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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

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
Determine the upper and lower 95 percent confidence interval estimates of μ in Example 7.3a.
Use the data of Example 7.3a to obtain a 99 percent confidence interval estimate of μ, along with 99 percent one-sided upper and lower intervals.
From past experience it is known that the weights of salmon grown at a commercial hatchery are normal with a mean that varies from season to season but with a standard deviation that remains fixed at
Let us again consider Example 7.3a but let us now suppose that when the value μ is transmitted at location A then the value received at location B is normal with mean μ and variance σ2 but with
Determine a 95 percent confidence interval for the average resting pulse of the members of a health club if a random selection of 15 members of the club yielded the data 54, 63, 58, 72, 49, 92, 70,
Simulation provides a powerful method for evaluating single and multidimensional integrals. For instance, let f be a function of an r-valued vector ( y1, . . . , yr ), and suppose that we want to
A standardized procedure is expected to produce washers with very small deviation in their thicknesses. Suppose that 10 such washers were chosen and measured.If the thicknesses of these washers were,
Two different types of electrical cable insulation have recently been tested to determine the voltage level at which failures tend to occur. When specimens were subjected to an increasing voltage
There are two different techniques a given manufacturer can employ to produce batteries. A random selection of 12 batteries produced by technique I and of 14 produced by technique II resulted in the
A sample of 100 transistors is randomly chosen from a large batch and tested to determine if they meet the current standards. If 80 of them meet the standards, then an approximate 95 percent
On October 14, 2003, the New York Times reported that a recent poll indicated that 52 percent of the population was in favor of the job performance of President Bush, with a margin of error of ±4
A certain manufacturer produces computer chips; each chip is independently acceptable with some unknown probability p. To obtain an approximate 99 percent confidence interval for p, whose length is
The successive items produced by a certain manufacturer are assumed to have useful lives that (in hours) are independent with a common density functionIf the sum of the lives of the first 10 items is
Let X1, X2, . . . , Xn be a random sample from a distribution having unknown mean θ. Thenare both unbiased estimators of θ since di (X1, X2, Xn) = X and +Xn d2(X1, X2, X): = n
Combining Independent Unbiased Estimators. Let d1 and d2 denote independent unbiased estimators of θ, having known variances σ2 1 and σ2 2 . That is, for i = 1, 2, E[d;] = 0, Var(d;) = 0}
Let X1, . . . , Xn denote a sample from a uniform (0, θ) distribution, whereθ is assumed unknown. Sincea “natural” estimator to consider is the unbiased estimator E[X;] = 2
Suppose that X1, . . . , Xn are independent Bernoulli random variables, each having probability mass function given by f (x|θ) = θx(1 − θ)1−x , x = 0, 1 where θ is unknown. Further, suppose
Suppose X1, . . . , Xn are independent normal random variables, each having unknown mean θ and known variance σ2 0 . If θ is itself selected from a normal population having known mean μ and known
Consider the likelihood function f (x1, . . . , xn|θ) and suppose that θ is uniformly distributed over some interval (a, b). The posterior density of θ given X1, . . . , Xn equals f(0|x1,...,xn) =
Suppose that if a signal of value s is sent from location A, then the signal value received at location B is normally distributed with mean s and variance 60. Suppose also that the value of a signal
1. Let X1, . . . , Xn be a sample from the distribution whose density function isDetermine the maximum likelihood estimator of θ. f(x) 0 (x-0) x0 otherwise
2. Determine the maximum likelihood estimator of θ when X1, . . . , Xn is a sample with density function > x > X- "10-x1-01 = (x)5
3. Let X1, . . . , Xn be a sample from a normal μ, σ2 population. Determine the maximum likelihood estimator of σ2 when μ is known. What is the expected value of this estimator?
4. The height of a radio tower is to be measured by measuring both the horizontal distance X from the center of its base to a measuring instrument and the vertical angle of the measuring device (see
5. Suppose that X1, . . . , Xn are normal with mean μ1; Y1, . . . , Yn are normal with mean μ2; and W1, . . . ,Wn are normal with mean μ1 + μ2. Assuming that all 3n random variables are
6. River floods are often measured by their discharges (in units of feet cubed per second). The value v is said to be the value of a 100-year flood if P{D ≥ v} = .01 where D is the discharge of the
7. A manufacturer of heat exchangers requires that the plate spacings of its exchangers be between .240 and .260 inches. A quality control engineer sampled 20 exchangers and measured the spacing of
8. An electric scale gives a reading equal to the true weight plus a random error that is normally distributed with mean 0 and standard deviation σ = .1 mg. Suppose that the results of five
9. The PCB concentration of a fish caught in Lake Michigan was measured by a technique that is known to result in an error of measurement that is normally distributed with a standard deviation of .08
10. The standard deviation of test scores on a certain achievement test is 11.3. If a random sample of 81 students had a sample mean score of 74.6, find a 90 percent confidence interval estimate for
11. Let X1, . . . , Xn, Xn+1 be a sample from a normal population having an unknown mean μ and variance 1. Let ¯Xn = ni=1 Xi /n be the average of the first n of them.(a) What is the distribution
12. If X1, . . . , Xn is a sample from a normal population whose mean μ is unknown but whose variance σ2 is known, show that (−∞, X + zασ/√n) is a 100(1 − α)percent lower confidence
13. A sample of 20 cigarettes is tested to determine nicotine content and the average value observed was 1.2 mg. Compute a 99 percent two-sided confidence interval for the mean nicotine content of a
14. In Problem 13, suppose that the population variance is not known in advance of the experiment. If the sample variance is .04, compute a 99 percent two-sided confidence interval for the mean
15. In Problem 14, compute a value c for which we can assert “with 99 percent confidence” that c is larger than the mean nicotine content of a cigarette
16. Suppose that when sampling from a normal population having an unknown mean μ and unknown variance σ2, we wish to determine a sample size n so as to guarantee that the resulting 100(1 − α)
17. The following data resulted from 24 independent measurements of the melting point of lead.Assuming that the measurements can be regarded as constituting a normal sample whose mean is the true
18. The following are scores on IQ tests of a random sample of 18 students at a large eastern university.130, 122, 119, 142, 136, 127, 120, 152, 141, 132, 127, 118, 150, 141, 133, 137, 129, 142(a)
19. Suppose that a random sample of nine recently sold houses in a certain city resulted in a sample mean price of $222,000, with a sample standard deviation of $22,000.Give a 95 percent upper
20. A company self-insures its large fleet of cars against collisions. To determine its mean repair cost per collision, it has randomly chosen a sample of 16 accidents.If the average repair cost in
21. A standardized test is given annually to all sixth-grade students in the state of Washington. To determine the average score of students in her district, a school supervisor selects a random
22. Each of 20 science students independently measured the melting point of lead.The sample mean and sample standard deviation of these measurements were(in degrees centigrade) 330.2 and 15.4,
23. A random sample of 300 CitiBank VISA cardholder accounts indicated a sample mean debt of $1,220 with a sample standard deviation of $840. Construct a 95 percent confidence interval estimate of
24. In Problem 23, find the smallest value v that “with 90 percent confidence,” exceeds the average debt per cardholder.
25. Verify the formula given in Table 7.1 for the 100(1−α) percent lower confidence interval for μ when σ is unknown.
26. The range of a new type of mortar shell is being investigated. The observed ranges, in meters, of 20 such shells are as follows:Assuming that a shell’s range is normally distributed, construct
27. Studies were conducted in Los Angeles to determine the carbon monoxide concentration near freeways. The basic technique used was to capture air samples in special bags and to then determine the
28. A set of 10 determinations, by a method devised by the chemist Karl Fischer, of the percentage of water in a methanol solution yielded the following data..50, .55, .53, .56, .54,.57, .52, .60,
29. Suppose that U1,U2, . . . is a sequence of independent uniform (0,1) random variables, and define N by N = min{n : U1 +· · ·+Un > 1}That is, N is the number of uniform (0, 1) random variables
30. An important issue for a retailer is to decide when to reorder stock from a supplier.A common policy used to make the decision is of a type called s, S: The retailer orders at the end of a period
31. A random sample of 16 full professors at a large private university yielded a sample mean annual salary of $90,450 with a sample standard deviation of $9,400. Determine a 95 percent confidence
32. Let X1, . . . , Xn, Xn+1 denote a sample from a normal population whose meanμ and variance σ2 are unknown. Suppose that we are interested in using the observed values of X1, . . . , Xn to
33. National Safety Council data show that the number of accidental deaths due to drowning in the United States in the years from 1990 to 1993 were (in units of one thousand) 5.2, 4.6, 4.3, 4.8. Use
34. The daily dissolved oxygen concentration for a water stream has been recorded over 30 days. If the sample average of the 30 values is 2.5 mg/liter and the sample standard deviation is 2.12
35. Verify the formulas given in Table 7.1 for the 100(1−α) percent lower and upper confidence intervals for σ2.
36. The capacities (in ampere-hours) of 10 batteries were recorded as follows:140, 136, 150, 144, 148, 152, 138, 141, 143, 151(a) Estimate the population variance σ2.(b) Compute a 99 percent
37. Find a 95 percent two-sided confidence interval for the variance of the diameter of a rivet based on the data given here.Assume a normal population. 6.68 6.66 6.62 6.72 6.76 6.67 6.70 6.72 6.78
38. The following are independent samples from two normal populations, both of which have the same standard deviation σ.16, 17, 19, 20, 18 and 3, 4, 8 Use them to estimate σ.
39. The amount of beryllium in a substance is often determined by the use of a photometric filtration method. If the weight of the beryllium is μ, then the value given by the photometric filtration
40. If X1, . . . , Xn is a sample from a normal population, explain how to obtain a 100(1 − α) percent confidence interval for the population variance σ2 when the population mean μ is known.
41. A civil engineer wishes to measure the compressive strength of two different types of concrete. A random sample of 10 specimens of the first type yielded the following data (in psi)whereas a
42. Independent random samples are taken from the output of two machines on a production line. The weight of each item is of interest. From the first machine, a sample of size 36 is taken, with
43. Do Problem 42 when it is known in advance that the population variances are 4 and 5.
44. The following are the burning times in seconds of floating smoke pots of two different types.Find a 99 percent confidence interval for the mean difference in burning times assuming normality with
45. If X1, . . . , Xn is a sample from a normal population having known mean μ1 and unknown variance σ2 1 , and Y1, . . . , Ym is an independent sample from a normal population having known mean
46. Two analysts took repeated readings on the hardness of city water. Assuming that the readings of analyst i constitute a sample from a normal population having variance σ2 i , i = 1, 2, compute a
47. A problem of interest in baseball is whether a sacrifice bunt is a good strategy when there is a man on first base and no outs. Assuming that the bunter will be out but will be successful in
48. A random sample of 1,200 engineers included 48 Hispanic Americans, 80 African Americans, and 204 females. Determine 90 percent confidence intervals for the proportion of all engineers that are(a)
49. To estimate p, the proportion of all newborn babies that are male, the gender of 10,000 newborn babies was noted. If 5,106 of them were male, determine (a) a 90 percent and (b) a 99 percent
50. An airline is interested in determining the proportion of its customers who are flying for reasons of business. If they want to be 90 percent certain that their estimate will be correct to within
51. A recent newspaper poll indicated that Candidate A is favored over Candidate B by a 53 to 47 percentage, with a margin of error of ±4 percent. The newspaper then stated that since the 6-point
52. A market research firm is interested in determining the proportion of households that are watching a particular sporting event. To accomplish this task, they plan on using a telephone poll of
53. In a recent study, 79 of 140 meteorites were observed to enter the atmosphere with a velocity of less than 25 miles per second. If we take ˆp = 79/140 as an estimate of the probability that an
54. A random sample of 100 items from a production line revealed 17 of them to be defective. Compute a 95 percent two-sided confidence interval for the probability that an item produced is defective.
55. Of 100 randomly detected cases of individuals having lung cancer, 67 died within 5 years of detection.(a) Estimate the probability that a person contracting lung cancer will die within 5
56. Derive 100(1 − α) percent lower and upper confidence intervals for p, when the data consist of the values of n independent Bernoulli random variables with parameter p.
57. Suppose the lifetimes of batteries are exponentially distributed with mean θ. If the average of a sample of 10 batteries is 36 hours, determine a 95 percent two-sided confidence interval for θ.
58. Determine 100(1−α) percent one-sided upper and lower confidence intervals forθ in Problem 57.
59. Let X1, X2, . . . , Xn denote a sample from a population whose mean value θ is unknown. Use the results of Example 7.7b to argue that among all unbiased estimators of θ of the formni=1 λiXi
60. Consider two independent samples from normal populations having the same variance σ2, of respective sizes n and m. That is, X1, . . . , Xn and Y1, . . . , Ym are independent samples from normal
61. Consider two estimators d1 and d2 of a parameter θ. If E[d1] = θ, Var(d1) = 6 and E[d2] = θ + 2, Var(d2) = 2, which estimator should be preferred?
62. Suppose that the number of accidents occurring daily in a certain plant has a Poisson distribution with an unknown mean λ. Based on previous experience in similar industrial plants, suppose that
63. The functional lifetimes in hours of computer chips produced by a certain semiconductor firm are exponentially distributed with mean 1/λ. Suppose that the prior distribution on λ is the gamma
64. Each item produced will, independently, be defective with probability p. If the prior distribution on p is uniform on (0, 1), compute the posterior probability that p is less than .2 given(a) a
65. The breaking strength of a certain type of cloth is to be measured for 10 specimens.The underlying distribution is normal with unknown mean θ but with a standard deviation equal to 3 psi.
It is known that disks produced by a certain company will be defective with probability .01 independently of each other. The company sells the disks in packages of 10 and offers a money-back
The color of one’s eyes is determined by a single pair of genes, with the gene for brown eyes being dominant over the one for blue eyes. This means that an individual having two blue-eyed genes
A communications system consists of n components, each of which will, independently, function with probability p. The total system will be able to operate effectively if at least one-half of its
Suppose that 10 percent of the chips produced by a computer hardware manufacturer are defective. If we order 100 such chips, will X, the number of defective ones we receive, be a binomial random
Let X be a binomial random variable with parameters n = 6, p = .4. Then, starting with P{X = 0} = (.6)6 and recursively employing Equation 5.1.4, we obtainThe text disk uses Equation 5.1.4 to compute
If X is a binomial random variable with parameters n = 100 and p = .75, find P{X = 70} and P{X ≤ 70}.
Suppose that the average number of accidents occurring weekly on a particular stretch of a highway equals 3. Calculate the probability that there is at least one accident this week.
Suppose the probability that an item produced by a certain machine will be defective is .1. Find the probability that a sample of 10 items will contain at most one defective item. Assume that the
Consider an experiment that consists of counting the number of α particles given off in a one-second interval by one gram of radioactive material. If we know from past experience that, on the
If the average number of claims handled daily by an insurance company is 5, what proportion of days have less than 3 claims? What is the probability that there will be 4 claims in exactly 3 of the
It has been established that the number of defective stereos produced daily at a certain plant is Poisson distributed with mean 4. Over a 2-day span, what is the probability that the number of
The components of a 6-component system are to be randomly chosen from a bin of 20 used components. The resulting system will be functional if at least 4 of its 6 components are in working condition.
An unknown number, say N, of animals inhabit a certain region.To obtain some information about the population size, ecologists often perform the following experiment: They first catch a number, say
If X is uniformly distributed over the interval [0, 10], compute the probability that (a) 2 < X < 9, (b) 1 < X < 4, (c) X < 5, (d) X > 6.

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