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Statistics And Probability With Applications For Engineers And Scientists Using MINITAB R And JMP 2nd Edition Irwin Guttman, Kalanka P. Jayalath, Bhisham C. Gupta - Solutions
5. Scores obtained by students in three sections of the Medical College Admission Test (MCAT) are independently normally distributed with means 10, 12, and 10 and standard deviations 2.6, 1.2, and 1.3, respectively. Determine the probability distribution of the total scores obtained by these
determine the probability distribution of the random variable U = 2X + 3Y .5.7 Approximation of the Binomial and Poisson Distributions by the Normal Distribution 193
3. The times required to finish two projects, say X and Y, are independently and normally distributed with means of 70 and 75 minutes and standard deviations of 8 and 10 minutes, respectively. Find the following probabilities: (a) P(X + Y ≥ 145),(b) P(−18 ≤ X − Y ≤ 16), (c) P(122 ≤ X +
2. Suppose that independent random variables X and Y are normally distributed with means of 12 and 15 and standard deviations of 4 and 5, respectively. Find the m.g.f.of the random variable X + 2Y .
1. Suppose that independent random variables, say X and Y, are normally distributed with means of 10 and 15, and standard deviations of 3 and 4, respectively. Find the following probabilities: (a) P(X + Y ≥ 33), (b) P(−8 ≤ X − Y ≤ 6), (c) P(20 ≤X + Y ≤ 28), (d) P(X − 2Y ≤ −10).
8. A car manufacturer claims that its new hybrid car can travel at speeds more than 60 mpg. The actual mpg is approximately normally distributed with a mean of 52 miles and a standard deviation of three miles. What is the probability that the manufacturer’s claim is valid? Comment on this claim.
7. The height of a certain population of female teenagers is approximately normally distributed with a mean of 155 cm and a standard deviation of 7 cm. Find the probability that a randomly selected teenager has height (a) between 145 and 165 cm,(b) more than 150 cm, and (c) less than 169 cm.
6. The total cholesterol level (LDL + HDL + 20% of triglyceride) of US males between 60 and 70 years of age is approximately normally distributed with a mean of 175 mg/100 ml and a standard deviation of 20 mg/100 ml. Find the probability that a randomly selected person from this population has
5. The amount of beverage in a 16-oz bottle is normally distributed with a mean of 16.2 oz and a standard deviation of 0.1 oz. Find the probability that the amount of beverage in a randomly select can is (a) between 15.5 and 16.2 oz, (b) more than 16.4 oz, (c) less than 16.1 oz.5.6 Distribution of
4. The postsurgery survival time of a breast cancer patient is normally distributed with a mean of eight years and a standard deviation of 1.5 years. Find the probabilities that a woman with breast cancer will survive after her surgery: (a) between five and seven years, (b) more than 11 years, (c)
3. The weights of Maine lobsters at the time of their catch are normally distributed with a mean of 1.8 lb and a standard deviation of 0.25 lb. What is the probability that a randomly selected lobster weighs (a) between 1.5 and 2 lb?, (b) more than 1.55 lb?, (c) less than 2.2 lb?
2. A random variable X is normally distributed with unknown mean μ and unknown standard deviation σ. Find μ and σ if it is known that the probability that X is less than 10 is 0.6950 and the probability that X exceeds 6 is 0.7939.
1. A random variable X is normally distributed with mean μ = 10 and standard deviationσ = 1.5. Find the following probabilities: (a) P(8 ≤ X ≤ 12), (b) P(X ≤ 12),(c) P(X ≥ 8.5).
8. A manufacturing company has designed a fuel-injection system for medium-size cars such that the cars yield X mpg uniformly distributed over an interval [45, 50].Find the mean and variance of X. Find the probability P(X >μ− σ).
7. The time X (in hours) taken by students to complete a standardized test is uniformly distributed over the interval [2, 4]. Find the mean and the standard deviation of X. Then, find the probability P(μ − 2σ
6. Suppose that a random variable X is distributed uniformly over an interval [15, 25].Find the following probabilities: (a) P(20 ≤ X ≤ 25), (b) P(X ≤ 25), (c) P(15 ≤X ≤ 25), (d) P(X ≤ 15).
5. The hourly wages of certain group of workers in a large manufacturing company are uniformly distributed on the interval [20, 32]. What percentage of these workers are making over $25 an hour?
4. A random variable X is uniformly distributed with mean 12 and variance three.Find the m.g.f. of the random variable X.
3. Referring to Problem 2, let the random variable X denote the time that the passenger has to wait. Find the mean, variance, and standard deviation of the random variable X.
2. The city buses stop at an assigned point every 15 minutes. A passenger who takes the bus from that point has arrived there but does not know when the last bus came.(a) What is the probability that the passenger will have to wait for more than eight minutes to take the next bus?180 5 Continuous
1. The time taken to download software from the internet is uniformly distributed between four and 10 minutes.(a) What is the probability that the downloading time for software is more than six minutes?(b) What is the probability that the downloading time for software is between five and eight
6. Suppose that X is a random variable having probability distribution with mean 16 and standard deviation 2.5. What can you say about the probability P(8.5 ≤ X ≤23.5)? (Hint: use Chebyshev’s inequality.)
5. Hospital records indicate that patients with heart surgery spend time (in days) in the hospital having a probability distribution with mean six days and standard deviation 1.5 days. Use Chebyshev’s inequality to find the lower bound of the percentage of patients who stay between three and nine
4. According to Chebyshev’s inequality, what can we say about the proportion of the data of a given data set that must fall within k standard deviations of the mean, for values of k as follows? (a) k = 3, (b) k = 4, (c) k = 8.
3. The following data give the number of defective parts manufactured in the last 20 shifts:6 18 10 12 11 10 11 9 14 7 13 6 9 10 7 12 8 16 12 17 5.4 The Uniform Distribution 175 Find the sample mean ¯X and the sample standard deviation S. Find the percentage of data points that fall in each of the
2. A random variable X is distributed with mean 16 and standard deviation 3. Using Chebyshev’s inequality, find lower or upper bounds for the following probabilities:(a) P(|X − 16| ≤ 6)(b) P(|X − 16| ≥ 7.5)(c) P(11.5 ≤ X ≤ 20.5)
1. The following data give the number of calls received by a receptionist in 20 randomly selected intervals of one hour each:16 19 19 15 15 19 17 18 18 20 18 18 17 17 16 16 17 18 18 20 Find the sample mean ¯X and the sample standard deviation S. Find the percentage of data points that fall in each
9. Refer to Problems 3 and 4 above and find the following probabilities:(a) P(μ −σ
8. The amount of time X (in minutes) by which the departure of a flight is rescheduled has the probability distribution f(x) =c(100 − x2), −10 < x < 10 0, otherwise Find the value ofc. Then, find the following:(a) The mean μ and standard deviation σ of X.(b) The mean and standard deviation
7. Suppose a random variable X has the p.d.f. f(x) given by f(x) = λe−λx, x>0, λ > 0 Find the m.g.f. of the random variable X and use your result to find the mean μand the variance σ2 of the random variable X.
5. The lifetime X (in units of 10 years) of a certain component of a home-heating furnace is a random variable with p.d.f.f(x) =cx2(1 − x), if 0 < x < 1 0, otherwise Determine the value ofc. Then find the probability that the life of such a component is more than eight years.
4. In Problem 3, find the mean μ and the variance σ2 of the random variable X.
3. The probability function of the amount of soft drink in a can is f(x) = 4cx for 11.5 < X < 12.5 oz. Determine the value of c such that f(x) represents a p.d.f..Then, find the following probabilities:(a) P(X >11.5), (b) P(X
2. Determine the value of c such that f(x) = c x2 for x > 1 represents the p.d.f. of a random variable X.
1. Suppose that the p.d.f. of a random variable X is f(x) = 2e−2x for x > 0. Determine the following probabilities:(a) P(X >4), (b) P(X
58. Of all customers buying cars at a car fair, 60% buy an American car. Let a random variable X represent the number of customers who bought an American car out of a total 50 cars sold at the fair.(a) Describe the probability distribution of the random variable X.(b) Determine the expected value
57. A pool of 15 applicants for the position of a manager for a firm consists of 10 applicants holding master’s degrees and five holding PhD degrees. An interviewer randomly selects eight applicants to interview. Determine the probabilities of the following events:(a) He/she selects three
56. The random variable X is distributed by the geometric distribution with p = 0.05.Determine the following probabilities: (a) P(X ≥ 24), (b) P(15 ≤ X ≤ 30), (c) P(X >28).
55. A quality control engineer is interested to find how many parts he/she needs to inspect to detect the first defective part. If the probability that a randomly selected part is defective is p and X is the number of parts inspected needed to detect the first defective part, then X is a random
54. The probability that a part manufactured at a plant being defective is 0.001. On a given day, the plant manufactured 10,000 parts. Find the following probabilities:(a) At least five parts are defective.(b) No more than eight parts are defective.(c) Between four and nine (inclusive) parts are
53. A circuit board of a very complex electronic system has 500 soldered joints. The probability that a joint becomes loose in one year use of the circuit board is 0.001.(a) What is the probability that in one year three joints become loose?(b) What is the probability that in one year at least two
52. In analyzing a large data set, a life insurance company estimated the probability that a person in the 70-80 years of age group dies due to natural causes in any given year Review Practice Problems 163 as 0.000002. If the company has 500,000 insurers in that age group, determine the probability
51. Refer to Problem 48. Find the mean and the variance of X + c (c is constant) and comment.
50. Let the random variable X have a discrete uniform distribution on the integers 0 ≤x ≤ 50. Determine the mean and the variance of X.
49. Let X be a random variable that is Bernoulli distributed with parameter p. Find the moment-generating function of X. Then, use the moment-generating function to find the mean and the variance of X.
48. Let X be a random variable having the uniform distribution on x = 1, 2, . . . , N. Find the mean and the variance of X.
47. Determine the mean and the variance of the following probability functions:(a) p(x) = x/21; x = 1, 2, 3, 4, 5, 6 and zero elsewhere.(b) p(x) = (x2 − 1)/50; x = 1, 2, 3, 4, 5 and zero elsewhere.
45. Determine the value of the constant c such that the following functions are valid probability functions:(a) p(x) = cx/20; x = 1, 2, 3, 4, 5, 6 and zero elsewhere.(b) p(x) = c(x2 + 1); x = 1, 2, 3, 4, 5 and zero elsewhere.(c) p(x) = c(x − 1); x = 1, 2, 3, 4, 5, 6 and zero elsewhere.
44. For the functions that are valid probability functions in Problem 43, find the mean and the variance of X.
43. Which of the following functions are valid probability functions? Explain.(a) p(x) = x/20; x = 1, 2, 3, 4, 5, 6 and zero elsewhere.(b) p(x) = x2/140; x = 1, 2, 3, 4, 5, 6, 7 and zero elsewhere.(c) p(x) = (x − 3)/5; x = 2, 3, 4, 5, 6 and zero elsewhere.
42. The number of patients admitted in an emergency room of a metropolitan hospital can be modeled as a Poisson random variable. Assume that on the average, five patients are admitted every hour.(a) What is the probability that exactly four patients are admitted in the next one hour?(b) What is the
41. A batch of 500 car batteries is scheduled to be shipped if a random sample of 20 from the batch has two or fewer defective batteries. If it is known that there are 60 defective batteries in the batch, find the probability that the batch will be shipped.
40. A six-sigma green belt quality control engineer found that on average, batches of 500 computer chips have exactly two defective chips.(a) Using the formula for the Poisson distribution, determine the probability that a box of 1000 chips will have exactly 5 defective chips.162 4 Discrete Random
39. Just before the 2006 midterm elections of the United States, one of the polling agencies found that 60% of the voters were against the Iraq war. Assume that this result is valid for all the voters in the entire country. Using the binomial distribution table(Table A.2), compute the probability
38. Indicate which of the following experiments can be studied using a binomial model.Justify your answer.(a) Drawing five ball-bearings with replacement from a box containing 25 ball-bearings, 10 of which are of diameter 10mm and 15 of diameter 20 mm, and observing the diameters of the drawn
37. On average, the number of customers arriving per every ten minutes at a teller’s window in a bank is four. Find the probability that during the next 10 min:(a) At least five customers will arrive at that teller’s window.(b) No more than two customers will arrive at that teller’s
36. A programmer makes two wrong entries every hour, on the average. Find the probability that during the next five hours she will make(a) Fewer than eight wrong entries.(b) At least four wrong entries.(c) Between three to five (inclusive) wrong entries.(d) More than one wrong entry.
35. An insurance company discovered that three policyholders out of every 1000 insured against a particular kind of accident file a claim every year. Suppose that the company Review Practice Problems 161 has 2000 persons who are insured against that kind of accident. Find the probability that(a)
34. An engineering club consists of five seniors and seven juniors. Suppose that five club members are selected randomly to form a committee. Find the probability that(a) The committee has at least two juniors.(b) The committee has three or more seniors.(c) The committee has no more than two
33. Let X denote the number of defective chips. Find the mean, variance, and the standard deviation of the random variable X.
32. A box of 100 computer chips contains eight defective chips. Suppose that a random sample of size 10 chips is selected without replacement from that box. Find the probability that the sample had(a) At least one defective chip.(b) All defective chips.(c) Nine defective chips.(d) No defective
31. The drug Xanax is used to control an anxiety problem. However, it is believed that 70% of the users get addicted to the drug. Suppose that we take a random sample of 15 Xanax users and find the number of persons addicted to Xanax. Find the probability that:(a) More than 10 are addicted to
30. Let X be a random variable distributed as binomial distribution with n = 25 and p = 0.35. Find the mean, variance, and the standard deviation of the random variable X.
29. In testing electric bulbs for a certain kind of projector, it is found that 40% of the bulbs burn out before the warranty period. Suppose that the engineering department of a school buys 12 such bulbs. Find the probability that:(a) Between four to six bulbs (inclusive) burn out before the
28. By using the moment-generating function (4.6.4) of a random variable X having the binomial distribution (4.6.2), show that the mean and variance of X are np and npq, respectively.
27. A lot contains N articles of which Np are defective. Articles are drawn successively at random and without replacement until k defectives are drawn. Let X be a random variable denoting the number of articles that must be drawn to achieve this objective.160 4 Discrete Random Variables and Some
26. Referring to Problem 25, show that P(Y >s+ t|Y >s) = P(Y >t) =∞y=t+1 pqy−1.(This result implies that the geometric distribution has no memory, for if the event of a failure has not occurred during the first s trials, then the probability that a failure will not occur in the next t trials
25. In the Problem 24 above, suppose k = 1. Show that if Y is a random variable denoting the number of trials required to obtain one failure, then the p.f. of Y is p(y) = pqy−1, y= 1, 2, . . .and that its mean and variance are 1/p and q/p2, respectively. This distribution is called the geometric
24. A device fails to operate on a single trial with probability p. Let X be a random variable denoting the number of trials required to obtain a total of k failures. If results of successive trials are independent, show that the probability function of X is given by p(x) =x − 1 k −
23. In Problem 22, suppose that another experiment requires four successful launches.What is the probability that six attempts will be required? What is the probability that x attempts will be required?
22. In 1970s, the proportion of successful launches at a test missile site has been 0.85.Suppose that an experiment is planned that requires three successful launches. What is the probability that exactly x attempts will be necessary? exactly five? exactly seven? fewer than six?
21. In testing a relay, suppose that the probability is p that it fails to make satisfactory contact in a single trial and that p remains unchanged over a very large number of trials. By assuming the outcomes of successive trials to be independent, what is the probability that(a) x trials have to
20. A lot of N articles has d defectives. If articles are taken at random from the lot one at a time, what is the probability, assuming sampling without replacement, that(a) Exactly x articles have to be examined to find the first defective?(b) Exactly x articles have to be examined to find the dth
19. If the probability is 0.6 that an engineer will pass the six-sigma black belt test in the first attempt, use the formula for the binomial distribution to find the probability that 6 of 10 engineers taking that test will pass in the first attempt.Review Practice Problems 159
17. The fraction of articles turned out by a machine that is defective is equal to p. The defectives occur “at random” during production. The articles are boxed m per box and cartoned n boxes per carton (assume production numbers are large).(a) If a box of articles is taken at random, what is
16. Suppose that a lot of 10,000 articles has 200 defectives and that a random sample of 100 articles is drawn from the lot (without replacement).(a) What is the probability of getting exactly x defectives in the sample?(b) Determine the binomial approximation for the probability in (a).(c)
15. An urn contains 10 white and 20 black balls. Balls are drawn one by one, without replacement, until five white ones have appeared. Let X be the number of draws necessary to find five white balls. What is the sample space of X? Find an expression for the probability of the event X = x.
14. Two coins are tossed n times. Find the probability of x, the number of times no heads appear; y, the number of times one head appears; and z, the number of times two heads appear (x + y + z = n).
13. Suppose that 10 people each throw two coins. What is the probability that:(a) Three people throw two heads, three people throw two tails, and four people throw one head and one tail?(b) No one threw a head and a tail?
12. Small brass pins are made by company ABC. Of the pins manufactured, 2% are undersized, 6% are oversized, and 92% are satisfactory. The pins are boxed 100 per box. A box is taken at random. Write down the expression for the probabilities of the following events:(a) The box contains x
11. A process for making plate glass produces an average of four “seeds” (small bubbles)scattered at random in the glass per 100 ft2. With the use of the Poisson distribution, what is the probability that(a) A piece of plate glass five ft by 10 ft will contain more than two seeds?(b) Six pieces
10. A bag of grass seed is known to contain 1% weed seeds. A sample of 100 seeds is drawn. Find the probabilities of 0, 1, 2, 3, . . . , 7 weed seeds being in the sample.158 4 Discrete Random Variables and Some Important Discrete Probability Distributions
9. It is known that 0.0005% of the insured males die from a certain kind of accident each year. What is the probability that an insurance company must pay off on more than three of 10,000 insured against such accidents in a given year?
8. What is the probability of throwing two heads three times in four throws of five coins?
7. Suppose that 13 cards are dealt from a thoroughly shuffled deck of ordinary playing cards.(a) What is the probability of getting x spades?(b) What is the probability of getting y hearts? Describe the sample space of y.(c) What is the probability of getting x spades and y hearts? Describe the
6. Suppose that 5% of the aspirins pressed by a certain type of machine are chipped.The tablets are boxed 12 per box. What percent of the boxes would you estimate:(a) To be free of chipped tablets?(b) To have not more than one chipped tablet?(c) To have exactly x chipped tablets?
5. What is the probability of drawing a 13-card hand containing no aces, kings, queens, or jacks?
4. If the probability of hitting a target is 0.2 and 10 shots are fired independently, what is the probability that the target will be hit at least once? At least twice?
3. In rolling five true dice, find the probability of obtaining at least one ace, exactly one ace, exactly two aces. (Here, ace implies one point.)
2. A lot contains 30 items, six of which are defective. What is the probability that a random sample of five items from the lot will contain no defective items? No more than one defective? More than two defectives? (Assume that sampling is without replacement.)
1. Suppose that a lot of 50 fuses, of which seven are known to be defective, is available for sampling. It is proposed to draw 10 fuses at random (without replacement) and test them. What is the probability that such random samples of 10 will contain 0, 1, 2, . . . , 7 defective fuses?
7. A quality control engineer in a manufacturing company detects that a recently purchased machine produces 5% of parts that do not meet the specifications. What is the probability that the third part that does not meet the specifications is the 40th part produced by that machine?
6. The probability that a baseball player hits a home run in any one inning is 0.30.What is the probability that he will hit a second home run in the eighth inning if he bats every inning?
5. The night shift in a manufacturing plant is known to produce 10% of its items defective. A quality control engineer inspects all the items that were manufactured in a given night shift. What is the probability that to find the fifth defective item, the engineer will have to inspect at least 31
4. Referring to Problem 3, find the probability that the 14th success occurs before the 12th failure.
3. Suppose that independent trials are carried out, each resulting in a success with probability 0.6. What is the probability that the ninth success occurs in the 20th trial?
2. Consider a sequence of independent Bernoulli trials with probability of success being 0.25. Determine the probability that (a) the fifth success occurs at the 16th trial,(b) the fourth success occurs at the 10th trial.
1. A manufacturing company of wind turbines found that the probability that a turbine is nonconforming is 0.03. Assume that the turbines are conforming or nonconforming independently. Find the probability that the third nonconforming turbine is the 100th turbine manufactured by that company.156 4
8. A random variable X is distributed by the binomial distribution with n = 15, p = 0.1.Find the following probabilities, first using the binomial distribution and then using the Poisson approximation to the binomial distribution. Compare your results.4.9 The Negative Binomial Distribution 153(a)
7. Suppose that the probability that an insurance company pays out a claim in a given six-month period against a car theft is 0.0003. Find the probability that of the 15,000 insurers against car theft, it will pay out at least 10 claims during any given year.
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