New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
mathematics
statistics

Applied Statistics And Probability For Engineers 6th Edition Douglas C. Montgomery, George C. Runger - Solutions

A die is rolled until the first time T that a six turns up.(a) What is the probability distribution for T? (b) Find P (T > 3).(c) Find P (T > 6|T > 3).
A worker for the Department of Fish and Game is assigned the job of estimating the number of trout in a certain lake of modest size. She proceeds as follows: She catches 100 trout, tags each of them, and puts them back in the lake. One month later, she catches 100 more trout, and notes that 10 of
The probability that, in a bridge deal, one of the four hands has all hearts is approximately 6.3 × 10−12. In a city with about 50,000 bridge players the resident probability expert is called on the average once a year (usually late at night) and told that the caller has just been dealt a
In a class of 80 students, the professor calls on 1 student chosen at random for a recitation in each class period. There are 32 class periods in a term. (a) Write a formula for the exact probability that a given student is called upon j times during the term. (b) Write a formula for the Poisson
For a certain experiment, the Poisson distribution with parameter λ = m has been assigned. Show that a most probable outcome for the experiment is the integer value k such that m − 1 ≤ k ≤ m. Under what conditions will there be two most probable values? Hint: Consider the
Reese Prosser never puts money in a 10-cent parking meter in Hanover. He assumes that there is a probability of .05 that he will be caught. The first offence costs nothing, the second costs 2 dollars, and subsequent offences cost 5 dollars each. Under his assumptions, how does the expected cost of
Feller5 discusses the statistics of flying bomb hits in an area in the south of London during the Second World War. The area in question was divided into 24 × 24 = 576 small areas. The total number of hits was 537. There were 229 squares with 0 hits, 211 with 1 hit, 93 with 2 hits, 35 with 3 hits,
The king’s coinmaster boxes his coins 500 to a box and puts 1 counterfeit coin in each box. The king is suspicious, but, instead of testing all the coins in 1 box, he tests 1 coin chosen at random out of each of 500 boxes. What is the probability that he finds at least one fake? What is it if the
In one of the first studies of the Poisson distribution, von Bortkiewicz7 considered the frequency of deaths from kicks in the Prussian army corps. From the study of 14 corps over a 20-year period, he obtained the data shown in Table 5.5. Fit a Poisson distribution to this data and see if you think
A manufactured lot of brass turnbuckles has S items of which D are defective. A sample of s items is drawn without replacement. Let X be a random variable that gives the number of defective items in the sample. Let p (d) = P(X = d).(a) Show that P (d) = (D/d) (S-D/s-d) / (S/s)Thus, X is
The students in a certain class were classified by hair colour and eye colour. The conventions used were: Brown and black hair were considered dark, and red and blonde hair were considered light; black and brown eyes were considered dark, and blue and green eyes were considered light. They
Choose a number U from the unit interval [0, 1] with uniform distribution. Find the cumulative distribution and density for the random variables(a) Y = U + 2.(b) Y = U3.
Choose a number U from the interval [0, 1] with uniform distribution. Find the cumulative distribution and density for the random variables(a) Y = 1/ (U + 1).(b) Y = log (U + 1).
Choose a number U from the interval [0, 1] with uniform distribution. Find the cumulative distribution and density for the random variables (a) Y = |U − 1/2|. (b) Y = (U − 1/2)2.
Let U, V be random numbers chosen independently from the interval [0, 1] with uniform distribution. Find the cumulative distribution and density of each of the variables (a) Y = U + V. (b) Y = |U − V |.
Find the cumulative distribution function F and the density function f for each of the random variables R, S, and T in Exercise 12.
Let X be a random variable with cumulative distribution function(a) What is the density function fX for X?(b) What is the probability that X
Find the generating functions, both ordinary h (z) and moment g (t), for the following discrete probability distributions. (a) The distribution describing a fair coin.(b) The distribution describing a fair die.(c) The distribution describing a die that always comes up 3.(d) The uniform distribution
Let p be a probability distribution on {0, 1, 2} with moments μ1 = 1, μ2 = 3/2. (a) Find its ordinary generating function h(z). (b) Using (a), find its moment generating function. (c) Using (b), find its first six moments. (d) Using (a), find p0, p1, and p2.
Let X be a discrete random variable with values in {0, 1, 2, . . . , n} and moment generating function g(t). Find, in terms of g (t), the generating functions for (a) −X. (b) X + 1. (c) 3X. (d) aX + b.
Let X and Y be random variables with values in {1, 2, 3, 4, 5, 6} with distribution functions PX and PY given byPX (j) = a j, PY (j) = b j .(a) Find the ordinary generating functions hX (z) and hY (z) for these distributions.(b) Find the ordinary generating function hZ(z) for the distribution Z = X
Show that if X is a random variable with mean μ and variance σ2, and if X* = (X − μ)/ σ is the standardized version of X, then
Let Z1, Z2, . . . , ZN describe a branching process in which each parent has j offspring with probability pj . Find the probability d that the process eventually dies out if (a) p0 = 1/2, p1 = 1/4, and p2 = 1/4. (b) p0 = 1/3, p1 = 1/3, and p2 = 1/3. (c) p0 = 1/3, p1 = 0, and p2 = 2/3. (d) pj =
We have seen that if the generating function for the offspring of a single parent is f(z), then the generating function for the number of offspring after two generations is given by h(z) = f(f(z)). Explain how this follows from the result of Exercise 4.
Let N be the expected total number of offspring in a branching process. Let m be the mean number of offspring of a single parent. Show that.And hence that N is finite if and only if m
Let X be a continuous random variable with values in [0, 2] and density fX. Find the moment generating function g(t) for X if (a) fX(x) = 1/2. (b) fX(x) = (1/2)x. (c) fX(x) = 1 − (1/2)x. (d) fX(x) = |1 − x|. (e) fX(x) = (3/8)x2.
Let X be a continuous random variable with values in [0,∞) and density fX. Find the moment generating functions for X if (a) fX(x) = 2e−2x. (b) fX(x) = e−2x + (1/2)e-x. (c) fX(x) = 4xe−2x. (d) fX(x) = λ(λ x)n−1e− λx/(n − 1)!.
Find the characteristic function kX(T) for each of the random variables X of Exercise 1.
Let X be a continuous random variable with values in [0, 1], uniform density function fX(x) ≡ 1 and moment generating function g(t) = (et − 1)/t. Find in terms of g(t) the moment generating function for (a) −X. (b) 1 + X. (c) 3X. (d) aX + b.
Let X1, X2, . . . , Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X1. (b) S2 = X1 + X2. (c) Sn = X1 + X2 + • • • + Xn. (d) An = Sn/n. (e) S*n = (Sn − nμ)/√nσ2.
Let X be a continuous random variable with mean μ(X) and variance σ2(X), and let X* = (X − μ)/ σ be its standardized version. Verify directly that μ(X*) = 0 and σ 2(X*) = 1.
Let {Xk}, 1 ≤ k ≤ n, be a sequence of random variables, all with mean μ and variance σ 2, and Yk = X*k be their standardized versions. Let Sn and Tn be the sum of the Xk and Yk, and Sn and Tn their standardized version. Show that S*n = T*n = Tn/√n.
Let Sn be the number of successes in n Bernoulli trials with probability .8 for success on each trial. Let An = Sn/n be the average number of successes. In each case give the value for the limit, and give a reason for your answer.
In an opinion poll it is assumed that an unknown proportion p of the people are in favor of a proposed new law and a proportion 1 − p are against it. A sample of n people is taken to obtain their opinion. The proportion p in favor in the sample is taken as an estimate of p. Using the Central
A rookie is brought to a baseball club on the assumption that he will have a .300 batting average. (Batting average is the ratio of the number of hits to the number of times at bat.) In the first year, he comes to bat 300 times and his batting average is .267. Assume that his at bats can be
A die is thrown until the first time the total sum of the face values of the die is 700 or greater. Estimate the probability that, for this to happen, (a) More than 210 tosses are required.(b) Less than 190 tosses are required.(c) Between 180 and 210 tosses, inclusive, are required.
Prove the Law of Large Numbers using the Central Limit Theorem.
Show that, if X and Y are random variables taking on only two values each, and if E(XY ) = E(X)E(Y ), then X and Y are independent. Discuss.
A coin is tossed until the first time a head turns up. If this occurs on the nth toss and n is odd you win 2n/n, but if n is even then you lose 2n/n. Then if your expected winnings exist they are given by the convergent series 1 − 1/2 + 1/3 − 1/4 + • • • called the alternating
(from Propp18) In the previous problem, let P be the probability that at the present time, each book is in its proper place, i.e., book i is ith from the top. Find a formula for P in terms of the pi’s. In addition, find the least upper bound on P, if the pi’s are allowed to vary. Hint: First
If the first roll in a game of craps is neither a natural nor craps, the player can make an additional bet, equal to his original one, that he will make his point before a seven turns up. If his point is four or ten he is paid off at 2: 1 odds; if it is a five or nine he is paid off at odds 3: 2;
Let X be the first time that a failure occurs in an infinite sequence of Bernoulli trials with probability p for success. Let pk = P(X = k) for k = 1, 2, . . . . Show that pk = pk−1q where q = 1 − p. Show that Pk pk = 1. Show that E(X) = 1/q. What is the expected number of tosses of a
A multiple choice exam is given. A problem has four possible answers, and exactly one answer is correct. The student is allowed to choose a subset of the four possible answers as his answer. If his chosen subset contains the correct answer, the student receives three points, but he loses one point
It has been said12 that a Dr. B. Muriel Bristol declined a cup of tea stating that she preferred a cup into which milk had been poured first. The famous statistician R. A. Fisher carried out a test to see if she could tell whether milk was put in before or after the tea. Assume that for the test
In the casino game of blackjack the dealer is dealt two cards, one face up and one face down, and each player is dealt two cards, both face down. If the dealer is showing an ace the player can look at his down cards and then make a bet called an insurance bet. (Expert players will recognize why it
(Feller14) A large number, N, of people are subjected to a blood test. This can be administered in two ways: (1) Each person can be tested separately, in this case N test are required, (2) the blood samples of k persons can be pooled and analyzed together. If this test is negative, this one test
The following related discrete problem also gives a good clue for the answer to Exercise 32. Randomly select with replacement t1, t2, . . . , tr from the set (1/n, 2/n, . . . , n/n). Let X be the smallest value of r satisfying t1 + t2 + €¢ €¢ €¢ + tr > 1 . Then E(X) =
Let X be the random variable of Exercise 2. (a) Calculate the function f(x) = P (|X − 10| ≥ x). (b) Now graph the function f(x), and on the same axes, graph the Chebyshev function g(x) = 100/(3x2). Show that f(x) ≤ g(x) for all x > 0, but that g(x) is not a very good
Show that, if X ≥0, then P(X ≥ a) ≤E(X)/a. Discuss.
Let X be a random variable with range [−1, 1] and let fX (x) be the density function of X. Find μ(X) and 2(X) if, for |x| < 1, (a) fX (x) = 1/2. (b) fX (x) = |x|. (c) fX (x) = 1 − |x|. (d) fX (x) = (3/2)x2.
Let X be a random variable with density function fX. Show, using elementary calculus, that the function Ф(a) = E((X − a)2) takes its minimum value when a = μ(X), and in that case (a) = σ2(X).
Let X, Y, and Z be independent random variables, each with mean μ and variance 2. (a) Find the expected value and variance of S = X + Y + Z. (b) Find the expected value and variance of A = (1/3) (X + Y + Z). (c) Find the expected value of S2 and A2.
The Pilsdorff Beer Company runs a fleet of trucks along the 100 mile road from Hangtown to Dry Gulch. The trucks are old, and are apt to break down at any point along the road with equal probability. Where should the company locate a garage so as to minimize the expected distance from a typical
Let X be a random variable that takes on nonnegative values and has distribution function F(x). Show thatHint: Integrate by parts.Illustrate this result by calculating E(X) by this method if X has an exponential distribution F(x) = 1 − e−λx for x ≥ 0, and F(x) = 0 otherwise.
Let X be a random variable distributed uniformly over [0, 20]. Define a new random variable Y by Y = [X] (the greatest integer in X). Find the expected value of Y. Do the same for Z = [X + .5]. Compute E (|X − Y |) and E (|X − Z|). (Note that Y is the value of X rounded off to the
Let X and Y be random variables. The covariance Cov(X, Y) is defined by (see Exercise 6.2.23) cov (X, Y) = E((X − μ(X))(Y − μ(Y))) . (a) Show that cov(X, Y) = E (XY) − E(X) E(Y). (b) Using (a), show that cov(X, Y) = 0, if X and Y are independent. (Caution: the
Let X and Y be independent random variables with uniform densities in [0, 1]. Let Z = X + Y and W = X − Y. Find (a) p (X, Y ) (see Exercise 18). (b) p (X,Z). (c) p (Y,W). (d) p (Z,W).
For correlated random variables X and Y it is natural to ask for the expected value for X given Y . For example, Galton calculated the expected value of the height of a son given the height of the father. He used this to show that tall men can be expected to have sons who are less tall on the
(from Hamming26) A game is played as follows: A random number X is chosen uniformly from [0, 1]. Then a sequence Y1, Y2, . . . of random numbers is chosen independently and uniformly from [0, 1]. The game ends the first time that Yi > X. You are then paid (i − 1) dollars. What is a fair
In a certain manufacturing process, the (Fahrenheit) temperature never varies by more than 2o from 62o. The temperature is, in fact, a random variable F with distribution(a) Find E (F) and V (F).(b) Define T = F − 62. Find E (T) and V (T), and compare these answers with those in part (a).(c)
A number is chosen at random from the integers 1, 2, 3, . . . , n. Let X be the number chosen. Show that E(X) = (n+1)/2 and V (X) = (n−1)(n+1)/12. Hint: The following identity may be useful:
Peter and Paul play Heads or Tails (see Example 1.4). Let Wn be Peter’s winnings after n matches. Show that E(Wn) = 0 and V (Wn) = n.
Suppose that n people have their hats returned at random. Let Xi = 1 if the ith person gets his or her own hat back and 0 otherwise. Let Sn = Σn i=1 Xi. Then Sn is the total number of people who get their own hats back. Show that (a) E (X2 i) = 1/n. (b) E (Xi • Xj) = 1/n (n − 1) for
Let Sn be the number of successes in n independent trials. Use the program BinomialProbabilities (Section 3.2) to compute, for given n, p, and j, the probability P (−j√npq < Sn − np < j√npq) . (a) Let p = .5, and compute this probability for j = 1, 2, 3 and n = 10, 30, 50.
Let X be a random variable taking on values a1, a2, . . . , pr with probabilities p1, p2, . . . , pr and with E(X) = μ. Define the spread of X as follows: This, like the standard deviation, is a way to quantify the amount that a random variable is spread out around its mean. Recall that the
Let X be a random variable with E(X) = μ and V (X) = σ2. Show that the function f(x) defined by
If X and Y are any two random variables, then the covariance of X and Y is defined by Cov(X, Y) = E ((X −E(X))(Y −E(Y ))). Note that Cov(X, X) = V (X). Show that, if X and Y are independent, then Cov(X, Y) = 0; and show, by an example, that we can have Cov(X, Y) = 0 and X and Y not
(Lamperti20) An urn contains exactly 5000 balls, of which an unknown number X are white and the rest red, where X is a random variable with a probability distribution on the integers 0, 1, 2, . . . , 5000. (a) Suppose we know that E(X) = μ. Show that this is enough to allow us to calculate
Referring to Exercise 6.1.30, find the variance for the number of boxes of Wheaties bought before getting half of the players’ pictures and the variance for the number of additional boxes needed to get the second half of the players’ pictures.
(a) Do you think the engineer is correct? Use a = 0.05.(b) What is the P-value for this test?(c) What is the power of the test in part (a) for a true difference in means of 0.04?(d) Find a 95% confidence interval on the difference in means. Provide a practical interpretation of this interval.
Two types of plastic are suitable for use by an electronics component manufacturer. The breaking strength of this plastic is important. It is known that σ1 = σ2 = 1.0 psi. From a random sample of size n1 = 10 and n2 = 12, we obtain and the company will not adopt plastic 1 unless its mean
Reconsider the situation in Exercise 10-2. Suppose that the true difference in means is really 12 psi. Find the power of the test assuming that a = 0.05. If it is really important to detect this difference, are the sample sizes employed in Exercise 10-2 adequate, in your opinion?
The burning rates of two different solid-fuel propellants used in aircrew escape systems are being studied. It is known that both propellants have approximately the same standard deviation of burning rate; that is σ1 = σ2 = 3 centimeters per second. Two random samples of n1 = 20 x1 =
Two machines are used to fill plastic bottles with dishwashing detergent. The standard deviations of fill volume are known to be σ1 = 0.10 fluid ounces and μ2 = 0.15 fluid ounces for the two machines, respectively. Two random samples of n1 = 12 bottles from machine 1 and n2 = 10 bottles
Reconsider the situation described in Exercise 10-5. (a) Test the hypothesis that both machines fill to the same mean volume. Use a = 0.05.(b) What is the P-value of the test in part (a)?(c) If the B-error of the test when the true difference in fill volume is 0.2 fluid ounces should not exceed
Two different formulations of an oxygenated motor fuel are being tested to study their road octane numbers. The variance of road octane number for formulation 1 is = 1.5, and for formulation 2 it is 2/2 = 1.2. Two random samples of size n1 = 15 and n2 = 20 are tested, and the mean road octane
Consider the situation described in Exercise 10-4. What sample size would be required in each population if we wanted the error in estimating the difference in mean burning rates to be less than 4 centimeters per second with 99% confidence?
Consider the road octane test situation described in Exercise 10-7. What sample size would be required in each population if we wanted to be 95% confident that the error in estimating the difference in mean road octane number is less than 1?
A polymer is manufactured in a batch chemical process. Viscosity measurements are normally made on each batch, and long experience with the process has indicated that the variability in the process is fairly stable with σ = 20. Fifteen batch viscosity measurements are given as follows: 724,
The concentration of active ingredient in a liquid laundry detergent is thought to be affected by the type of catalyst used in the process. The standard deviation of active concentration is known to be 3 grams per liter, regardless of the catalyst type. Ten observations on concentration are taken
Consider the polymer batch viscosity data in Exercise 10-10. If the difference in mean batch viscosity is 10 or less, the manufacturer would like to detect it with a high probability.(a) Formulate and test an appropriate hypothesis using a =0.10. What are your conclusions?(b) Calculate the P-value
For the laundry detergent problem in Exercise 10-11, test the hypothesis that the mean active concentrations are the same for both types of catalyst. Use a = 0.05. What is the P-value for this test? Compare your answer to that found in part (b) of Exercise 10-11, and comment on why they are the
Reconsider the laundry detergent problem in Exercise 10-11. Suppose that the true mean difference in active concentration is 5 grams per liter. What is the power of the test to detect this difference if a = 0.05? If this difference is really important, do you consider the sample sizes used by the
Consider the polymer viscosity data in Exercise 10- 10. Does the assumption of normality seem reasonable for both samples?
Consider the concentration data in Exercise 10-11. Does the assumption of normality seem reasonable?
The diameter of steel rods manufactured on two different extrusion machines is being investigated. Two random samples of sizes n1 = 15 and n2 = 17 are selected, and the sample means and sample variances are = 8.73, s2/1 = 0.35,x = 8.68, and s2/2 = 0.40, respectively. Assume that σ2/1= σ
An article in Fire Technology investigated two different foam expanding agents that can be used in the nozzles of fire-fighting spray equipment. A random sample of five observations with an aqueous film-forming foam (AFFF) had a sample mean of 4.7 and a standard deviation of 0.6. A random sample of
Two catalysts may be used in a batch chemical process. Twelve batches were prepared using catalyst 1, resulting in an average yield of 86 and a sample standard deviation of 3. Fifteen batches were prepared using catalyst 2, and they resulted in an average yield of 89 with a standard deviation of 2.
The deflection temperature under load for two different types of plastic pipe is being investigated. Two random samples of 15 pipe specimens are tested, and the deflection temperatures observed are as follows (in oF): Type 1: 206, 188, 205, 187, 194, 193, 207, 185, 189, 213, 192, 210, 194, 178,
In semiconductor manufacturing, wet chemical etching is often used to remove silicon from the backs of wafers prior to metallization. The etch rate is an important characteristic in this process and known to follow a normal distribution. Two different etching solutions have been compared, using two
Two suppliers manufacture a plastic gear used in a laser printer. The impact strength of these gears measured in foot-pounds is an important characteristic. A random sample of 10 gears from supplier 1 results in and s1 = 12, while another random sample of 16 gears from the second supplier results
Reconsider the situation in Exercise 10-22, part (a). Construct a confidence interval estimate for the difference in mean impact strength, and explain how this interval could be used to answer the question posed regarding supplier to- supplier differences.
A photoconductor film is manufactured at a nominal thickness of 25 mils. The product engineer wishes to increase the mean speed of the film, and believes that this can be achieved by reducing the thickness of the film to 20 mils. Eight samples of each film thickness are manufactured in a pilot
The melting points of two alloys used in formulating solder were investigated by melting 21 samples of each material. The sample mean and standard deviation for alloy 1 was x1 = 420oF and s1 = 4oF, while for alloy 2 they were x2 = 426oF, and s2 = 3oF. Do the sample data support the claim that both
Referring to the melting point experiment in Exercise 10-25, suppose that the true mean difference in melting points is 3oF. How large a sample would be required to detect this difference using an a = 0.05 level test with probability at least 0.9? Use σ1 = σ2 = 4 as an initial estimate of
Two companies manufacture a rubber material intended for use in an automotive application. The part will be subjected to abrasive wear in the field application, so we decide to compare the material produced by each company in a test. Twenty-five samples of material from each company are tested in
The thickness of a plastic film (in mils) on a substrate material is thought to be influenced by the temperature at which the coating is applied. A completely randomized experiment is carried out. Eleven substrates are coated at 125oF, resulting in a sample mean coating thickness of and a sample
Reconsider the coating thickness experiment in Exercise 10-28. How could you have answered the question posed regarding the effect of temperature on coating thickness by using a confidence interval? Explain your answer.
Reconsider the abrasive wear test in Exercise 10-27. Construct a confidence interval that will address the questions in parts (a) and (c) in that exercise.
The overall distance traveled by a golf ball is tested by hitting the ball with Iron Byron, a mechanical golfer with a swing that is said to emulate the legendary champion, Byron Nelson. Ten randomly selected balls of two different brands are tested and the overall distance measured. The data

Showing 1000 - 1100 of 88243

First
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved