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Questions and Answers of Statistics

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
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 =
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
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.
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,
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
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?
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
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
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
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
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
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
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
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
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
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
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,
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
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
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);
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
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
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
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
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
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
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
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
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; =
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
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
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) =
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
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
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
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)
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
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
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.
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
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
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
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
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.
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
(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
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
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
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
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)
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
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
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
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
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 =
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
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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