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introduction to probability statistics
Introduction To Probability And Statistics For Engineers And Scientists 3rd Edition Sheldon M. Ross - Solutions
In an attempt to show that proofreader A is superior to proofreader B, both proofreaders were given the same manuscript to read. If proofreader A found 28 errors, and proofreader B found 18, with 10 of these errors being found by both, can we conclude that A is the superior proofreader?
An industrial concern runs two large plants. If the number of accidents during the last 8 weeks at plant 1 were 16, 18, 9, 22, 17, 19, 24, 8 while the number of accidents during the last 6 weeks at plant 2 were 22, 18, 26, 30, 25, 28, can we conclude, at the 5 percent level of significance, that
Management’s claim that the mean number of defective computer chips produced daily is not greater than 25 is in dispute. Test this hypothesis, at the 5 percent level of significance, if a sample of 5 days revealed 28, 34, 32, 38, and 22 defective chips.
Suppose that method 1 resulted in 20 unacceptable transistors out of 100 produced; whereas method 2 resulted in 12 unacceptable transistors out of 100 produced.Can we conclude from this, at the 10 percent level of significance, that the two methods are equivalent?
Historical data indicate that 4 percent of the components produced at a certain manufacturing facility are defective. A particularly acrimonious labor dispute has recently been concluded, and management is curious about whether it will result in any change in this figure of 4 percent. If a random
In Example 8.6a, np0 = 300(.02) = 6, andConsequently, the p-value that results from the data X = 10 isSuppose now that we want to test the null hypothesis that p is equal to some specified value; that is, we want to test npo(1 - Po) = 5.88.
A computer chip manufacturer claims that no more than 2 percent of the chips it sends out are defective. An electronics company, impressed with this claim, has purchased a large quantity of such chips. To determine if the manufacturer’s claim can be taken literally, the company has decided to
There are two different choices of a catalyst to stimulate a certain chemical process. To test whether the variance of the yield is the same no matter which catalyst is used, a sample of 10 batches is produced using the first catalyst, and 12 using the second.If the resulting data is S2 1= .14 and
A machine that automatically controls the amount of ribbon on a tape has recently been installed. This machine will be judged to be effective if the standard deviationσ of the amount of ribbon on a tape is less than .15 cm. If a sample of 20 tapes yields a sample variance of S2 = .025 cm2, are we
An industrial safety program was recently instituted in the computer chip industry. The average weekly loss (averaged over 1 month) in man-hours due to accidents in 10 similar plants both before and after the program are as follows:Determine, at the 5 percent level of significance, whether the
Reconsider Example 8.4a, but now suppose that the population variances are unknown but equal.
Twenty-two volunteers at a cold research institute caught a cold after having been exposed to various cold viruses. A random selection of 10 of these volunteers was given tablets containing 1 gram of vitamin C. These tablets were taken four times a day.The control group consisting of the other 12
Two new methods for producing a tire have been proposed. To ascertain which is superior, a tire manufacturer produces a sample of 10 tires using the first method and a sample of 8 using the second. The first set is to be road tested at location A and the second at location B. It is known from past
In a single-server queueing system in which customers arrive according to a Poisson process, the long-run average queueing delay per customer depends on the service distribution through its mean and variance. Indeed, if μ is the mean service time, and σ2 is the variance of a service time, then
The manufacturer of a new fiberglass tire claims that its average life will be at least 40,000 miles. To verify this claim a sample of 12 tires is tested, with their lifetimes(in 1,000s of miles) being as follows:Test the manufacturer’s claim at the 5 percent level of significance. Tire 1 2 4 56
A public health official claims that the mean home water use is 350 gallons a day. To verify this claim, a study of 20 randomly selected homes was instigated with the result that the average daily water uses of these 20 homes were as follows:Do the data contradict the official’s claim? 340 344
Among a clinic’s patients having blood cholesterol levels ranging in the medium to high range (at least 220 milliliters per deciliter of serum), volunteers were recruited to test a new drug designed to reduce blood cholesterol. A group of 50 volunteers was given the drug for 1 month and the
All cigarettes presently on the market have an average nicotine content of at least 1.6 mg per cigarette. A firm that produces cigarettes claims that it has discovered a new way to cure tobacco leaves that will result in the average nicotine content of a cigarette being less than 1.6 mg. To test
Suppose in Example 8.3a that we know in advance that the signal value is at least as large as 8. What can be concluded in this case?
For the problem of Example 8.3a, how many signals need be sent so that the .05 level test of H0 : μ = 8 has at least a 75 percent probability of rejection whenμ = 9.2?
For the problem presented in Example 8.3a, let us determine the probability of accepting the null hypothesis that μ = 8 when the actual value sent is 10. To do so, we compute ---2--5 = (-or)-
In Example 8.3a, suppose that the average of the 5 values received is X = 8.5. In this case,Sinceit follows that the p-value is .576 and thus the null hypothesis H0 that the signal sent has value 8 would be accepted at any significance level α On the other hand, if the average of the data values
It is known that if a signal of value μ is sent from location A, then the value received at location B is normally distributed with mean μ and standard deviation 2.That is, the random noise added to the signal is an N(0, 4) random variable. There is reason for the people at location B to suspect
21. Can we use the runs test if we consider whether each data value is less than or greater than some predetermined value rather than the value s-med?
20. The following data represent the successive quality levels of 25 articles: 100, 110, 122, 132, 99, 96, 88, 75, 45, 211, 154, 143, 161, 142, 99, 111, 105, 133, 142, 150, 153, 121, 126, 117, 155. Does it appear that these data are a random sample from some population?
19. A production run of 50 items resulted in 11 defectives, with the defectives occurring on the following items (where the items are numbered by their order of production): 8, 12, 13, 14, 31, 32, 37, 38, 40, 41, 42. Can we conclude that the successive items did not constitute a random sample?
17. In a 10-year study of the dispersal patterns of beavers (Sun and Muller-Schwarze,“Statistical resampling methods in biology: A case study of beaver dispersal patterns,” American Journal of Mathematical and Management Sciences, 16, pp. 463–502, 1996) a total of 332 beavers were trapped in
16. In a 1943 experiment (Whitlock and Bliss, “A bioassay technique for antihelminthics,”Journal of Parasitology, 29, pp. 48–58, 10), albino rats were used to study the effectiveness of carbon tetrachloride as a treatment for worms. Each rat received an injection of worm larvae. After 8 days,
14. Determine the p-value in Problem 13 by(a) using the normal approximation;(b) using a simulation study.
13. Fifteen cities, of roughly equal size, are chosen for a traffic safety study. Eight of them are randomly chosen, and in these cities a series of newspaper articles dealing with traffic safety is run over a 1-month period. The number of traffic accidents reported in the month following this
11. Let X1, . . . , Xn be a sample from the continuous distribution F having median m;and suppose we are interested in testing the hypothesis H0 : m = m0 against the one-sided alternative H1 : m > m0. Present the one-sided analog of the signed rank test. Explain how the p-value would be computed.
7. Determine the p-value when using the signed rank statistic in Problems 1 and 2.
5. In 1987, the national median salary of all U.S. physicians was $124,400. A recent random sample of 14 physicians showed 1990 incomes of (in units of $1,000)125.5, 130.3, 133.0, 102.6, 198.0, 232.5, 106.8, 114.5, 122.0, 100.0, 118.8, 108.6, 312.7, 125.5 Use these data to test the hypothesis that
4. To test the hypothesis that the median weight of 16-year-old females from Los Angeles is at least 110 pounds, a random sample of 200 such females was chosen. If 120 females weighed less than 110 pounds, does this discredit the hypothesis? Use the 5 percent level of significance. What is the
3. The published figure for the median systolic blood pressure of middle-aged men is 128. To determine if there has been any change in this value, a random sample of 100 men has been selected. Test the hypothesis that the median is equal to 128 if(a) 60 men have readings above 128;(b) 70 men have
Use the sign test to determine if the medicine has an effect on blood pressure.What is the p-value?
The lifetime of 19 successively produced storage batteries is as follows:145 152 148 155 176 134 184 132 145 162 165 185 174 198 179 194 201 169 182 The sample median is the 10th smallest value—namely, 169. The data indicating whether the successive values are less than or equal to or greater
The following is the result of the last 30 games played by an athletic team, with W signifying a win and L a loss.W W W LW W LW W LW LW W LW W W W LW LW W W LW LW L Are these data consistent with pure randomness?
Running the text disk program on the data of Example 12.4c yields Figure 12.4, which is quite close to the exact value of .1225. Running the program using the data of Example 12.4d yields Figure 12.5, which is again quite close to the exact value of .0364.
Suppose that in testing whether 2 production methods yield identical results, 9 items are produced using the first method and 13 using the second. If, among all 22 items, the sum of the ranks of the 9 items produced by method 1 is 72, what conclusions would you draw?
Suppose we are interested in determining whether a certain population has an underlying probability distribution that is symmetric about 0. If a sample of size 20 from this population results in a signed rank test statistic of value 142, what conclusion can we draw at the 10 percent level of
A financial institution has decided to open an office in a certain community if it can be established that the median annual income of families in the community is greater than $90,000. To obtain information, a random sample of 80 families was chosen, and the family incomes determined. If 28 of
25. Data are said to be from a lognormal distribution with parameters μ and σ if the natural logarithms of the data are normally distributed with mean μ and standard deviation σ. Use the Kolmogorov–Smirnov test with significance level .05 to decide whether the following lifetimes (in days) of
17. Repeat Problem 16 with all of the data values doubled—that is, with these data:20 80 30 270
14. In Problem 4, test the hypothesis that the daily number of failures has a Poisson distribution.
13. A sample of size 120 had a sample mean of 100 and a sample standard deviation of 15. Of these 120 data values, 3 were less than 70; 18 were between 70 and 85;30 were between 85 and 100; 35 were between 100 and 115; 32 were between 115 and 130; and 2 were greater than 130. Test the hypothesis
12. Use simulation to determine the p-value and compare it with the result you obtained using the chi-square approximation in Problem 1. Let the number of simulation runs be(a) 1,000;(b) 5,000;(c) 10,000.
11. Sometimes reported data fit a model so well that it makes one suspicious that the data are not being accurately reported. For instance, a friend of mine has reported that he tossed a fair coin 40,000 times and obtained 20,004 heads and 19,996 tails. Is such a result believable? Explain your
9. In a certain region, insurance data indicate that 82 percent of drivers have no accidents in a year, 15 percent have exactly 1 accident, and 3 percent have 2 or more accidents. In a random sample of 440 engineers, 366 had no accidents, 68 had exactly 1 accident, and 6 had 2 or more. Can you
5. Among 100 vacuum tubes tested, 41 had lifetimes of less than 30 hours, 31 had lifetimes between 30 and 60 hours, 13 had lifetimes between 60 and 90 hours, and 15 had lifetimes of greater than 90 hours. Are these data consistent with the hypothesis that a vacuum tube’s lifetime is exponentially
3. Determine the birth and death dates of 100 famous individuals and, using the four-category approach of Example 11.2a, test the hypothesis that the death month is not affected by the birth month. Use the chi-square approximation.
1. According to the Mendelian theory of genetics, a certain garden pea plant should produce either white, pink, or red flowers, with respective probabilities 14, 12, 14.To test this theory, a sample of 564 peas was studied with the result that 141 produced white, 291 produced pink, and 132 produced
Consider an experiment having six possible outcomes whose probabilities are hypothesized to be .1, .1, .05, .4, .2, and .15. This is to be tested by performing 40 independent replications of the experiment. If the resultant number of times that each of the six outcomes occurs is 3, 3, 5, 18, 4, 7,
A contractor who purchases a large number of fluorescent lightbulbs has been told by the manufacturer that these bulbs are not of uniform quality but rather have been produced in such a way that each bulb produced will, independently, either be of quality level A, B, C, D, or E, with respective
In recent years, a correlation between mental and physical well-being has increasingly become accepted. An analysis of birthdays and death days of famous people could be used as further evidence in the study of this correlation. To use these data, we are supposing that being able to look forward to
An insurance company has 25,000 automobile policy holders. If the yearly claim of a policy holder is a random variable with mean 320 and standard deviation 540, approximate the probability that the total yearly claim exceeds 8.3 million
Civil engineers believe that W, the amount of weight (in units of 1,000 pounds) that a certain span of a bridge can withstand without structural damage resulting, is normally distributed with mean 400 and standard deviation 40. Suppose that the weight (again, in units of 1,000 pounds) of a car is a
The ideal size of a first-year class at a particular college is 150 students.The college, knowing from past experience that, on the average, only 30 percent of those accepted for admission will actually attend, uses a policy of approving the applications of 450 students. Compute the probability
The weights of a population of workers have mean 167 and standard deviation 27.(a) If a sample of 36 workers is chosen, approximate the probability that the sample mean of their weights lies between 163 and 170.(b) Repeat part (a) when the sample is of size 144.
An astronomer wants to measure the distance from her observatory to a distant star. However, due to atmospheric disturbances, any measurement will not yield the exact distanced. As a result, the astronomer has decided to make a series of measurements and then use their average value as an estimate
The time it takes a central processing unit to process a certain type of job is normally distributed with mean 20 seconds and standard deviation 3 seconds. If a sample of 15 such jobs is observed, what is the probability that the sample variance will exceed 12?
Suppose that 45 percent of the population favors a certain candidate in an upcoming election. If a random sample of size 200 is chosen, find(a) the expected value and standard deviation of the number of members of the sample that favor the candidate;(b) the probability that more than half the
According to the U.S. Department of Agriculture’s World Livestock Situation, the country with the greatest per capita consumption of pork is Denmark. In 1994, the amount of pork consumed by a person residing in Denmark had a mean value of 147 pounds with a standard deviation of 62 pounds. If a
1. Plot the probability mass function of the sample mean of X1, . . . , Xn, when(a) n = 2;(a) n = 3.Assume that the probability mass function of the Xi is P{X = 0} = .2, P{X = 1} = .3, P{X = 3} = .5 In both cases, determine E[X ] and Var(X ).
2. If 10 fair dice are rolled, approximate the probability that the sum of the values obtained (which ranges from 20 to 120) is between 30 and 40 inclusive.
3. Approximate the probability that the sum of 16 independent uniform (0, 1)random variables exceeds 10.
4. A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you bet 1 on a specified number, you either win 35 if the roulette ball lands on that number or lose 1 if it does not. If you continually make such bets, approximate the probability that (a) you are winning after 34 bets;(b) you
5. A highway department has enough salt to handle a total of 80 inches of snowfall.Suppose the daily amount of snow has a mean of 1.5 inches and a standard deviation of .3 inches.(a) Approximate the probability that the salt on hand will suffice for the next 50 days.(b) What assumption did you make
6. Fifty numbers are rounded off to the nearest integer and then summed. If the individual roundoff errors are uniformly distributed between −.5 and .5, what is the approximate probability that the resultant sum differs from the exact sum by more than 3?
7. A six-sided die, in which each side is equally likely to appear, is repeatedly rolled until the total of all rolls exceeds 400. Approximate the probability that this will require more than 140 rolls.
8. The amount of time that a certain type of battery functions is a random variable with mean 5 weeks and standard deviation 1.5 weeks. Upon failure, it is immediately replaced by a new battery. Approximate the probability that 13 or more batteries will be needed in a year.
9. The lifetime of a certain electrical part is a random variable with mean 100 hours and standard deviation 20 hours. If 16 such parts are tested, find the probability that the sample mean is(a) less than 104;(b) between 98 and 104 hours.
10. A tobacco company claims that the amount of nicotine in its cigarettes is a random variable with mean 2.2 mg and standard deviation .3 mg. However, the sample mean nicotine content of 100 randomly chosen cigarettes was 3.1 mg. What is the approximate probability that the sample mean would have
11. The lifetime (in hours) of a type of electric bulb has expected value 500 and standard deviation 80. Approximate the probability that the sample mean of n such bulbs is greater than 525 when(a) n = 4;(b) n = 16;(c) n = 36;(d) n = 64.
12. An instructor knows from past experience that student exam scores have mean 77 and standard deviation 15. At present the instructor is teaching two separate classes — one of size 25 and the other of size 64.(a) Approximate the probability that the average test score in the class of size 25
13. If X is binomial with parameters n = 150, p = .6, compute the exact value of P{X ≤ 80} and compare with its normal approximation both (a) making use of and (b) not making use of the continuity correction.
14. Each computer chip made in a certain plant will, independently, be defective with probability .25. If a sample of 1,000 chips is tested, what is the approximate probability that fewer than 200 chips will be defective?
15. A club basketball team will play a 60-game season. Thirty-two of these games are against class A teams and 28 are against class B teams. The outcomes of all the games are independent. The team will win each game against a class A opponent with probability .5, and it will win each game against a
16. Argue, based on the central limit theorem, that a Poisson random variable having mean λ will approximately have a normal distribution with mean and variance both equal to λ when λ is large. If X is Poisson with mean 100, compute the exact probability that X is less than or equal to 116 and
17. Use the text disk to compute P{X ≤ 10} when X is a binomial random variable with parameters n = 100, p = .1. Now compare this with its (a) Poisson and(b) normal approximation. In using the normal approximation, write the desired probability as P{X Poisson (5) 0.20 0.15 0.10- 0.05- 0.0 0 5 10
18. The temperature at which a thermostat goes off is normally distributed with variance σ2. If the thermostat is to be tested five times, find(a) P{S2/σ2 ≤ 1.8}(b) P{.85 ≤ S2/σ2 ≤ 1.15}where S2 is the sample variance of the five data values.
19. In Problem 18, how large a sample would be necessary to ensure that the probability in part (a) is at least .95?
20. Consider two independent samples—the first of size 10 from a normal population having variance 4 and the second of size 5 from a normal population having variance 2. Compute the probability that the sample variance from the second sample exceeds the one from the first. (Hint: Relate it to the
21. Twelve percent of the population is left-handed. Find the probability that there are between 10 and 14 left-handers in a random sample of 100 members of this population. That is, find P{10 ≤ X ≤ 14}, where X is the number of left-handers in the sample.
22. Fifty-two percent of the residents of a certain city are in favor of teaching evolution in high school. Find or approximate the probability that at least 50 percent of a random sample of size n is in favor of teaching evolution, when(a) n = 10;(b) n = 100;(c) n = 1,000;(d) n = 10,000.
23. The following table gives the percentages of individuals, categorized by gender, that follow certain negative health practices. Suppose a random sample of 300 men is chosen. Approximate the probability that(a) at least 150 of them rarely eat breakfast;(b) fewer than 100 of them smoke. Sleeps 6
24. (Use the table from Problem 23.) Suppose a random sample of 300 women is chosen. Approximate the probability that(a) at least 60 of them are overweight by 20 percent or more;(b) fewer than 50 of them sleep 6 hours or less nightly.
25. (Use the table from Problem 23.) Suppose random samples of 300 women and of 300 men are chosen. Approximate the probability that more women than men rarely eat breakfast.
26. The following table uses 1989 data concerning the percentages of male and female full-time workers whose annual salaries fall in different salary groupings. Suppose random samples of 1,000 men and 1,000 women were chosen. Use the table to approximate the probability that(a) at least half of the
27. In 1995 the percentage of the labor force that belonged to a union was 14.9. If five workers had been randomly chosen in that year, what is the probability that none of them would have belonged to a union? Compare your answer to what it would be for the year 1945, when an all time high of 35.5
28. The sample mean and sample standard deviation of all San Francisco student scores on the most recent Scholastic Aptitude Test examination in mathematics were 517 and 120. Approximate the probability that a random sample of 144 students would have an average score exceeding(a) 507;(b) 517;(c)
29. The average salary of newly graduated students with bachelor’s degrees in chemical engineering is $43,600, with a standard deviation of $3,200. Approximate the probability that the average salary of a sample of 12 recently graduated chemical engineers exceeds $45,000.
Suppose that n independent trials, each of which is a success with probability p, are performed. What is the maximum likelihood estimator of p?
Two proofreaders were given the same manuscript to read. If proofreader 1 found n1 errors, and proofreader 2 found n2 errors, with n1,2 of these errors being found by both proofreaders, estimate N, the total number of errors that are in the manuscript
Suppose X1, . . . , Xn are independent Poisson random variables each having mean λ. Determine the maximum likelihood estimator of λ.
The number of traffic accidents in Berkeley, California, in 10 randomly chosen nonrainy days in 1998 is as follows:4, 0, 6, 5, 2, 1, 2, 0, 4, 3 Use these data to estimate the proportion of nonrainy days that had 2 or fewer accidents that year.
Kolmogorov’s law of fragmentation states that the size of an individual particle in a large collection of particles resulting from the fragmentation of a mineral compound will have an approximate lognormal distribution, where a random variable X is said to have a lognormal distribution if log(X )
Suppose X1, . . . , Xn constitute a sample from a uniform distribution on (0, θ), where θ is unknown. Their joint density is thusThis density is maximized by choosing θ as small as possible. Since θ must be at least as large as all of the observed values xi , it follows that the smallest
Suppose that when a signal having value μ is transmitted from location A the value received at location B is normally distributed with mean μ and variance 4. That is, if μ is sent, then the value received is μ + N where N, representing noise, is normal with mean 0 and variance 4. To reduce
It follows, under the assumption that the values received are independent, that a 95 percent confidence interval for μ isHence, we are “95 percent confident” that the true message value lies between 7.69 and 10.31. (9 - 1.96 -3,9 + 1.96-2) = (7.69, 10.31)
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