomedical research activity it was determined that the survival time, in weeks, of an animal when subjected to a certain exposure of gamma radiation has a gamma distribution with a =5 and 3 =10.(a) What is the mean survival time of a randomly selected animal of the type used in the
The lifetime, in weeks, of a certain type of transistor is known to follow a gamma distribution with mean 10 weeks and standard deviation √50 weeks. (a) What is the probability that the transistor will last at most 50 weeks? (b) What is the probability that the transistor will not survive
Computer response time is an important application of the gamma and exponential distributions. Suppose that a study of a certain computer system reveals that the response time, in seconds, has an exponential distribution with a mean of 3 seconds.(a) What is the probability that response time
Rate data often follow a lognormal distribution. Average power usage (dB per hour) for a particular company is studied and is known to have a lognormal distribution with parameters μ = 4 and a = 2. What is the probability that the company uses more than 270 dB during any particular hour?
For Exercise 6.54, what is the mean power usage (average dBs per hour)? What is the variance?
The number of automobiles that arrive at a certain intersection per minute has a Poisson distribution with a mean of 5. Interest centers around the time that elapses before 10 automobiles appear at the intersection.(a) What is the probability that more than 10 automobiles appear at the intersection
Consider the information in Exercise 6.56.(a) What is the probability that more than 1 minute elapses between arrivals?(b) What is the mean number of minutes that elapse between arrivals?
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