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introduction to probability statistics

Introduction To Probability And Statistics For Engineers And Scientists 3rd Edition Sheldon M. Ross - Solutions

For the data set 7 2 3(a) Calculate the deviations (x x ) and check to see that they add up to 0.(b) Calculate the sample variance and the standard deviation.
Some properties of the standard deviation.1. If a fixed number c is added to all measurements in a data set, the deviations remain unchanged (see Exercise 2.52). Consequently, and s remain unchanged.2. If all measurements in a data set are multiplied by a fixed numberd, the deviations get
For the data set of Exercise 2.22, calculate the interquartile range.
Presidents also take midterms! After two years of the President’s term, members of Congress are up for election. The following table gives the number of net seats lost, by the party of the President, in the House of Representatives since the end of World War II.Net House Seats Lost in Midterm
Refer to Exercise 2.3 and the data on extracurricular activities. Find the sample mean and standard deviation.
Refer to Example 5 and the data on hours of sleep(a) Obtain the five-number summary: minimum,, , , and maximum.(b) Make a boxplot of the hours of sleep.
Refer to Exercise 2.27 and the data on the consumer price index for various cities. Find the increase, for each city, by subtracting the 2001 value from the 2007 value.(a) Find the sample mean and standard deviation of these differences.(b) What proportion of the increases lie between
Refer to Exercise 2.27 and the data on the consumer price index for various cities. Find the increase, for each city, by subtracting the 2001 value from the 2007 value.(a) Obtain the five-number summary: minimum, , , , and maximum. Which city had the largest increase? Were there any decreases?(b)
Refer to the data on throwing speed in Exercise 2.42. Make separate boxplots to compare males and females.
Two cities provided the following information on public school teachers’ salaries.Minimum Q1 Median Q3 Maximum City A 38,400 44,000 48,300 50,400 56,300 City B 39,600 46,500 51,200 55,700 61,800(a) Construct a boxplot for the salaries in City A.(b) Construct a boxplot, on the same graph, for the
The weights (oz) of nineteen babies born in Madison, Wisconsin, are summarized in the computer output.Descriptive Statistics: Weight Variable N Mean Median StDev Weight 19 118.05 117.00 15.47 Referring to Exercise 2.82 obtain the z score for a baby weighing(a) 102 oz(b) 144 oz
Sample z score. The z scale (or standard scale)measures the position of a data point relative to the mean and in units of the standard deviation.Specifically, When two measurements originate from different sources, converting them to the z scale helps to draw a sensible interpretation of their
Refer to the data on number of returns in Example 3.(a) Calculate and s.(b) Find the proportions of the observations that are in the intervals and(c) Compare the results of part (b) with the empirical guidelines.
Refer to the data on lizards in Exercise 2.19.(a) Calculate and s.(b) Find the proportion of the observations that are in the intervals and(c) Compare the results of part (b) with the empirical guidelines.
Refer to the data on bone mineral content in Exercise 2.41.(a) Calculate and s.(b) Find the proportion of the observations that are in the intervals and(c) Compare the results of part (b) with the empirical guidelines.
give and(a) Find the proportion of the observations in the intervals and(b) Compare your findings in part (a) with those suggested by the empirical guidelines for bell-shaped distributions.
Calculations with the test scores data of Exercise
For the extracurricular data of Exercise 2.3, calculate the interquartile range.
21. For the model of Problem 19:(a) Do the methods of extraction appear to differ?(b) Do the storage conditions affect the content? Test at the α = .05 level of significance.
3. Explain why we cannot efficiently test the hypothesis H0 : μ1 = μ2 = · · · = μm by running t -tests on all of them2pairs of samples.
If n = 4, m0 = 2, and the data values are X1 = 4.2, X2 = 1.8, X3 = 5.3, X4 = 1.7, then the rankings of |Xi − 2| are .2, .3, 2.2, 3.3. Since the first of these values—namely, .2—comes from the data point X2, which is less than 2, it follows that 1.0 y 0.8 0.6 0.4- 0.2- 20 0.0 1 2 3 4 x 5 6
Let us reconsider the problem presented in Example 11.2b. A simulation study yielded the resultand so the critical value should be 9.52381, which is remarkably close to χ2 .05,4 = 9.488 given as the critical value by the chi-square approximation. This is most interesting since the rule of thumb
For the data of Example 12.3a,Hence the p-value is 2P4(3), which is computed as follows:Program 12.3 will use the recursion in Equations 12.3.5 and 12.3.6 to compute the p-value of the signed rank test data. The input needed is the sample size n and the value of test statistic T . min in (3,4,5 3)
Suppose the weekly number of accidents over a 30-week period is as follows:Test the hypothesis that the number of accidents in a week has a Poisson distribution 8 0 0 1 3 402 1 8 020 1 93 12 5 45 33 34 7 4 0 1 2 1 2
An experiment designed to compare two treatments against corrosion yielded the following data in pieces of wire subjected to the two treatments.(The data represent the maximum depth of pits in units of one thousandth of an inch.)The ordered values are 58.5, 59.4, 65.2∗, 66.2, 67.1∗, 68,
A sample of 300 people was randomly chosen, and the sampled individuals were classified as to their gender and political affiliation, Democrat, Republican, or Independent. The following table, called a contingency table, displays the resulting dataThus, for instance, the contingency table indicates
Suppose we wanted to determine P(2, 1, 3). We use Equation 12.4.3 as follows:andHence,which checks since in order for the sum of the ranks of the two X values to be less than or equal to 3, the largest of the values X1, X2, Y1, must be Y1, which, when H0 is true, has probability 1/3. ||P(2, 1, 3) =
A company operates four machines on three separate shirts daily. The following contingency table presents the data during a 6-month time period, concerning the machine breakdowns that resulted.Suppose we are interested in determining whether a machine’s breakdown probability during a particular
In Example 12.4a, the sizes of the two samples are 5 and 6, respectively, and the sum of the ranks of the first sample is 21. Running Program 12.4 yields the result: p-value = .1255
A randomly chosen group of 20,000 nonsmokers and one of 10,000 smokers were followed over a 10-year period. The following data relate the numbers of them that developed lung cancer during that period.Test the hypothesis that smoking and lung cancer are independent. Use the 1 percent level of
A recent study reported that 500 female office workers were randomly chosen and questioned in each of four different countries. One of the questions related to whether these women often received verbal or sexual abuse on the job. The following data resulted.Based on these data, is it plausible that
In Example 12.4a, n = 5,m = 6, and the test statistic’s value is 21. Since n(n+m+1) 2 nm(n+m+1) 30 = 30 12
Suppose we want to test the hypothesis that a given population distribution is exponential with mean 100; that is, F (x) = 1 − e−x/100. If the (ordered) values from a sample of size 10 from this distribution arewhat conclusion can be drawn? 66, 72, 81, 94, 112, 116, 124, 140, 145, 155
2. To ascertain whether a certain die was fair, 1,000 rolls of the die were recorded, with the following results.Test the hypothesis that the die is fair (that is, that pi = 16 , i = 1, . . . ,6) at the 5 percent level of significance. Use the chi-square approximation. Outcome Number of Occurrences
Suppose that a sequence of sixty 1’s and sixty 0’s resulted in 75 runs. Sincewe see that the approximate p-value is 3,540 = 61 and = 5.454 119
4. It is believed that the daily number of electrical power failures in a certain Midwestern city is a Poisson random variable with mean 4.2. Test this hypothesis if over 150 days the number of days having i power failures is as follows: Failures Number of Days 0 1 0 5 22 2 3 23 4 32 5698 22 19 7
1. A new medicine against hypertension was tested on 18 patients. After 40 days of treatment, the following changes of the diastolic blood pressure were observed -5, -1, +2, +8, -25, +1, +5, -12, -16 -9, -8, -18, -5, -22, +4, -21, -15, -11
6. The past output of a machine indicates that each unit it produces will beA new machine, designed to perform the same job, has produced 500 units with the following results.Can the difference in output be ascribed solely to chance? top grade with probability .40 high grade with probability .30
2. An engineering firm is involved in selecting a computer system, and the choice has been narrowed to two manufacturers. The firm submits eight problems to the two computer manufacturers and has each manufacturer measure the number of seconds required to solve the design problem with the
7. The neutrino radiation from outer space was observed during several days. The frequencies of signals were recorded for each sidereal hour and are as given below:Test whether the signals are uniformly distributed over the 24-hour period. Frequency of Neutrino Radiation from Outer Space Hour
8. Neutrino radiation was observed over a certain period and the number of hours in which 0, 1, 2,… signals were received was recorded.Test the hypothesis that the observations come from a population having a Poisson distribution with mean .3. Number of Signals per Hour 0 1 2 3 4 5 6 or more
10. A study was instigated to see if southern California earthquakes of at least moderate size (having values of at least 4.4 on the Richter scale) are more likely to occur on certain days of the week than on others. The catalogs yielded the following data on 1,100 earthquakes.Test, at the 5
6. An experiment was initiated to study the effect of a newly developed gasoline detergent on automobile mileage. The following data, representing mileage per gallon before and after the detergent was added for each of eight cars, resultedFind the p-value of the test of the hypothesis that mileage
8. Twelve patients having high albumin content in their blood were treated with a medicine. Their blood content of albumin was measured before and after treatment. The measured values are shown in the table.Is the effect of the medicine significant at the 5 percent level?(a) Use the sign test.(b)
9. An engineer claims that painting the exterior of a particular aircraft affects its cruising speed. To check this, the next 10 aircraft off the assembly line were flown to determine cruising speed prior to painting, and were then painted and reflown.The following data resultedDo the data uphold
10. Ten pairs of duplicate spectrochemical determinations for nickel are presented below. The readings in column 2 were taken with one type of measuring instrument and those in column 3 were taken with another typeTest the hypothesis, at the 5 percent level of significance, that the two measuring
15. A random sample of 500 families was classified by region and income (in units of$1,000). The following data resulted.Determine the p-value of the test that a family’s income and region are independent. Income South North 0-10 42 53 10-20 55 90 20-30 47 88 >30 36 89
16. The following data relate the mother’s age and the birthweight (in grams) of her child.Test the hypothesis that the baby’s birthweight is independent of the mother’s age. Birthweight More Than 2,500 Grams 40 135 Maternal Age Less Than 2,500 Grams 20 years or less Greater than 20 10 15
12. In a study of bilingual coding, 12 bilingual (French and English) college students are divided into two groups. Each group reads an article written in French, and each answers a series of 25 multiple-choice questions covering the content of the article. For one group the questions are written
18. The number of infant mortalities as a function of the baby’s birthweight (in grams)for 72,730 live white births in New York in 1974 is as follows:Test the hypothesis that the birthweight is independent of whether or not the baby survives its first year. Outcome at the End of 1 Year
19. An experiment designed to study the relationship between hypertension and cigarette smoking yielded the following data.Test the hypothesis that whether or not an individual has hypertension is independent of how much that person smokes. Nonsmoker Moderate Smoker Hypertension 20 No hypertension
15. The following are the burning times in seconds of floating smoke pots of two different types:We are interested in testing the hypothesis that the burning time distributions are the same.(a) Determine the exact p-value.(b) Determine the p-value yielded by the normal approximation.(c) Run a
20. The following table shows the number of defective, acceptable, and superior items in samples taken both before and after the introduction of a modification in the manufacturing processIs this change significant at the .05 level? Defective Acceptable Superior Before 25 After 9 218 22 103 14
21. A sample of 300 cars having cellular phones and one of 400 cars without phones were tracked for 1 year. The following table gives the number of these cars involved in accidents over that year.Use the above to test the hypothesis that having a cellular phone in your car and being involved in an
22. To study the effect of fluoridated water supplies on tooth decay, two communities of roughly the same socioeconomic status were chosen. One of these communities had fluoridated water while the other did not. Random samples of 200 teenagers from both communities were chosen, and the numbers of
18. The m sample problem: Consider m independent random samples of respective sizes n1, . . . , nm from the respective population distributions F1, . . . , Fm; and consider the problem of testing H0 : F1 = F2 = · · · = Fm. To devise a test, let Ri denote the sum of the ranks of the ni elements
23. To determine if a malpractice lawsuit is more likely to follow certain types of surgery, random samples of three different types of surgeries were studied, and the following data resulted.Test the hypothesis that the percentages of the surgical operations that lead to lawsuits are the same for
24. In a famous article (S. Russell, “A red sky at night…,” Metropolitan Magazine London, 61, p. 15, 1926) the following data set of frequencies of sunset colors and whether each was followed by rain was presented.Test the hypothesis that whether it rains tomorrow is independent of the color
22. The following table (taken from Quinn, W. H., Neal, T. V., and Antu˜nez de Mayolo, 1987, “El Ni˜no occurrences over the past four-and-a-half centuries,”Journal of Geophysical Research, 92 (C13), pp. 14,449–14,461) gives the years and magnitude (either moderate or strong) of major El
11. Suppose that 20 items are to be put on test that is to be terminated when the 10th failure occurs. If the lifetime distribution is exponential with mean 10 hours, compute the following quantities.(a) The mean length of the testing period.(b) The variance of the testing period.
10. Suppose 30 items are put on test that is scheduled to stop when the 8th failure occurs. If the failure times are, in hours, .35, .73, .99, 1.40, 1.45, 1.83, 2.20, 2.72, test, at the 5 percent level of significance, the hypothesis that the mean life is equal to 10 hours. Assume that the
9. Consider a test of H0 : θ = θ0 versus H1 : θ = θ0 for the model of Section 14.3.1. Suppose that the observed value of 2τ /θ0 is v. Show that the hypothesis should be rejected at significance level α whenever α is less than the p-value given bywhere χ2 2r is a chi-square random variable
8. When 30 transistors were simultaneously put on a life test that was to be terminated when the 10th failure occurred, the observed failure times were (in hours) 4.1, 7.3, 13.2, 18.8, 24.5, 30.8, 38.1, 45.5, 53, 62.2. Assume an exponential life distribution.(a) What is the maximum likelihood
7. For the model of Section 14.3.1, explain how the following figure can be used to show thatwhere IM j=1, = 1
12. Vacuum tubes produced at a certain plant are assumed to have an underlying exponential life distribution having an unknown mean θ. To estimate θ it has been decided to put a certain number n of tubes on test and to stop the test at the 10th failure. If the plant officials want the mean length
5. A continuous life distribution is said to be an IFR (increasing failure rate)distribution if its failure rate function λ(t ) is nondecreasing in t .(a) Show that the gamma distribution with densityis IFR.(b) Show, more generally, that the gamma distribution with parameters α, λ is IFR
4. Suppose the life distribution of an item has failure rate function λ(t ) = t 3, 0
3. The lung cancer rate of a t -year-old male smoker, λ(t ), is such thatAssuming that a 40-year-old male smoker survives all other hazards, what is the probability that he survives to (a) age 50, (b) age 60, without contracting lung cancer? In the foregoing we are assuming that he remains a
2. If X and Y are independent random variables having failure rate functions λx (t )and λy (t ), show that the failure rate function of Z = min(X , Y ) is hz (t) = x(t) + hy(t)
1. A random variable whose distribution function is given byis said to have a Weibull distribution with parameters α, β. Compute its failure rate function. F(t) = 1-exp{-at), t0
Test the hypothesis, at the 5 percent level of significance, that the lifetimes of items produced at two given plants have the same exponential life distribution if a sample of size 10 from the first plant has a total lifetime of 420 hours whereas a sample of 15 from the second plant has a total
Suppose that 20 items having an exponential life distribution with an unknown rate λ are put on life test at various times. When the test is ended, there have been 10 observed failures — their lifetimes being (in hours) 5, 7, 6.2, 8.1, 7.9, 15, 18 , 3.9, 4.6, 5.8. The 10 items that did not fail
14. Using the fact that a Poisson process results when the times between successive events are independent and identically distributed exponential random variables, show thatwhen X is a Poisson random variable with mean x/2 and Fχ2 2n is the chi-square distribution function with 2n degrees of
15. From a sample of items having an exponential life distribution with unknown mean θ, items are tested in sequence. The testing continues until either the rth failure occurs or after a time T elapses.(a) Determine the likelihood function.(b) Verify that the maximum likelihood estimator of θ is
30. If U is uniformly distributed on (0, 1), show that −log U has an exponential distribution with mean 1. Now use Equation 14.3.7 and the results of the previous problems to establish Equation 14.5.7.
28. If X is a continuous random variable having distribution function F , show that(a) F (X ) is uniformly distributed on (0, 1);(b) 1 − F (X ) is uniformly distributed on (0, 1).
27. If U is uniformly distributed on (0, 1) — that is, U is a random number —show that [−(1/α) log U]1/β is a Weibull random variable with parameters(α, β).The next three problems are concerned with verifying Equations 14.5.5 and 14.5.7.
25. If the following are the sample data from a Weibull population having unknown parameters α and β, determine the least square estimates of these quantities, using either of the methods presented.Data: 15.4, 16.8, 6.2, 10.6, 21.4, 18.2, 1.6, 12.5, 19.4, 17
24. Show that if X is a Weibull random variable with parameters (α, β), then Var(X) =-2/B - (1 + 2/3) - (r (1 + 2/3)
23. If X is a Weibull random variable with parameters (α, β), show thatand make the change of variables E[X]=ar(1+1/) where (y) is the gamma function defined by r ( y ) = e- 0 exxdx Hint: Write E[X] = Stat expl-at" } dt
22. Suppose that the life distributions of two types of transistors are both exponential.To test the equality of means of these two distributions, n1 type 1 transistors are simultaneously put on a life test that is scheduled to end when there have been a total of r1 failures. Similarly, n2 type 2
21. The following data represent failure times, in minutes, for two types of electrical insulation subject to a certain voltage stress.Test the hypothesis that the two sets of data come from the same exponential distribution. Type I 212, 88.5, 122.3, 116.4, 125, 132, 66 Type II 34.6, 54, 162, 49,
20. What is the Bayes estimate of λ = 1/θ in Problem 18 if the prior distribution onλ is exponential with mean 1/30?
19. In Problem 17, suppose that prior to the testing phase and based on past experience one felt that the value of λ = 1/θ could be thought of as the outcome of a gamma random variable with parameters 1, 100. What is the Bayes estimate of λ?
18. Suppose that the remission time, in weeks, of leukemia patients that have undergone a certain type of chemotherapy treatment is an exponential random variable having an unknown mean θ. A group of 20 such patients is being monitored and, at present, their remission times are (in weeks) 1.2,
17. A testing laboratory has facilities to simultaneously life test 5 components. The lab tested a sample of 10 components from a common exponential distribution by initially putting 5 on test and then replacing any failed component by one still waiting to be tested. The test was designed to end
16. Verify that the maximum likelihood estimate corresponding to Equation 14.3.9 is given by Equation 14.3.10.
6. Show that the uniform distribution on (a,b) is an IFR distribution.
A company claims that the mean lifetimes of the semiconductors it produces is at least 25 hours. To substantiate this claim, an independent testing service has decided to sequentially test, one at a time, the company’s semiconductors for 600 hours. If 30 semiconductors failed during this period,
If a one-at-a-time sequential test yields 10 failures in the fixed time of T = 500 hours, then the maximum likelihood estimate of θ is 500/10 = 50 hours. A 95 percent confidence interval estimate of θ is 0 (1,000/X025.20, 1,000/X975,20) Running Program 5.8.1b yields that X.025.20 = 34.17,
29. Let X(i) denote ith smallest of a sample of size n from a continuous distribution function F . Also, let U(i) denote the ith smallest from a sample of size n from a uniform (0, 1) distribution.(a) Argue that the density function of U(i) is given by[Hint: In order for the ith smallest of n
26. Show that if X is a Weibull random variable with parameters (α, β), then αX β is an exponential random variable with mean 1.
One often hears that the death rate of a person that smokes is, at each age, twice that of a nonsmoker. What does this mean? Does it mean that a nonsmoker has twice the probability of surviving a given number of years as does a smoker of the same age?
A sample of 50 transistors is simultaneously put on a test that is to be ended when the 15th failure occurs. If the total time on test of all transistors is equal to 525 hours, determine a 95 percent confidence interval for the mean lifetime of a transistor.Assume that the underlying distribution
A producer of batteries claims that the lifetimes of the items it manufactures are exponentially distributed with a mean life of at least 150 hours. To test this claim, 100 batteries are simultaneously put on a test that is slated to end when the 20th failure occurs. If, at the end of the
13. A one-at-a-time sequential life testing scheme is scheduled to run for 300 hours.A total of 16 items fail within that time. Assuming an exponential life distribution with unknown mean θ (measured in hours):(a) Determine the maximum likelihood estimate of θ.b) Test at the .05 level of
13. The following data represent the results of inspecting all personal computers produced at a given plant during the last 12 days.Does the process appear to have been in control? Determine control limits for future production. Day Number of Units Number Defective 1 80 23456789012 110 90 80 100 12
12. The following data present the number of defective bearing and seal assemblies in samples of size 100.Does it appear that the process was in control throughout? If not, determine revised control limits if possible. Sample Number Number of Defectives Sample Number Number of Defectives 1 3 4 5215
11. Samples of n = 6 items are taken from a manufacturing process at regular intervals. A normally distributed quality characteristic is measured, and X and S values are calculated for each sample. After 50 subgroups have been analyzed, we have(a) Compute the control limit for the X - and S-control
10. The following data refer to the amounts by which the diameters of 14-inch ball bearings differ from 14-inch in units of .001 inches. The subgroup size is n = 5.(a) Set up trial control limits for X - and S-control charts.(b) Does the process appear to have been in control throughout the

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