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applied statistics and probability for engineers

Questions and Answers of Applied Statistics And Probability For Engineers

18. Let X1, X2, . . . , Xn be a random sample from a pdf f(x) that is symmetric about , so that is an unbiased estimator of . If n is large, it can be shown that V( ) 1/(4n[ f( )]2).a. Compare V( )
17. In Chapter 3, we defined a negative binomial rv as the number of failures that occur before the rth success in a sequence of independent and identical success/failure trials. The probability mass
16. Suppose the true average growth of one type of plant during a 1-year period is identical to that of a second type, but the variance of growth for the first type is 2, whereas for the second
15. Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution with pdf f(x; ) x ex2/(2) x 0a. It can be shown that E(X2) 2. Use this fact to construct an unbiased
13. Consider a random sample X1, . . . , Xn from the pdf f(x; ) .5(1 x) 1 x 1 where 1 1 (this distribution arises in particle physics). Show that ˆ 3X is an unbiased estimator of
c. How would you use the observed values x1 and x2 to estimate the standard error of your estimator?d. If n1 n2 200, x1 127, and x2 176, use the estimator of part (a) to obtain an estimate of
b. What is the standard error of the estimator in part (a)?
a. Show that (X1/n1) (X2/n2) is an unbiased estimator for p1 p2. [Hint: E(Xi) nipi for i 1, 2.]
11. Of n1 randomly selected male smokers, X1 smoked filter cigarettes, whereas of n2 randomly selected female smokers, X2 smoked filter cigarettes. Let p1 and p2 denote the probabilities that a
10. Using a long rod that has length , you are going to lay out a square plot in which the length of each side is . Thus the area of the plot will be 2. However, you do not know the value of , so you
9. Each of 150 newly manufactured items is examined and the number of scratches per item is recorded (the items are supposed to be free of scratches), yielding the following data:
8. In a random sample of 80 components of a certain type, 12 are found to be defective.a. Give a point estimate of the proportion of all such components that are not defective.b. A system is to be
7.a. A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected and the amount of gas (therms) used during the month of January is determined for each
6. Consider the accompanying observations on stream flow(1000s of acre-feet) recorded at a station in Colorado for the period April 1–August 31 over a 31-year span (from an article in the 1974
5. As an example of a situation in which several different statistics could reasonably be used to calculate a point estimate, consider a population of N invoices. Associated with each invoice is its
4. The article from which the data of Exercise 1 was extracted also gave the accompanying strength observations for cylinders:6.1 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.3 7.8 8.1 7.4 8.5 8.9 9.8 9.7 14.1
3. Consider the following sample of observations on coating thickness for low-viscosity paint (“Achieving a Target Value for a Manufacturing Process: A Case Study,” J. of Quality Technology,
2. A sample of 20 students who had recently taken elementary statistics yielded the following information on brand of calculator owned (T Texas Instruments, H Hewlett Packard, C Casio, S
1. The accompanying data on flexural strength (MPa) for concrete beams of a certain type was introduced in Example 1.2.5.9 7.2 7.3 6.3 8.1 6.8 7.0 7.6 6.8 6.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.7 7.4 7.7
38. Each of n specimens is to be weighed twice on the same scale. Let Xi and Yi denote the two observed weights for the ith specimen. Suppose Xi and Yi are independent of one another, each normally
37. When the sample standard deviation S is based on a random sample from a normal population distribution, it can be shown that E(S) 2/(n1)(n/2)/((n 1)/2)Use this to obtain an unbiased
36. When the population distribution is normal, the statistic median{⏐X1 ⏐, . . . , ⏐Xn ⏐}/.6745 can be used to estimate . This estimator is more resistant to the effects of outliers
35. Let X1, . . . , Xn be a random sample from a pdf that is symmetric about . An estimator for that has been found to perform well for a variety of underlying distributions is the Hodges–Lehmann
34. The mean squared error of an estimator ˆ is MSE(ˆ) E(ˆ )2. If ˆ is unbiased, then MSE(ˆ) V(ˆ), but in general MSE(ˆ) V(ˆ) (bias)2. Consider the estimatorˆ 2 KS2, where
33. At time t 0, there is one individual alive in a certain population. A pure birth process then unfolds as follows. The time until the first birth is exponentially distributed with parameter .
32.a. Let X1, . . . , Xn be a random sample from a uniform distribution on [0, ]. Then the mle of is ˆ Y max(Xi).Use the fact that Y y iff each Xi y to derive the cdf of Y. Then show
31. An estimator ˆ is said to be consistent if for any ! 0, P(⏐ˆ ⏐ !) 0 0 as n 0 . That is, ˆ is consistent if, as the sample size gets larger, it is less and less likely that ˆ
30. At time t 0, 20 identical components are put on test. The lifetime distribution of each is exponential with parameter .The experimenter then leaves the test facility unmonitored.On his return
29. Consider a random sample X1, X2, . . . , Xn from the shifted exponential pdf f(x; , ) {e(x) x 0 otherwise Taking 0 gives the pdf of the exponential distribution considered
28. Let X1, X2, . . . , Xn represent a random sample from the Rayleigh distribution with density function given in Exercise 15. Determinea. The maximum likelihood estimator of and then calculate
27. Let X1, . . . , Xn be a random sample from a gamma distribution with parameters and .a. Derive the equations whose solution yields the maximum likelihood estimators of and . Do you think
26. Refer to Exercise 25. Suppose we decide to examine another test spot weld. Let X shear strength of the weld.Use the given data to obtain the mle of P(X 400).[Hint: P(X 400) ((400 )/).]
25. The shear strength of each of ten test spot welds is determined, yielding the following data (psi):392 376 401 367 389 362 409 415 358 375a. Assuming that shear strength is normally distributed,
24. Refer to Exercise 20. Instead of selecting n 20 helmets to examine, suppose I examine helmets in succession until I have found r 3 flawed ones. If the 20th helmet is the third flawed one (so
23. Two different computer systems are monitored for a total of n weeks. Let Xi denote the number of breakdowns of the first system during the ith week, and suppose the Xis are independent and drawn
22. Let X denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test.Suppose the pdf of X is f(x; ) ( 1)x 0 x 1 0 otherwise where 1 .
21. Let X have a Weibull distribution with parameters and, so E(X) (1 1/)V(X) 2{(1 2/) [(1 1/)]2}a. Based on a random sample X1, . . . , Xn, write equations for the method of
20. A random sample of n bike helmets manufactured by a certain company is selected. Let X the number among the n that are flawed, and let p P(flawed). Assume that only X is observed, rather than
19. An investigator wishes to estimate the proportion of students at a certain university who have violated the honor code.Having obtained a random sample of n students, she realizes that asking
18. Let X1, X2, . . . , Xn be a random sample from a pdf f(x) that is symmetric about , so that is an unbiased estimator of . If n is large, it can be shown that V( ) 1/(4n[ f( )]2).a. Compare V( )
17. In Chapter 3, we defined a negative binomial rv as the number of failures that occur before the rth success in a sequence of independent and identical success/failure trials. The probability mass
16. Suppose the true average growth of one type of plant during a 1-year period is identical to that of a second type, but the variance of growth for the first type is 2, whereas for the second
15. Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution with pdf f(x; ) x ex2/(2) x 0a. It can be shown that E(X2) 2. Use this fact to construct an unbiased
14. A sample of n captured Pandemonium jet fighters results in serial numbers x1, x2, x3, . . . , xn. The CIA knows that the aircraft were numbered consecutively at the factory starting with and
13. Consider a random sample X1, . . . , Xn from the pdf f(x; ) .5(1 x) 1 x 1 where 1 1 (this distribution arises in particle physics). Show that ˆ 3X is an unbiased estimator of
12. Suppose a certain type of fertilizer has an expected yield per acre of 1 with variance 2, whereas the expected yield for a second type of fertilizer is 2 with the same variance 2.Let S2 1 and
c. How would you use the observed values x1 and x2 to estimate the standard error of your estimator?d. If n1 n2 200, x1 127, and x2 176, use the estimator of part (a) to obtain an estimate of
b. What is the standard error of the estimator in part (a)?
a. Show that (X1/n1) (X2/n2) is an unbiased estimator for p1 p2. [Hint: E(Xi) nipi for i 1, 2.]
11. Of n1 randomly selected male smokers, X1 smoked filter cigarettes, whereas of n2 randomly selected female smokers, X2 smoked filter cigarettes. Let p1 and p2 denote the probabilities that a
10. Using a long rod that has length , you are going to lay out a square plot in which the length of each side is . Thus the area of the plot will be 2. However, you do not know the value of , so you
9. Each of 150 newly manufactured items is examined and the number of scratches per item is recorded (the items are supposed to be free of scratches), yielding the following data:
8. In a random sample of 80 components of a certain type, 12 are found to be defective.a. Give a point estimate of the proportion of all such components that are not defective.b. A system is to be
7.a. A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected and the amount of gas (therms) used during the month of January is determined for each
6. Consider the accompanying observations on stream flow(1000s of acre-feet) recorded at a station in Colorado for the period April 1–August 31 over a 31-year span (from an article in the 1974
5. As an example of a situation in which several different statistics could reasonably be used to calculate a point estimate, consider a population of N invoices. Associated with each invoice is its
4. The article from which the data of Exercise 1 was extracted also gave the accompanying strength observations for cylinders:6.1 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.3 7.8 8.1 7.4 8.5 8.9 9.8 9.7 14.1
3. Consider the following sample of observations on coating thickness for low-viscosity paint (“Achieving a Target Value for a Manufacturing Process: A Case Study,” J. of Quality Technology,
2. A sample of 20 students who had recently taken elementary statistics yielded the following information on brand of calculator owned (T Texas Instruments, H Hewlett Packard, C Casio, S
1. The accompanying data on flexural strength (MPa) for concrete beams of a certain type was introduced in Example 1.2.5.9 7.2 7.3 6.3 8.1 6.8 7.0 7.6 6.8 6.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.7 7.4 7.7
Let X and Y be independent standard normal random variables, and define a new rv by U .6X .8Y.a. Determine Corr(X, U).b. How would you alter U to obtain Corr(X, U) for a specified value of ?
A more accurate approximation to E[h(X1, . . . , Xn)] in Exercise 93 is h(1, . . . , n) 1 22 1 . . . 1 22 n Compute this for Y h(X1, X2, X3, X4) given in Exercise 93, and compare it to the
Let X1, . . . , Xn be independent rv’s with mean values 1, . . . , n and variances 2 1, . . . , 2 n. Consider a function h(x1, . . . , xn), and use it to define a new rv Y h(X1, . . . , Xn).
Let A denote the percentage of one constituent in a randomly selected rock specimen, and let B denote the percentage of a second constituent in that same specimen.Suppose D and E are measurement
A rock specimen from a particular area is randomly selected and weighed two different times. Let W denote the actual weight and X1 and X2 the two measured weights. Then X1 W E1 and X2 W E2, where
a. Show that Cov(X, Y Z) Cov(X, Y) Cov(X, Z).b. Let X1 and X2 be quantitative and verbal scores on one aptitude exam, and let Y1 and Y2 be corresponding scores on another exam. If Cov(X1, Y1) 5,
a. Let X1 have a chi-squared distribution with parameter 1(see Section 4.4), and let X2 be independent of X1 and have a chi-squared distribution with parameter 2. Use the technique of Example 5.21 to
Suppose a randomly chosen individual’s verbal score X and quantitative score Y on a nationally administered aptitude examination have joint pdf f(x, y) {2 5(2x 3y) 0 x 1, 0 y 1 0
a. Use the general formula for the variance of a linear combination to write an expression for V(aX Y). Then let a Y/X, and show that 1. [Hint: Variance is always 0, and Cov(X, Y) X Y
A student has a class that is supposed to end at 9:00 A.M.and another that is supposed to begin at 9:10 A.M. Suppose the actual ending time of the 9 A.M. class is a normally distributed rv X1 with
Refer to Exercise 58, and suppose that the Xis are independent with each one having a normal distribution. What is the probability that the total volume shipped is at most 100,000 ft3?
If the amount of soft drink that I consume on any given day is independent of consumption on any other day and is normally distributed with 13 oz and 2 and if I currently have two six-packs of
Let denote the true pH of a chemical compound. A sequence of n independent sample pH determinations will be made. Suppose each sample pH is a random variable with expected value and standard
Suppose the proportion of rural voters in a certain state who favor a particular gubernatorial candidate is .45 and the proportion of suburban and urban voters favoring the candidate is .60. If a
We have seen that if E(X1) E(X2) . . . E(Xn) , then E(X1 . . . Xn) n. In some applications, the number of Xis under consideration is not a fixed number n but instead is an rv N. For
The mean weight of luggage checked by a randomly selected tourist-class passenger flying between two cities on a certain airline is 40 lb, and the standard deviation is 10 lb.The mean and standard
Suppose that for a certain individual, calorie intake at breakfast is a random variable with expected value 500 and standard deviation 50, calorie intake at lunch is random with expected value 900
Let X1, X2, . . . , Xn be random variables denoting n independent bids for an item that is for sale. Suppose each Xi is uniformly distributed on the interval [100, 200]. If the seller sells to the
A health-food store stocks two different brands of a certain type of grain. Let X the amount (lb) of brand A on hand and Y the amount of brand B on hand. Suppose the joint pdf of X and Y is f(x,
In cost estimation, the total cost of a project is the sum of component task costs. Each of these costs is a random variable with a probability distribution. It is customary to obtain information
A restaurant serves three fixed-price dinners costing $12,$15, and $20. For a randomly selected couple dining at this restaurant, let X the cost of the man’s dinner and Y the cost of the
In an area having sandy soil, 50 small trees of a certain type were planted, and another 50 trees were planted in an area having clay soil. Let X the number of trees planted in sandy soil that
Suppose the expected tensile strength of type-A steel is 105 ksi and the standard deviation of tensile strength is 8 ksi. For type-B steel, suppose the expected tensile strength and standard
I have three errands to take care of in the Administration Building. Let Xi the time that it takes for the ith errand(i 1, 2, 3), and let X4 the total time in minutes that I spend walking to
In Exercise 66, the weight of the beam itself contributes to the bending moment. Assume that the beam is of uniform thickness and density so that the resulting load is uniformly distributed on the
Consider a random sample of size n from a continuous distribution having median 0 so that the probability of any one observation being positive is .5. Disregarding the signs of the observations, rank
Three different roads feed into a particular freeway entrance. Suppose that during a fixed time period, the number of cars coming from each road onto the freeway is a random variable, with expected
Two airplanes are flying in the same direction in adjacent parallel corridors. At time t 0, the first airplane is 10 km ahead of the second one. Suppose the speed of the first plane(km/hr) is
One piece of PVC pipe is to be inserted inside another piece. The length of the first piece is normally distributed with mean value 20 in. and standard deviation .5 in. The length of the second piece
If two loads are applied to a cantilever beam as shown in the accompanying drawing, the bending moment at 0 due to the loads is a1X1 a2X2.a. Suppose that X1 and X2 are independent rv’s with means 2
Suppose that when the pH of a certain chemical compound is 5.00, the pH measured by a randomly selected beginning chemistry student is a random variable with mean 5.00 and standard deviation .2. A
Suppose your waiting time for a bus in the morning is uniformly distributed on [0, 8], whereas waiting time in the evening is uniformly distributed on [0, 10] independent of morning waiting time.a.
Refer to Exercise 3.a. Calculate the covariance between X1 the number of customers in the express checkout and X2 the number of customers in the superexpress checkout.b. Calculate V(X1 X2). How
Manufacture of a certain component requires three different machining operations. Machining time for each operation has a normal distribution, and the three times are independent of one another. The
Exercise 26 introduced random variables X and Y, the number of cars and buses, respectively, carried by a ferry on a single trip. The joint pmf of X and Y is given in the table in Exercise 7. It is
Five automobiles of the same type are to be driven on a 300-mile trip. The first two will use an economy brand of gasoline, and the other three will use a name brand. Let X1, X2, X3, X4, and X5 be
Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a certain service facility.Suppose they are independent, normal rv’s with expected values 1, 2, and 3
A shipping company handles containers in three different sizes: (1) 27 ft3 (3 3 3), (2) 125 ft3, and (3) 512 ft3. Let Xi (i 1, 2, 3) denote the number of type i containers shipped during a
Suppose the distribution of the time X (in hours) spent by students at a certain university on a particular project is gamma with parameters 50 and 2. Because is large, it can be shown that
A binary communication channel transmits a sequence of“bits” (0s and 1s). Suppose that for any particular bit transmitted, there is a 10% chance of a transmission error (a 0 becoming a 1 or a 1

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