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statistics for engineers and scientists
Principles Of Statistics For Engineers And Scientists 1st Edition William Navidi - Solutions
5. The number of hits on a certain website follows a Poisson distribution with a mean rate of 4 per minute.a. What is the probability that five messages are received in a given minute?b. What is the probability that 9 messages are received in 1.5 minutes?c. What is the probability that fewer than
3. Suppose that 0.2% of diodes in a certain application fail within the first month of use. Let X represent the number of diodes in a random sample of 1000 that fail within the first month. Finda. P(X = 4)b. P(X ≤ 1)c. P(1 ≤ X < 4)d. µXe. σX
2. The concentration of particles in a suspension is 4 per mL. The suspension is thoroughly agitated, and then 2 mL is withdrawn. Let X represent the number of particles that are withdrawn. Finda. P(X = 6)b. P(X ≤ 3)c. P(X > 2)d. µXe. σX
1. Let X ∼ Poisson(3). Finda. P(X = 2)b. P(X = 0)c. P(X < 3)d. P(X > 2)e. µXf. σX
12. Refer to Exercise 11 for the definition of a k out of n system. For a certain 4 out of 6 system, assume that on a rainy day each component has probability 0.7 of functioning and that on a nonrainy day each component has probability 0.9 of functioning.a. What is the probability that the system
10. A distributor receives a large shipment of components. The distributor would like to accept the shipment if 10% or fewer of the components are defective and to return it if more than 10% of the components are defective. She decides to sample 10 components and to return the shipment if more
9. Of the bolts manufactured for a certain application, 85% meet the length specification and can be used immediately, 10% are too long and can be used after being cut, and 5% are too short and must be scrapped.a. Find the probability that a randomly selected bolt can be used (either immediately or
8. In a large shipment of automobile tires, 10% have a flaw. Four tires are chosen at random to be installed on a car.a. What is the probability that none of the tires have a flaw?b. What is the probability that exactly one of the tires has a flaw?c. What is the probability that one or more of the
7. A fair coin is tossed eight times.a. What is the probability of obtaining exactly five heads?b. Find the mean number of heads obtained.c. Find the variance of the number of heads obtained.d. Find the standard deviation of the number of heads obtained.
6. A large industrial firm allows a discount on any invoice that is paid within 30 days. Of all invoices, 10% receive the discount. In a company audit, 12 invoices are sampled at random.a. What is the probability that exactly four of them receive the discount?b. What is the probability that fewer
4. Ten percent of the items in a large lot are defective. A sample of six items is drawn from this lot.a. Find the probability that none of the sampled items are defective.b. Find the probability that one or more of the sampled items is defective.c. Find the probability that exactly one of the
3. Find the following probabilities:a. P(X = 7) when X ∼ Bin(13, 0.4)b. P(X ≥ 2) when X ∼ Bin(8, 0.4)c. P(X < 5) when X ∼ Bin(6, 0.7)d. P(2 ≤ X ≤ 4) when X ∼ Bin(7, 0.1)
2. Let X ∼ Bin(5, 0.3). Finda. P(X < 3)b. P(X ≥ 1)c. P(1 ≤ X ≤ 3)d. P(2 < X < 5)e. P(X = 0)f. P(X = 3) g. µX h. σ2 X
1. Let X ∼ Bin(10, 0.6). Finda. P(X = 3)b. P(X = 6)c. P(X ≤ 4)d. P(X > 8)e. µXf. σ2 X
15. In the article “An Investigation of the Ca–CO3–CaF2– K2SiO3–SiO2–Fe Flux System Using the Submerged Arc Welding Process on HSLA-100 and AISI-1018 Steels” (G. Fredrickson, M.S. thesis, Colorado School of Mines, 1992), the carbon equivalent P of a weld metal is defined to be a
14. An environmental scientist is concerned with the rate at which a certain toxic solution is absorbed into the skin. Let X be the volume in microliters of solution absorbed by 1 in2 of skin in 1 min. Assume that the probability density function of X is well approximated by the function f (x) = (2
13. Refer to Exercise 12. A competing process produces rings whose diameters (in centimeters) vary according to the probability density function f (x) = 15[1 − 25(x − 10.05)2 ]/4 9.85 < x < 10.25 0 otherwise Specifications call for the diameter to be 10.0 ± 0.1 cm. Which process is better,
12. A process that manufactures piston rings produces rings whose diameters (in centimeters) vary according to the probability density function f (x) = 3[1 − 16(x − 10)2 ] 9.75
11. The repair time (in hours) for a certain machine is a random variable with probability density function f (x) = xe−x x > 0 0 x ≤ 0a. What is the probability that the repair time is less than 2 hours?b. What is the probability that the repair time is between 1.5 and 3 hours?c. Find the mean
10. The pressure P, temperature T , and volume V of one mole of an ideal gas are related by the equation T = 0.1203PV, when P is measured in kilopascals, T is measured in kelvins, and V is measured in liters.a. Assume that P is measured to be 242.52 kPa with a standard deviation of 0.03 kPa and V
9. The period T of a simple pendulum is given by T = 2π √L/g, where L is the length of the pendulum and g is the acceleration due to gravity. Assume g = 9.80 m/s2 exactly, and L is measured to be 0.742 m with a standard deviation of 0.005 m. Estimate T , and find the standard deviation of the
6. Let A and B be events with P(A) = 0.4 and P(A ∩ B) = 0.2. For what value of P(B) will A and B be independent?
4. The lifetime, in years, of a certain type of fuel cell is a random variable with probability density function f (x) = 81 (x + 3)4 x > 0 0 x ≤ 0a. What is the probability that a fuel cell lasts more than 3 years?b. What is the probability that a fuel cell lasts between 1 and 3
2. In a certain type of automobile engine, the cylinder head is fastened to the block by 10 bolts, each of which should be torqued to 60 N·m. Assume that the torques of the bolts are independent. If each bolt is torqued correctly with probability 0.99, what is the probability that all the bolts
14. One way to measure the water content of a soil is to weigh the soil both before and after drying it in an oven. The water content is W = (M1 − M2)/M1, where M1 is the mass before drying and M2 is the mass after drying. Assume that M1 = 1.32 ± 0.01 kg and M2 = 1.04 ± 0.01 kg.a. Estimate W,
13. The acceleration g due to gravity is estimated by dropping an object and measuring the time it takes to travel a certain distance. Assume the distance s is known to be exactly 5 m, and the time is measured to be t = 1.01 ± 0.02 s. Estimate g, and find the standard deviation of the estimate.
11. The number of miles traveled per gallon of gasoline for a certain car has a mean of 25 and a standard deviation of 2. The tank holds 20 gallons.a. Find the mean number of miles traveled per tank.b. Assume the distances traveled are independent for each gallon of gas. Find the standard deviation
6. The period of a pendulum is estimated by measuring the starting and stopping times and taking their difference. If the starting and stopping times are measured independently, each with standard deviation 0.2 s, what is the standard deviation of the estimated period?
5. A piece of plywood is composed of five layers. The layers are a simple random sample from a popula tion whose thickness has mean 3.50 mm and standard deviation 0.10 mm.a. Find the mean thickness of a piece of plywood.b. Find the standard deviation of the thickness of a piece of plywood.
4. The force, in N, exerted by gravity on a mass of m kg is given by F = 9.8m. Objects of a certain type have mass whose mean is 2.3 kg with a standard deviation of 0.2 kg. Find the mean and standard deviation of F.
3. A process that fills plastic bottles with a beverage has a mean fill volume of 2.013 L and a standard deviation of 0.005 L. A case contains 24 bottles. Assuming that the bottles in a case are a simple random sample of bottles filled by this method, find the mean and standard deviation of the
In a study, test X has been administered to a group of highability examinees, and test Y to a low-ability group. Both groups also have been tested with test V. In a second study, all examinees have been tested with test V. The examinees with relatively high scores on V have been tested with test X,
We have two test forms, X and Y, each with five items. Both tests are analyzed with the Rasch model. The items are numbered consecutively in the table below. We notice that two items, items 3 and 5, are common to X and Y.Compute the estimated item parameters of the items of Y on the scale defined
V is a subtest of test X. Assume that test X is the sum of subtest V and k − 1 tests parallel to subtest V. Prove that k(= σT(X)/σT(V) ) = /σXV (Equation 11.9).
We have two groups of persons. One group is administered test form Y, the other is administered test form X. Suppose we know that the group that answered test form Y is somewhat better than the other group. What can you say about the x score equivalent to y = 50 in Exercise 11.1? Explain your
We have two tests forms X and Y. In a large random selection of persons, the mean score on Y is 60.0 and the standard deviation is 16.0. In a second random selection, the mean score on X is 55.0 and the standard deviation is 17.0. We want to equate Y with X. With which score on X does the score y =
In an item bank we have items conforming to the 2PL model.The items have the item parameters: b1 = −0.5, a1 = 1.0, b2 =−0.25, a2 = 2.0, b3 = 0.0, a3 = 0.7, b4 = 0.25, a4 = 1.0, b5 = 0.5, and a5 = 1.5. We test a person and the present point estimate of ability θ is 0.20. Which item should be
We have five Rasch items with b1 = −0.5, b2 = −0.3, b3 = 0.0, b4 = 0.25, and b5 = 0.5. We want to construct a two-item test that discriminates relatively well at θ1 = −0.5 and at θ2 = 0.5.Which combination of two items from the item bank with five items is best, given the criterion that the
Two test items were administered to a reference group R and a focal group F. The proportions correct are p1(R) = 0.70 p1(F) = 0.65 p2(R) = 0.70 p2(F) = 0.60 Is the second item biased against the focal group?
We have a discrete distribution of θ with values –1, −0.5, 0.0, 0.5, and 1. The following is known:Compute the reliability of the test when maximum likelihood is used for the estimation of θ. Value 0 Frequency f(0) I(0) -1.0 0.1 7.864 -0.5 0.2 9.400 0.0 0.4 10.000 0.5 0.2 9.400 1.0 0.1 7.864
We have three items with item parameters:b1 = 0.5, a1 = 1.0, c1 = 0.0 b2 = 0.5, a2 = 2.0, c2 = 0.0 b3 = 0.5, a3 = 2.0, c3 = 0.25 Compute the item informations at θ = 0.0
Given is a test with three Rasch items. The item parameters are b1 = −0.5, b2 = 0.0, and b3 = 0.5. A person has answered items 1 and 2 correctly, and item 3 incorrectly. Compute the likelihood for θ = −1.0, −0.5, 0.0, 0.5, 1.0.a. Consider the four intervals defined by the five values of
We have the responses of two homogeneous groups of persons on two items. The response probabilities are P1(θ1) = 0.3775, P1(θ2) = 0.6225, P2(θ1) = 0.4378, and P2(θ2) = 0.7112. Estimate the person parameters θ1 and θ2 on the basis of the response probabilities for the first item, assuming that
Compute the probability of a correct response for a Rasch item with item parameter equal to 0.0 and person parameterθ = −2.0 (0.5) 2.0.
Compute coefficient κ for the data in the following table: - 1 + 10 20 60 10 +
We have a test with a mean equal to 75.0, a standard deviation equal to 8.0, and a reliability equal to 0.25. With the test we want to decide which examinees are masters and who are nonmasters. The criterion of mastery is 70.0 on the truescore scale. The errors of classifying masters and nonmasters
What happens if the base rate of belonging to group B in Exercise 7.2 is 0.5 instead of 0.2?
Given is a ten-item test with the following frequency distribution in two groups A and B:We want to use the test in order to classify persons in the future. Both kinds of errors are equally serious. At what test score should we take the decision to classify a person as a “B person” assuming
In a study, test X is administrated to all persons. Test Y is administrated to a selection of persons. Within each group with the same score on X persons are randomly chosen for selection into the group that is administered test Y. The correlation between X and Y equals 0.8. The variance of the
Compare rit and rir for tests with all item variances equal to 0.25 and all interitem covariances equal to 0.05. Compute the correlations for test lengths 10, 20, and 40.
A test consists of three items. The probabilities correct for person p are P1(ζp) = 0.6, P2(ζp) = 0.7, and P3(ζp) = 0.8. Compute the error variance on the total score scale. Also compute the error variance under the binomial model assumption. Comment on the difference.
For a person p the probability of a correct answer to two items is P1(ζp) = 0.7 and P2(ζp) = 0.8, respectively. Compute the probabilities of all possible response patterns.6.5 What information would you like to obtain in order to verify whether the assumptions made by Keats, see Equation 6.14 and
In a testing procedure, each examinee responds to a different set of ten items, randomly selected from a large item pool.The test mean equals 7.5, and standard deviation equals 1.5.What might be concluded about the test reliability?
Compute the probability that a person with a domain score equal to 0.8 answers at least 8 out of 10 items correct, assuming that the items have been randomly selected from a large item pool.6.2a. Compute the proportion correct and the item–rest correlation of item 8 in the table of Exercise 5.1.
Three judges rated 50 examinees each. The variances of the ratings are practically equal for all three judges. The pooled within-judges variance equals 100.0. The judges have different means. Judge 1 has a mean equal to 32.0, judge 2 has a mean equal to 35.0, and judge 3 has a mean equal to 38.0.
Derive the formulas for the relative and absolute error variance for the crossed p × i × j design.
Derive the formula for the correlation between two judges who both judge the responses to ni items. Use the notation of the variance components from generalizability theory(cf. Maxwell and Pilliner, 1968).
The following table gives the expected mean squares for the nested j : (i × p) design. Give the coefficients of the variance components in terms of np, ni, and nj. EMS of the Nested j: (i x p) Design EMS, EMS + a + bo +co + do EMSpl +e 1-44-x EMS ---
Compute the generalizability coefficient for (a) 30 items and 4 judges and (b) 60 items and 2 judges, using the estimated variance components from Exercise 5.2.
A test consisting of 15 open-answer items is given to 500 examinees. The responses are judged by four judges in a completely crossed design. The mean squares from an ANOVA are given in the table below. Compute the variance components and the generalizability coefficient for 15 items and 4 judges.
We have the following table:Compute the item variances and the variance of total scores.Next, compute coefficient α.Compute the mean squares for items, persons, and interaction. Compute the variance components and discuss the implications of the values of these components. Finally, compute the
Two tests X and Y are available. The tests have equal observedscore variances: = = 25.0. The reliability of test X is 0.8, the reliability of test Y is 0.6. Their intercorrelation is zero.Compute the reliability of the composite test X + Y. Also, compute the reliability of the composite after
A test has a mean score equal to 40.0, a standard deviation equal to 10.0, and a reliability equal to 0.5. Which difference score do you expect after a retest when the first score of a person equals 30?
We have three tests X1, X2, and X3 measuring the same construct. Their correlations with test Y equal 0.80, 0.70, and 0.60. Their covariances with Y are equal to 0.20. The means of the tests are 16.0, 16.0, and 20.0, respectively. Are these tests parallel tests, tau-equivalent, essential
In a test, several items cover the same subject. Which assumption of classical test theory might be violated? What should we do when we want to estimate reliability with coefficient α?
Given are two tests X and Y with = 16.0, = 16.0, ρXX′ =ρYY′ = 0.8, and ρXY = 0.7.a. Compute the observed-score variance, the true-score variance, and the reliability of the difference scores X – Y.b. Compare the variance of the raw score differences with of Equation 4.18.
Two tests X1 and X2 are congeneric measurement instruments.Their correlations with other variables Y1, Y2, and so on, differ.Is there a pattern to be found in the correlations?
Prove that for parallel test items coefficient alpha equals the Spearman–Brown formula for the reliability of a lengthened test.
Use the variance–covariance matrix from Exercise 4.1 for estimating test reliability according to the model of congeneric tests. Use Equation 4.6 for the estimation of the ai.
A test X is given with three subtests, X1, X2, and X3. The variance–covariance matrix for the subtests is given in the table below. Estimate reliability with coefficient α. X X X3 X 8.0 6.0 8.0 X 6.0 12.0 12.0 X 8.0 12.0 17.0
Let ρXY be the validity of test X with respect to test Y. Write the validity of test X lengthened by a factor k, in terms of ρXY,σX, σY, and ρXX′. What happens when k becomes very large?
The reliability of test X equals 0.49. What is the maximum correlation that can be obtained between test X and a criterion? Explain your answer. Suggestion: Use the formula for the correction for attenuation.
Compute the ratio of the standard error of estimation and the standard error of measurement for ρxx′ = 0.5 and ρxx′ = 0.9.Compute the Kelley estimate of true score for an observed score equal to 30, and μX = 40, ρxx′ = 0.5, respectively, ρxx′ = 0.9.
The reliability of a test is 0.5. Compute test reliability if the test is lengthened with a factor k = 2, 3, 4,…, 14 (k = 2(1)14, for short).
The reliability of a test is 0.75. The standard deviation of observed scores is 10.0. Compute the standard error of measurement.
Two intelligence tests are administered close after one another. What kind of problem do you expect?
A large testing agency administers test X to all candidates at the same time in the morning. Other test centers organize sessions at different moments. Give alternative definitions of true score.
In a tennis tournament, five persons play in all different combinations. Player A wins all games; B wins from C, D, and E; C wins from D and E; and D wins from E. The number of games won is taken as the total score. Which property has this score in terms of Stevens’ classification?
Two researchers evaluate the same educational program.Researcher A uses an easy test as a pretest and posttest, researcher B uses a relatively difficult test. Is it likely that their results will differ? If that is the case, in which way are the results expected to differ?
The error in the length of a part (absolute value of the difference between the actual length and the target length), in mm, is a random variable with probability density function 0 < x
The concentration of a reactant is a random variable with probability density function (1.2(x+x) f(x)= {0 0 < x < 1 otherwisea. What is the probability that the concentration is greater than 0.5?b. Find the mean concentration.c. Find the probability that the concentration is within 0.1 of the
The diameter of a rivet (in mm) is a random variable with probability density function f(x)= 6(x-12)(13-x) 0 12 < x 13 otherwisea. What is the probability that the diameter is less than 12.5 mm?b. Find the mean diameter.c. Find the standard deviation of the diameters.d. Find the cumulative
The lifetime of a transistor in a certain application has a lifetime that is random with probability density function f (t) = 0.1e−0.1t t > 0 0 t ≤ 0a. Find the mean lifetime.b. Find the standard deviation of the lifetimes.c. Find the cumulative distribution function of the lifetime.d. Find the
Elongation (in %) of steel plates treated with aluminum are random with probability density function f (x) = x 250 20 < x < 30 0 otherwisea. What proportion of steel plates have elongations greater than 25%?b. Find the mean elongation.c. Find the variance of the elongations.d. Find the standard
Resistors labeled 100 have true resistances that are between 80 and 120 . Let X be the mass of a randomly chosen resistor. The probability density function of X is given by f (x) = x − 80 800 80 < x < 120 0 otherwisea. What proportion of resistors have resistances less than 90 ?b. Find
After manufacture, computer disks are tested for errors. Let X be the number of errors detected on a randomly chosen disk. The following table presents values of the cumulative distribution function F(x) of X. x F(x) 0 0.41 1 0.72 2 0.83 3 0.95 4 1.00a. What is the probability that two or fewer
A certain type of component is packaged in lots of four. Let X represent the number of properly functioning components in a randomly chosen lot. Assume that the probability that exactly x components function is proportional to x; in other words, assume that the probability mass function of X is
The element titanium has five stable occurring isotopes, differing from each other in the number of neutrons an atom contains. If X is the number of neutrons in a randomly chosen titanium atom, the probability mass function of X is given as follows: x 24 25 26 27 28 p(x) 0.0825 0.0744 0.7372
A chemical supply company ships a certain solvent in 10-gallon drums. Let X represent the number of drums ordered by a randomly chosen customer. Assume X has the following probability mass function: x 12345 p(x) 0.4 0.2 0.2 0.1 0.1a. Find the mean number of drums ordered.b. Find the variance of the
Computer chips often contain surface imperfections. For a certain type of computer chip, the probability mass function of the number of defects X is presented in the following table. x 0 1 234 p(x) 0.4 0.3 0.15 0.10 0.05a. Find P(X ≤ 2).b. Find P(X > 1).c. Find µX .d. Find σ2 X .
Determine whether each of the following random variables is discrete or continuous.a. The number of heads in 100 tosses of a coin.b. The length of a rod randomly chosen from a day’s production.c. The final exam score of a randomly chosen student from last semester’s engineering statistics
6. A system consists of four components connected as shown in the following diagram:Assume A, B, C, and D function independently. If the probabilities that A, B, C, and D fail are 0.10, 0.05, 0.10, and 0.20, respectively, what is the probability that the system functions? A B D
5. A geneticist is studying two genes. Each gene can be either dominant or recessive. A sample of 100 individuals is categorized as follows.Gene 2 Gene 1 Dominant Recessive Dominant 56 24 Recessive 14 6a. What is the probability that a randomly sampled individual, Gene 1 is dominant?b. What is the
4. Refer to Exercise 3. Let E1 be the event that the wafer comes from Lot A, and let E2 be the event that the wafer is conforming. Are E1 and E2 independent? Explain.
3. A population of 600 semiconductor wafers contains wafers from three lots. The wafers are categorized by lot and by whether they conform to a thickness specification. The following table presents the number of wafers in each category. A wafer is chosen at random from the population. Lot
2. A drag racer has two parachutes, a main and a backup, that are designed to bring the vehicle to a stop after the end of a run. Suppose that the main chute deploys with probability 0.99 and that if the main fails to deploy, the backup deploys with probability 0.98.a. What is the probability that
1. Suppose that start-up companies in the area of biotechnology have probability 0.2 of becoming profitable and that those in the area of information technology have probability 0.15 of becoming profitable. A venture capitalist invests in one firm of each type. Assume the companies function
8. A flywheel is attached to a crankshaft by 12 bolts, numbered 1 through 12. Each bolt is checked to determine whether it is torqued correctly. Let A be the event that all the bolts are torqued correctly, let B be the event that the #3 bolt is not torqued correctly, let C be the event that
7. Sixty percent of large purchases made at a certain computer retailer are personal computers, 30% are laptop computers, and 10% are peripheral devices such as printers. As part of an audit, one purchase record is sampled at random.a. What is the probability that it is a personal computer?b. What
6. Human blood may contain either or both of two antigens, A and B. Blood that contains only the A antigen is called type A, blood that contains only the B antigen is called type B, blood that contains both antigens is called type AB, and blood that contains neither antigen is called type O. At a
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