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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

1. Consider a trial in which a jury must decide between the hypothesis that the defendant is guilty and the hypothesis that he or she is innocent.(a) In the framework of hypothesis testing and the
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
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
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
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
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
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;
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
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
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
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
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
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.
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
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)
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
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
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
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
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
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.
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,
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
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
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
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 >
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,
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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.
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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.
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)
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
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
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
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
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
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
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
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
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 ≤
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
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
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
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
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
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
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
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
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
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,
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
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
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
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

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