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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

6. Twenty overweight individuals, each more than 40 pounds overweight, were randomly assigned to one of two diets. After 10 weeks, the total weight losses (in pounds) of the individuals on each of
5. Four standard chemical procedures are used to determine the magnesium content in a certain chemical compound. Each procedure is used four times on a given compound with the following data
4. A machine shop contains 3 ovens that are used to heat metal specimens. Subject to random fluctuations, they are all supposed to heat to the same temperature. To test this hypothesis, temperatures
2. We want to know what type of filter should be used over the screen of a cathoderay oscilloscope in order to have a radar operator easily pick out targets on the presentation. A test to accomplish
1. A purification process for a chemical involves passing it, in solution, through a resin on which impurities are adsorbed. A chemical engineer wishing to test the efficiency of 3 different resins
The life of a particular type of generator is thought to be influenced by the material used in its construction and also by the temperature at the location where it is utilized. The following table
The following data* represent the number of different macroinvertebrate species collected at 6 stations, located in the vicinity of a thermal discharge, from 1970 to 1977.To test the hypotheses that
The following data from Example 10.4a give the scores obtained when four different reading tests were given to each of five students. Use it to estimate the parameters of the model. Student
Four different standardized reading achievement tests were administered to each of 5 students, with the scores shown in the table resulting. Each value in this set of 20 data points is affected by
A college administrator claims that there is no difference in first-year grade point averages for students entering the college from any of three different city high schools. The following data give
Let us do the computations of Example 10.3a by hand. The first thing to note is that subtracting a constant from each data value will not affect the value of the test statistic. So we subtract 220
An auto rental firm is using 15 identical motors that are adjusted to run at a fixed speed to test 3 different brands of gasoline. Each brand of gasoline is assigned to exactly 5 of the motors. Each
58. Twelve first-time heart attack victims were given a test that measures internal anger. The following data relates their scores and whether they had a second heart attack within 5 years(a) Explain
57. A study of 64 prematurely born infants was interested in the relation between the gestational age (in weeks) of the infant at birth and whether the infant was breast-feeding at the time of
55. Suppose in Problem 54 that job satisfaction was related solely to years on the job, with the following data resulting.(a) Estimate the regression parameters α and β.(b) What is the qualitative
54. A recently completed study attempted to relate job satisfaction to income (in 1,000s) and seniority for a random sample of 9 municipal workers. The job satisfaction value given for each worker is
53. The following data relate the systolic blood pressure to the age (x1) and weight(x2) of a set of individuals of similar body type and lifestyle.(a) Test the hypothesis that, when an
52. The cost of producing power per kilowatt hour is a function of the load factor and the cost of coal in cents per million Btu. The following data were obtained from 12 mills.(a) Estimate the
51. Consider the following data set.(a) Fit a linear relationship between y and x1, x2.(b) Determine the variance of the error term.(c) Determine an interval that, with 95 percent confidence, will
49. The time to failure of a machine component is related to the operating voltage(x1), the motor speed in revolutions per minute (x2), and the operating temperature (x3). A designed experiment is
48. The tensile strength of a certain synthetic fiber is thought to be related to x1, the percentage of cotton in the fiber, and x2, the drying time of the fiber. A test of 10 pieces of fiber
47. (a) Fit a multiple linear regression equation to the following data set.(b) Test the hypothesis that β0 = 0.(c) Test the hypothesis that β3 = 0.(d) Test the hypothesis that the mean response at
46. The following data refer to Stanford heart transplants. It relates the survival time of patients that have received heart transplants to their age when the transplant occurred and to a so-called
45. Fit a multiple linear regression equation to the following data set. x1 x2 x3 X4 y 1 11 16 4 275 234567 2 10 9 9 4 32 183 140 8 1 1 82 8 9 10 765432 7 2 1 97 1 -1 122 5 4 -2 146 9 -3 246 3 16 -4
44. The sediment load in a stream is related to the size of the contributing drainage area (x1) and the average stream discharge (x2). Estimate this relationship using the following data. Area (x 103
43. The peak discharge of a river is an important parameter for many engineering design problems. Estimates of this parameter can be obtained by relating it to the watershed area (x1) and watershed
42. The following data set refers to Example 9.8c.(a) Estimate the number of accidents in a month when the number of cars using the highway is 3,500.(b) Use the modeland redo part (a). Number of Cars
41. The following data relate the proportions of coal miners who exhibit symptoms of pneumoconiosis to the number of years of working in coal mines.Estimate the probability that a coal miner who has
39. The following data represent the relation between the number of cans damaged in a boxcar shipment of cans and the speed of the boxcar at impact(a) Analyze as a simple linear regression model.(b)
38. A new drug was tested on mice to determine its effectiveness in reducing cancerous tumors. Tests were run on 10 mice, each having a tumor of size 4 grams, by varying the amount of the drug used
37. The following data yield the amount of hydrogen present (in parts per million)in core drillings of fixed size at the following distances (in feet) from the base of a vacuum-cast ingot.(a) Draw a
35. The proportion of a given heat rise that has dissipated a time t after the source is cut off is of the formfor some unknown constant α. Given the dataestimate the value of α. Estimate the value
34. The chlorine residual in a swimming pool at various times after being cleaned is as given:Fit a curve of the formWhat would you predict for the chlorine residual 15 hours after a cleaning? Time
33. In 1957 the Dutch industrial engineer J. R. DeJong proposed the following model for the time it takes to perform a simple manual task as a function of the number of times the task has been
32. It has been determined that the relation between stress (S) and the number of cycles to failure (N) for a particular type alloy is given bywhere A andmare unknown constants. An experiment is run
31. The weight and systolic blood pressure of randomly selected males in age-group 25 to 30 are shown in the following table(a) Estimate the regression coefficients.(b) Do the data support the claim
30. Prove the identity R= xY Sxx SYY
29. The regression modelis called regression through the origin since it presupposes that the expected response corresponding to the input level x = 0 is equal to 0. Suppose that (xi , Yi ), i = 1, .
28. Glass plays a key role in criminal investigations, because criminal activity often results in the breakage of windows and other glass objects. Since glass fragments often lodge in the clothing of
27. A screw manufacturer is interested in giving out data to his customers on the relation between nominal and actual lengths. The following results (in inches)were observed.(a) Estimate the
26. The determination of the shear strength of spot welds is relatively difficult, whereas measuring the weld diameter of spot welds is relatively simple. As a result, it would be advantageous if
25. It is difficult and time consuming to measure directly the amount of protein in a liver sample. As a result, medical laboratories often make use of the fact that the amount of protein is related
18. The following data give the average price of all books reviewed in the journal Science in the years from 1990 to 1996.Give an interval that, with 95 percent confidence, will contain the average
17. The following data represent the relationship between the number of alignment errors and the number of missing rivets for 10 different aircrafts.(a) Plot a scatter diagram.(b) Estimate the
14. A study has shown that a good model for the relationship between X and Y , the first and second year batting averages of a randomly chosen major league baseball player, is given by the
13. Suppose in the simple linear regression model Y = α + βx + e that 0 (a) Show that if x (b) Show that if x > α/(1 − β), thenand conclude that E[Y ] is always between x and α/(1 − β). x
12. The following data set presents the heights of 12 male law school classmates whose law school examination scores were roughly equal. It also gives their annual salaries 5 years after graduation.
11. The following table relates the number of sunspots that appeared each year from 1970–1983 to the number of auto accident deaths during that year. Test the hypothesis that the number of auto
10. Verify that SSR = Sxx Sxx Sr Syy Sy xY
8. Verify Equation 9.3.3, which states that Var(A): ' 73 n(x; - x) i=1
7. The following table gives the 1996 SAT mean math and verbal scores in each state and the District of Columbia, along with the percentage of the states’ graduating high school students that took
6. Infrared spectroscopy is often used to determine the natural rubber content of mixtures of natural and synthetic rubber. For mixtures of known percentages, the infrared spectroscopy gave the
4. The following data indicate the relationship between x, the specific gravity of a wood sample, and Y , its maximum crushing strength in compression parallel to the grain.(a) Plot a scatter
3. The corrosion of a certain metallic substance has been studied in dry oxygen at 500 degrees Centigrade. In this experiment, the gain in weight after various periods of exposure was used as a
2. The following data relate the number of units of a good that were ordered as a function of the price of the good at six different locationsHow many units do you think would be ordered if the price
1. The following data relate x, the moisture of a wet mix of a certain product, to Y , the density of the finished product.(a) Draw a scatter diagram.(b) Fit a linear curve to the data. Y 569 5 7.4
A common assumption for whether an animal becomes sick when exposed to a chemical at dosage level x is to assume a threshold model, which supposes that each animal has a random threshold and will
If in Example 9.10d we were interested in determining an interval in which a single steel sheet, produced with a carbon content of .15 percent and at an annealing temperature of 1,150◦F, would lie,
The diameter of a tree at its breast height is influenced by many factors.The data in Table 9.5 relate the diameter of a particular type of eucalyptus tree to its age, average rainfall at its site,
The data in Table 9.4 relate the suicide rate to the population size and the divorce rate at eight different locationsFit a multiple linear regression model to these data. That is, fit a model of the
Fit a polynomial to the following data. x Y 1 20.6 30.8 3 55 2345678 71.4 97.3 6 131.8 7 156.3 8 197.3 9 238.7 10 291.7
Consider the relationship between Y , the number of accidents on a heavily traveled highway, and x, the number of cars traveling on the highway. After a little thought it would probably seem to most
The following data represent travel times in a downtown area of a certain city. The independent, or input, variable is the distance to be traveled.Assuming a linear relationship of the formbetween Y
To develop a feel as to why the estimators should be obtained by minimizing the weighted sum of squares rather than the ordinary sum of squares, consider the following situation. Suppose that X1, . .
The following table gives the percentages of a chemical that were used up when an experiment was run at various temperatures (in degrees celsius). Use it to estimate the percentage of the chemical
In Example 9.4c, which relates the height of a son to that of his father, the output from Program 9.2 yielded thatThus,In other words, 96 percent of the variation of the heights of the 10 individuals
The data of Table 9.1 relate the number of motor vehicle deaths occurring in 12 counties in the northwestern United States in the years 1988 and 1989.A glance at Figure 9.8 indicates that in 1989
To illustrate Galton’s thesis of regression to the mean, the British statistician Karl Pearson plotted the heights of 10 randomly chosen sons versus that of their fathers.The resulting data (in
An individual claims that the fuel consumption of his automobile does not depend on how fast the car is driven. To test the plausibility of this hypothesis, the car was tested at various speeds
The following data relate x, the moisture of a wet mix of a certain product, to Y, the density of the finished product.Fit a linear curve to these data. Also determine SSR. xi yi 5 7.4 6 9.3 7 10.6
The raw material used in the production of a certain synthetic fiber is stored in a location without a humidity control. Measurements of the relative humidity in the storage location and the moisture
Consider the following 10 data pairs (xi , yi ), i = 1, . . . , 10, relating y, the percent yield of a laboratory experiment, to x, the temperature at which the experiment was run.A plot of yi versus
44. Cars pass a certain street location according to a Poisson process with rate λ. A woman who wants to cross the street at that location waits until she can see that no cars will come by in the
43. Customers arrive at a two-server service station according to a Poisson process with rate λ. Whenever a new customer arrives, any customer that is in the system immediately departs. A new
42. Customers arrive to a single server system according to a Poisson process with rate λ. An arrival that finds the server idle immediately begins service; an arrival that finds the server busy
41. In Exercise 40 what is the probability that the first event of the combined process is from the process?
*40. Show that if are independent Poisson processes with rate , , thenI mage is a Poisson process with rate whereI mage.
39. A certain scientific theory supposes that mistakes in cell division occur according to a Poisson process with rate 2.5 per year, and that an individual dies when 196 such mistakes have
38. Let be independent Poisson processes with respective rates , and set The stochastic process is called a bivariate Poisson process.(a) Find .(b) Find .
37. LetI mage be a Poisson process with rate λ. For andI mage,(a) find ;(b) find .
*36. Let denote the price of a security at time t. A popular model for the process supposes that the price remains unchanged until a “shock” occurs, at which time the price is multiplied by a
35. LetI mage be the time of the first event ofI mage, a Poisson process with rate λ. For another way to show thatI mage is exponential with rate λ, let be its failure rate function. Using that
34. Two individuals, A and B, both require kidney transplants. If she does not receive a new kidney, then A will die after an exponential time with rate , and B after an exponential time with rate
33. Let X and Y be independent exponential random variables with respective rates λ and μ.(a) Argue that, conditional on , the random variables and are independent.(b) Use part (a) to conclude that
32. Let X be a uniform random variable on (0, 1), and consider a counting process where events occur at times , for .(a) Does this counting process have independent increments?(b) Does this counting
31. Suppose are independent nonnegative continuous random variables, with W being exponential with rate λ, and with having density function , .(a) Show that That is, given that , the random
30. The lifetimes of A's dog and cat are independent exponential random variables with respective rates and . One of them has just died. Find the expected additional lifetime of the other pet.
29. Let X and Y be independent exponential random variables with respective rates λ and μ, where . Let .(a) Show that the conditional density function of X, given that , is
28. Consider n components with independent lifetimes, which are such that component i functions for an exponential time with rate. Suppose that all components are initially in use and remain so until
27. Show, in Example 5.7, that the distributions of the total cost are the same for the two algorithms.
26. Each entering customer must be served first by server 1, then by server 2, and finally by server 3. The amount of time it takes to be served by server i is an exponential random variable with
25. Customers can be served by any of three servers, where the service times of server i are exponentially distributed with rate. Whenever a server becomes free, the customer who has been waiting the
24. There are two servers available to process n jobs. Initially, each server begins work on a job. Whenever a server completes work on a job, that job leaves the system and the server begins
*23. A flashlight needs two batteries to be operational. Consider such a flashlight along with a set of n functional batteries—battery 1, battery 2 battery n. Initially, battery 1 and 2 are
22. Suppose in Exercise 21 you arrive to find two others in the system, one being served by server 1 and one by server 2. What is the expected time you spend in the system? Recall that if server 1
21. In a certain system, a customer must first be served by server 1 and then by server 2. The service times at server i are exponential with rate . An arrival finding server 1 busy waits in line for
20. Consider a two-server system in which a customer is served first by server 1, then by server 2, and then departs. The service times at server i are exponential random variables with rates .When
19. In a mile race between A and B, the time it takes A to complete the mile is an exponential random variable with rate and is independent of the time it takes B to complete the mile, which is an
*18. Let and be independent exponential random variables, each having rate μ. Let Find(a) ,(b) ,(c) ,(d) .
17. A set of n cities is to be connected via communication links. The cost to construct a link between cities i and j is . Enough links should be constructed so that for each pair of cities there is

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