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applied statistics and probability for engineers

Probability And Statistics For Engineering And The Sciences 7th Edition Dave Ellis, Jay L Devore - Solutions

What is the approximate probability thata. Between 35 and 70 tickets are given out on a particular day? [Hint: When is large, a Poisson rv has approximately a normal distribution.]b. The total number of tickets given out during a 5-day week is between 225 and 275?
The number of parking tickets issued in a certain city on any given weekday has a Poisson distribution with parameter
Suppose the sediment density (g/cm) of a randomly selected specimen from a certain region is normally distributed with mean 2.65 and standard deviation .85 (suggested in “Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants,” Water Research, 1984: 1169–1174).a. If a
Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of 1.2.a. If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 9 pins is at least 51?b. What is the (approximate) probability that the
The lifetime of a certain type of battery is normally distributed with mean value 10 hours and standard deviation 1 hour.There are four batteries in a package. What lifetime value is such that the total lifetime of all batteries in a package exceeds that value for only 5% of all packages?
The time taken by a randomly selected applicant for a mortgage to fill out a certain form has a normal distribution with mean value 10 min and standard deviation 2 min. If five individuals fill out a form on one day and six on another, what is the probability that the sample average amount of time
The breaking strength of a rivet has a mean value of 10,000 psi and a standard deviation of 500 psi.a. What is the probability that the sample mean breaking strength for a random sample of 40 rivets is between 9900 and 10,200?b. If the sample size had been 15 rather than 40, could the probability
There are 40 students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of 6 min and a standard deviation of 6 min.a. If grading times are
Let X1, X2, . . . , X100 denote the actual net weights of 100 randomly selected 50-lb bags of fertilizer.a. If the expected weight of each bag is 50 and the variance is 1, calculate P(49.9 X 50.1) (approximately)using the CLT.b. If the expected weight is 49.8 lb rather than 50 lb so that on
Refer to Exercise 46. Suppose the distribution of diameter is normal.a. Calculate P(11.99 X 12.01) when n 16.b. How likely is it that the sample mean diameter exceeds 12.01 when n 25?
The inside diameter of a randomly selected piston ring is a random variable with mean value 12 cm and standard deviation .04 cm.a. If X is the sample mean diameter for a random sample of n 16 rings, where is the sampling distribution of X centered, and what is the standard deviation of the X
Carry out a simulation experiment using a statistical computer package or other software to study the sampling distribution of X when the population distribution is lognormal with E(ln(X)) 3 and V(ln(X)) 1. Consider the four sample sizes n 10, 20, 30, and 50, and in each case use 500
Carry out a simulation experiment using a statistical computer package or other software to study the sampling distribution of X when the population distribution is Weibull with 2 and 5, as in Example 5.19. Consider the four sample sizes n 5, 10, 20, and 30, and in each case use 500
Suppose the amount of liquid dispensed by a certain machine is uniformly distributed with lower limit A 8 oz and upper limit B 10 oz. Describe how you would carry out simulation experiments to compare the sampling distribution of the (sample) fourth spread for sample sizes n 5, 10, 20, and 30.
A company maintains three offices in a certain region, each staffed by two employees. Information concerning yearly salaries (1000s of dollars) is as follows:Office 1 1 2 233 Employee 1 2 3 456 Salary 29.7 33.6 30.2 33.6 25.8 29.7a. Suppose two of these employees are randomly selected from among
Let X be the number of packages being mailed by a randomly selected customer at a certain shipping facility.Suppose the distribution of X is as follows:x | 1 2 34 p(x) | .4 .3 .2 .1a. Consider a random sample of size n 2 (two customers), and let X be the sample mean number of packages
The first five contain no money, the next three each contains $5, and there is a $10 bill in each of the last two. A sample of size 3 is selected with replacement (so we have a random sample), and you get the largest amount in any of the envelopes selected. If X1, X2, and X3 denote the amounts in
A box contains ten sealed envelopes numbered 1, . . . ,
It is known that 80% of all brand A zip drives work in a satisfactory manner throughout the warranty period (are “successes”). Suppose that n 10 drives are randomly selected.Let X the number of successes in the sample. The statistic X/n is the sample proportion (fraction) of
There are two traffic lights on my way to work. Let X1 be the number of lights at which I must stop, and suppose that the distribution of X1 is as follows:x1 | 012 1.1, 2 .49 p(x1) | .2 .5 .3 Let X2 be the number of lights at which I must stop on the way home; X2 is independent of X1. Assume
A particular brand of dishwasher soap is sold in three sizes: 25 oz, 40 oz, and 65 oz. Twenty percent of all purchasers select a 25-oz box, 50% select a 40-oz box, and the remaining 30% choose a 65-oz box. Let X1 and X2 denote the package sizes selected by two independently selected purchasers.a.
Show that if Y aX b (a 0), then Corr(X, Y) 1 or 1.Under what conditions will 1?
a. Use the rules of expected value to show that Cov(aX b, cYd) ac Cov(X, Y).b. Use part (a) along with the rules of variance and standard deviation to show that Corr(aXb, cYd) Corr(X, Y)when a and c have the same sign.c. What happens if a and c have opposite signs?
a. Recalling the definition of 2 for a single rv X, write a formula that would be appropriate for computing the variance of a function h(X, Y) of two random variables.[Hint: Remember that variance is just a special expected value.]b. Use this formula to compute the variance of the recorded score
Use the result of Exercise 28 to show that when X and Y are independent, Cov(X, Y) Corr(X, Y) 0.
Reconsider the minicomputer component lifetimes X and Y as described in Exercise 12. Determine E(XY). What can be said about Cov(X, Y) and ?
a. Compute the covariance between X and Y in Exercise 9.b. Compute the correlation coefficient for this X and Y.
a. Compute the covariance for X and Y in Exercise 22.b. Compute for X and Y in the same exercise.
Compute the correlation coefficient for X and Y of Example 5.16 (the covariance has already been computed).
[Hint: Consider the continuous case with f(x, y) fX(x) fY(y).]
Show that if X and Y are independent rv’s, then E(XY) E(X) E(Y). Then apply this in Exercise
Annie and Alvie have agreed to meet for lunch between noon(0:00 P.M.) and 1:00 P.M. Denote Annie’s arrival time by X, Alvie’s by Y, and suppose X and Y are independent with pdf’s fX(x) {3x2 0 x 1 0 otherwise fY(y) {2y 0 y 1 0 otherwise What is the expected amount of time that the
Consider a small ferry that can accommodate cars and buses.The toll for cars is $3, and the toll for buses is $10. Let X and Y denote the number of cars and buses, respectively, carried on a single trip. Suppose the joint distribution of X and Y is as given in the table of Exercise 7. Compute the
A surveyor wishes to lay out a square region with each side having length L. However, because of measurement error, he instead lays out a rectangle in which the north–south sides both have length X and the east–west sides both have length Y. Suppose that X and Y are independent and that each is
Six individuals, including A and B, take seats around a circular table in a completely random fashion. Suppose the seats?
The difference between the number of customers in line at the express checkout and the number in line at the superexpress checkout in Exercise 3 is X1 X2. Calculate the expected difference.
An instructor has given a short quiz consisting of two parts.For a randomly selected student, let X the number of points earned on the first part and Y the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table.
Let X1, X2, and X3 be the lifetimes of components 1, 2, and 3 in a three-component system.a. How would you define the conditional pdf of X3 given that X1 x1 and X2 x2?b. How would you define the conditional joint pdf of X2 and X3 given that X1 x1?
Let X1, X2, X3, X4, X5, and X6 denote the numbers of blue, brown, green, orange, red, and yellow M&M candies, respectively, in a sample of size n. Then these Xis have a multinomial distribution. According to the M&M web site, the color proportions are p1 .24, p2 .13, p3 .16, p4 .20, p5
The joint pdf of pressures for right and left front tires is given in Exercise 9.a. Determine the conditional pdf of Y given that X x and the conditional pdf of X given that Y y.b. If the pressure in the right tire is found to be 22 psi, what is the probability that the left tire has a pressure
Refer to Exercise 1 and answer the following questions:a. Given that X 1, determine the conditional pmf of Y—i.e., pY⏐X (0⏐1), pY⏐X (1⏐1), and pY⏐X (2⏐1).b. Given that two hoses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the
An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first selecting a direction and then a distance from the center in that direction). Let X the x coordinate of the point selected and Y the y coordinate
a. For f(x1, x2, x3) as given in Example 5.10, compute the joint marginal density function of X1 and X3 alone (by integrating over x2).b. What is the probability that rocks of types 1 and 3 together make up at most 50% of the sample? [Hint: Use the result of part (a).]c. Compute the marginal pdf of
Consider a system consisting of three components as pictured. The system will continue to function as long as the first component functions and either component 2 or component 3 functions. Let X1, X2, and X3 denote the lifetimes of components 1, 2, and 3, respectively. Suppose the Xis are
Suppose that you have ten lightbulbs, that the lifetime of each is independent of all the other lifetimes, and that each lifetime has an exponential distribution with parameter .a. What is the probability that all ten bulbs fail before time t?b. What is the probability that exactly k of the ten
You have two lightbulbs for a particular lamp. Let X the lifetime of the first bulb and Y the lifetime of the second bulb (both in 1000s of hours). Suppose that X and Y are independent and that each has an exponential distribution with parameter 1.a. What is the joint pdf of X and Y?b. What
Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y:f(x, y) {xex(1y) x 0 and y 0 0 otherwisea. What is the probability that the lifetime X of the first component exceeds 3?b. What are the marginal pdf’s of X and Y? Are the two lifetimes
Two different professors have just submitted final exams for duplication. Let X denote the number of typographical errors on the first professor’s exam and Y denote the number of such errors on the second exam. Suppose X has a Poisson distribution with parameter , Y has a Poisson distribution
Annie and Alvie have agreed to meet between 5:00 P.M.and 6:00 P.M. for dinner at a local health-food restaurant.Let X Annie’s arrival time and Y Alvie’s arrival time.Suppose X and Y are independent with each uniformly distributed on the interval [5, 6].a. What is the joint pdf of X and Y?b.
Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable—X for the right tire and Y for the left tire, with joint pdf f(x, y) {K(x2 y2) 20 x 30, 20 y 30 0 otherwisea. What is the
A stockroom currently has 30 components of a certain type, of which 8 were provided by supplier 1, 10 by supplier 2, and 12 by supplier 3. Six of these are to be randomly selected for a particular assembly. Let X the number of supplier 1s components selected, Y the number of supplier 2s
The joint probability distribution of the number X of cars and the number Y of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table.a. What is the probability that there is exactly one car and exactly one bus during a cycle?b. What is the
Let X denote the number of Canon digital cameras sold during a particular week by a certain store. The pmf of X is x | 01 2 3 4 pX(x) | .1 .2 .3 .25 .15 Sixty percent of all customers who purchase these cameras also buy an extended warranty. Let Y denote the number of purchasers during this week
The number of customers waiting for gift-wrap service at a department store is an rv X with possible values 0, 1, 2, 3, 4 and corresponding probabilities .1, .2, .3, .25, .15. A randomly selected customer will have 1, 2, or 3 packages for wrapping with probabilities .6, .3, and .1, respectively.
Return to the situation described in Exercise 3.a. Determine the marginal pmf of X1, and then calculate the expected number of customers in line at the express checkout.b. Determine the marginal pmf of X2.c. By inspection of the probabilities P(X1 4), P(X2 0), and P(X1 4, X2 0), are X1 and
A certain market has both an express checkout line and a superexpress checkout line. Let X1 denote the number of customers in line at the express checkout at a particular time of day, and let X2 denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint
When an automobile is stopped by a roving safety patrol, each tire is checked for tire wear, and each headlight is checked to see whether it is properly aimed. Let X denote the number of headlights that need adjustment, and let Y denote the number of defective tires.a. If X and Y are independent
45. If a X b, show that a E(X) b.?
47. Use Appendix Table A.1 to obtain the following probabilities:a. B(4; 15, .3)b. b(4; 15, .3)c. b(6; 15, .7)d. P(2 X 4) when X Bin(15, .3)e. P(2 X) when X Bin(15, .3)f. P(X 1) when X Bin(15, .7)g. P(2 X 6) when X Bin(15, .3)
48. When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Let X the number of defective boards in a random sample of size n 25, so X Bin(25, .05).
51. Refer to the previous exercise.
44. A result called Chebyshev’s inequality states that for any probability distribution of an rv X and any number k that is at least 1, P(⏐X ⏐ k) 1/k2. In words, the probability that the value of X lies at least k standard deviations from its mean is at most 1/k2.
43. Write a general rule for E(X c) where c is a constant.What happens when you let c , the expected value of X?
42. Suppose E(X) 5 and E[X(X 1)] 27.5. What isa. E(X2)? [Hint: E[X(X 1)] E[X2 X] E(X2) E(X)]?b. V(X)?c. The general relationship among the quantities E(X), E[X(X 1)], and V(X)?
41. Use the definition in Expression (3.13) to prove that V(aXb) a2 2 X. [Hint: With h(X) aXb, E[h(X)] a b where E(X).]
39. A chemical supply company currently has in stock 100 lb of a certain chemical, which it sells to customers in 5-lb lots.Let X the number of lots ordered by a randomly chosen customer, and suppose that X has pmf Compute E(X) and V(X). Then compute the expected number of pounds left after the
38. Let X the outcome when a fair die is rolled once. If before the die is rolled you are offered either (1/3.5) dollars or h(X) 1/X dollars, would you accept the guaranteed amount or would you gamble? [Note: It is not generally true that 1/E(X) E(1/X).]
37. The n candidates for a job have been ranked 1, 2, 3, . . . , n. Let X the rank of a randomly selected candidate, so that X has pmf p(x) { 1/n x 1, 2, 3, . . . , n 0 otherwise(this is called the discrete uniform distribution). Compute E(X) and V(X) using the shortcut formula. [Hint: The
35. A small market orders copies of a certain magazine for its magazine rack each week. Let X demand for the magazine, with pmf Suppose the store owner actually pays $1.00 for each copy of the magazine and the price to customers is $2.00. If magazines left at the end of the week have no salvage
a. What is the expected number of calls among the 25 that involve a fax message?
b. What is the standard deviation of the number among the 25 calls that involve a fax message?
c. What is the probability that the number of calls among the 25 that involve a fax transmission exceeds the expected number by more than 2 standard deviations?
60. A toll bridge charges $1.00 for passenger cars and $2.50 for other vehicles. Suppose that during daytime hours, 60% of all vehicles are passenger cars. If 25 vehicles cross the bridge during a particular daytime period, what is the resulting expected toll revenue? [Hint: Let X the number of
62.a. For fixed n, are there values of p (0 p 1) for which V(X) 0? Explain why this is so.b. For what value of p is V(X) maximized? [Hint: Either graph V(X) as a function of p or else take a derivative.]
63.a. Show that b(x; n, 1 p) b(n x; n, p).b. Show that B(x; n, 1 p) 1 B(n x 1; n, p).[Hint: At most x S’s is equivalent to at least (n x) F’s.]c. What do parts (a) and (b) imply about the necessity of including values of p greater than .5 in Appendix Table A.1?
64. Show that E(X) np when X is a binomial random variable.[Hint: First express E(X) as a sum with lower limit x 1.Then factor out np, let y x 1 so that the sum is from y 0 to y n 1, and show that the sum equals 1.]
65. Customers at a gas station pay with a credit card (A), debit card (B), or cash (C). Assume that successive customers make independent choices, with P(A) .5, P(B) .2, and P(C) .3.a. Among the next 100 customers, what are the mean and variance of the number who pay with a debit card?Explain
67. Refer to Chebyshev’s inequality given in Exercise 44.Calculate P(⏐X ⏐ k) for k 2 and k 3 when X Bin(20, .5), and compare to the corresponding upper bound. Repeat for X Bin(20, .75).
57. Suppose that 90% of all batteries from a certain supplier have acceptable voltages. A certain type of flashlight requires two type-D batteries, and the flashlight will work only if both its batteries have acceptable voltages. Among ten randomly?
52. Suppose that 30% of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other 70% want a used copy. Consider randomly selecting 25 purchasers.a. What are the mean value and standard deviation of the number who want a new copy of the book?b.
53. Exercise 30 (Section 3.3) gave the pmf of Y, the number of traffic citations for a randomly selected individual insured by a particular company. What is the probability that among 15 randomly chosen such individualsa. At least 10 have no citations?b. Fewer than half have at least one
56. The College Board reports that 2% of the 2 million high school students who take the SAT each year receive special accommodations because of documented disabilities (Los Angeles Times, July 16, 2002). Consider a random sample of 25 students who have recently taken the test.
a. What is the probability that exactly 1 received a special accommodation?
b. What is the probability that at least 1 received a special accommodation?
c. What is the probability that at least 2 received a special accommodation?d. What is the probability that the number among the 25 who received a special accommodation is within 2 standard deviations of the number you would expect to be accommodated?e. Suppose that a student who does not receive a
68. A certain type of digital camera comes in either a 3-megapixel version or a 4-megapixel version. A camera store has received a shipment of 15 of these cameras, of which 6 have 3-megapixel resolution. Suppose that 5 of these cameras are randomly selected to be stored behind the counter;the other
34. Suppose that the number of plants of a particular type found in a rectangular region (called a quadrat by ecologists) in a certain geographic area is an rv X with pmf p(x) { c/x3 x 1, 2, 3, . . .0 otherwise Is E(X) finite? Justify your answer (this is another distribution that statisticians
33. Let X be a Bernoulli rv with pmf as in Example 3.18.a. Compute E(X2).b. Show that V(X) p(1 p).c. Compute E(X79).
11. An automobile service facility specializing in engine tune-ups knows that 45% of all tune-ups are done on four-cylinder automobiles, 40% on six-cylinder automobiles, and 15% on eight-cylinder automobiles. Let X the number of cylinders on the next car to be tuned.a. What is the pmf of X?b.
12. Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table.
a. What is the probability that the flight will accommodate all ticketed passengers who show up?
b. What is the probability that not all ticketed passengers who show up can be accommodated?
c. If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are
13. A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time.Suppose the pmf of X is as given in the accompanying table.Calculate the probability of each of the following events.a. {at most three lines are in use}b. {fewer than three lines
15. Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose a computer manufacturer receives computer boards in lots of five. Two boards are selected from each lot for inspection. We can represent possible outcomes of the selection process
4. Let X the number of nonzero digits in a randomly selected zip code. What are the possible values of X? Give three possible outcomes and their associated X values.
6. Starting at a fixed time, each car entering an intersection is observed to see whether it turns left (L), right (R), or goes straight ahead (A). The experiment terminates as soon as a car is observed to turn left. Let X the number of cars observed. What are possible X values? List five
7. For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete.a. X the number of unbroken eggs in a randomly chosen standard egg cartonb. Y the number of students on a class list for a particular course who are absent
16. Some parts of California are particularly earthquake-prone.Suppose that in one metropolitan area, 30% of all homeowners are insured against earthquake damage. Four homeowners?a. Find the probability distribution of X. [Hint: Let S denote a homeowner who has insurance and F one who does not.Then
b. Draw the corresponding probability histogram.

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