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statistics for engineers and scientists
Introduction To Probability And Statistics For Engineers And Scientists 4th Edition Sheldon M. Ross - Solutions
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 equal to 1,740, what is a 95 percent confidence interval for the population mean θ? f(x) ===-x10 0
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 percent. What does this mean? Can we infer how many people were questioned?
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 following capacities (in ampere hours):Determine a 90 percent level two-sided confidence interval
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 stress in a laboratory experiment, failures for the two types of cable insulation occurred at the
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, in inches,.123 .133.124 .125.126 .128.120 .124.130 .126 what is a 90 percent confidence interval
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, 73, 69, 104, 48, 66, 80, 64, 77. Also determine a 95 percent lower confidence interval for this mean.
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 σ2 being unknown. If 9 successive values are, as in Example 7.3a, 5, 8.5, 12, 15, 7, 9, 7.5, 6.5,
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 0.3 pounds. If we want to be 95 percent certain that our estimate of the present season’s mean
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.
Determine the upper and lower 95 percent confidence interval estimates of μ in Example 7.3a.
Kolmogorov’s law of fragmentation states that the size of an individual particle in a large collection of particles resulting from the fragmentation of a mineral compound will have an approximate lognormal distribution, where a random variable X is said to have a lognormal distribution if log(X )
The number of traffic accidents in Berkeley, California, in 10 randomly chosen nonrainy days in 1998 is as follows:4, 0, 6, 5, 2, 1, 2, 0, 4, 3 Use these data to estimate the proportion of nonrainy days that had 2 or fewer accidents that year.
Two proofreaders were given the same manuscript to read. If proofreader 1 found n1 errors, and proofreader 2 found n2 errors, with n1,2 of these errors being found by both proofreaders, estimate N, the total number of errors that are in the manuscript.
A certain component is critical to the operation of an electrical system and must be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its standard deviation is 30 hours, how many of the components must be in stock so that the probability that the
The average salary of newly graduated students with bachelor’s degrees in chemical engineering is $53,600, with a standard deviation of $3,200. Approximate the probability that the average salary of a sample of 12 recently graduated chemical engineers exceeds $55,000.
The sample mean and sample standard deviation of all San Francisco student scores on the most recent Scholastic Aptitude Test examination in mathematics were 517 and 120. Approximate the probability that a random sample of 144 students would have an average score exceeding(a) 507;(b) 517;(c)
In 1995 the percentage of the labor force that belonged to a union was 14.9. If five workers had been randomly chosen in that year, what is the probability that none of them would have belonged to a union? Compare your answer to what it would be for the year 1945, when an all-time high of 35.5
The following table uses 1989 data concerning the percentages of male and female full-time workers whose annual salaries fall in different salary groupings. Suppose random samples of 1,000 men and 1,000 women were chosen. Use the table to approximate the probability that(a) at least half of the
(Use the table from Problem 23.) Suppose random samples of 300 women and of 300 men are chosen. Approximate the probability that more women than men rarely eat breakfast.
(Use the table from Problem 23.) Suppose a random sample of 300 women is chosen. Approximate the probability that(a) at least 60 of them are overweight by 20 percent or more;(b) fewer than 50 of them sleep 6 hours or less nightly.
The following table gives the percentages of individuals, categorized by gender, that follow certain negative health practices. Suppose a random sample of 300 men is chosen. Approximate the probability that(a) at least 150 of them rarely eat breakfast;(b) fewer than 100 of them smoke. Sleeps 6
Fifty-two percent of the residents of a certain city are in favor of teaching evolution in high school. Find or approximate the probability that at least 50 percent of a random sample of size n is in favor of teaching evolution, when(a) n = 10;(b) n = 100;(c) n = 1,000;(d) n = 10,000.
Twelve percent of the population is left-handed. Find the probability that there are between 10 and 14 left-handers in a random sample of 100 members of this population. That is, find P{10 ≤ X ≤ 14}, where X is the number of left-handers in the sample.
Consider two independent samples—the first of size 10 from a normal population having variance 4 and the second of size 5 from a normal population having variance 2. Compute the probability that the sample variance from the second sample exceeds the one from the first. (Hint: Relate it to the
In Problem 18, howlarge a sample would be necessary to ensure that the probability in part (a) is at least .95?
The temperature at which a thermostat goes off is normally distributed with varianceσ2. If the thermostat is to be tested five times, find(a) P{S2/σ2 ≤ 1.8}(b) P{.85 ≤ S2/σ2 ≤ 1.15}where S2 is the sample variance of the five data values.
Use the text disk to compute P{X ≤ 10} when X is a binomial random variable with parameters n = 100, p = .1. Now compare this with its (a) Poisson and(b) normal approximation. In using the normal approximation, write the desired probability as P{X Poisson (5) 0.20 0.15 0.10 0.05 0.0 0 5 10 15 20
Argue, based on the central limit theorem, that a Poisson random variable having mean λ will approximately have a normal distribution with mean and variance both equal to λ when λ is large. If X is Poisson with mean 100, compute the exact probability that X is less than or equal to 116 and
A club basketball team will play a 60-game season. Thirty-two of these games are against class A teams and 28 are against class B teams. The outcomes of all the games are independent. The team will win each game against a class A opponent with probability .5, and it will win each game against a
Each computer chip made in a certain plant will, independently, be defective with probability .25. If a sample of 1,000 chips is tested, what is the approximate probability that fewer than 200 chips will be defective?
If X is binomial with parameters n=150, p=.6, compute the exact value of P{X ≤ 80} and compare with its normal approximation both (a) making use of and (b) not making use of the continuity correction.
An instructor knows from past experience that student exam scores have mean 77 and standard deviation 15. At present the instructor is teaching two separate classes — one of size 25 and the other of size 64.(a) Approximate the probability that the average test score in the class of size 25 lies
The lifetime (in hours) of a type of electric bulb has expected value 500 and standard deviation 80. Approximate the probability that the sample mean of n such bulbs is greater than 525 when(a) n = 4;(b) n = 16;(c) n = 36;(d) n = 64.
A tobacco company claims that the amount of nicotine in its cigarettes is a random variable with mean 2.2 mg and standard deviation .3 mg. However, the sample mean nicotine content of 100 randomly chosen cigarettes was 3.1 mg. What is the approximate probability that the sample mean would have been
The lifetime of a certain electrical part is a random variable with mean 100 hours and standard deviation 20 hours. If 16 such parts are tested, find the probability that the sample mean is(a) less than 104;(b) between 98 and 104 hours.
The amount of time that a certain type of battery functions is a random variable with mean 5 weeks and standard deviation 1.5 weeks. Upon failure, it is immediately replaced by a new battery. Approximate the probability that 13 or more batteries will be needed in a year.
A six-sided die, in which each side is equally likely to appear, is repeatedly rolled until the total of all rolls exceeds 400. Approximate the probability that this will require more than 140 rolls.
Fifty numbers are rounded off to the nearest integer and then summed. If the individual roundoff errors are uniformly distributed between −.5 and .5, what is the approximate probability that the resultant sum differs from the exact sum by more than 3?
A highway department has enough salt to handle a total of 80 inches of snowfall.Suppose the daily amount of snowhas a mean of 1.5 inches and a standard deviation of .3 inches.(a) Approximate the probability that the salt on hand will suffice for the next 50 days.(b) What assumption did you make in
A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you bet 1 on a specified number, you either win 35 if the roulette ball lands on that number or lose 1 if it does not. If you continually make such bets, approximate the probability that(a) you are winning after 34 bets;(b) you are
Approximate the probability that the sum of 16 independent uniform (0, 1) random variables exceeds 10.
If 10 fair dice are rolled, approximate the probability that the sum of the values obtained (which ranges from 20 to 120) is between 30 and 40 inclusive.
Plot the probability mass function of the sample mean of X1, . . . , Xn, when(a) n = 2;(a) n = 3.Assume that the probability mass function of the Xi is P{X = 0} = .2, P{X = 1} = .3, P{X = 3} = .5 In both cases, determine E[X ] and Var(X ).
According to the U.S. Department of Agriculture’sWorld Livestock Situation, the country with the greatest per capita consumption of pork is Denmark. In 1994, the amount of pork consumed by a person residing in Denmark had a mean value of 147 pounds with a standard deviation of 62 pounds. If a
Suppose that 45 percent of the population favors a certain candidate in an upcoming election. If a random sample of size 200 is chosen, find(a) the expected value and standard deviation of the number of members of the sample that favor the candidate;(b) the probability that more than half the
The time it takes a central processing unit to process a certain type of job is normally distributed with mean 20 seconds and standard deviation 3 seconds. If a sample of 15 such jobs is observed, what is the probability that the sample variance will exceed 12?
An astronomer wants to measure the distance from her observatory to a distant star.However, due to atmospheric disturbances, any measurement will not yield the exact distanced. As a result, the astronomer has decided to make a series of measurements and then use their average value as an estimate
The weights of a population of workers have mean 167 and standard deviation 27.(a) If a sample of 36 workers is chosen, approximate the probability that the sample mean of their weights lies between 163 and 170.(b) Repeat part (a) when the sample is of size 144.
The ideal size of a first-year class at a particular college is 150 students.The college, knowing from past experience that, on the average, only 30 percent of those accepted for admission will actually attend, uses a policy of approving the applications of 450 students. Compute the probability
Civil engineers believe that W, the amount of weight (in units of 1,000 pounds) that a certain span of a bridge can withstand without structural damage resulting, is normally distributed with mean 400 and standard deviation 40. Suppose that the weight (again, in units of 1,000 pounds) of a car is a
An insurance company has 25,000 automobile policy holders. If the yearly claim of a policy holder is a random variable with mean 320 and standard deviation 540, approximate the probability that the total yearly claim exceeds 8.3 million.
Let be the standard normal distribution function. If, for constants a and b >0characterize the distribution of X. x- b P{X x)=0 = 0 (x)
If Tn has a t -distribution with n degrees of freedom, show that T 2 n has an F -distribution with 1 and n degrees of freedom.
If T has a t-distribution with 8 degrees of freedom, find (a) P{T ≥ 1},(b) P{T ≤ 2}, and (c) P{−1 < T < 1}.
Show that (1/2) =√π (Hint: Evaluate∞0 e−xx−1/2 dx by letting x = y2/2, dx = y dy.)
If X and Y are independent chi-square random variables with 3 and 6 degrees of freedom, respectively, determine the probability that X + Y will exceed 10.
If X is a chi-square random variable with 6 degrees of freedom, find(a) P{X ≤ 6};(b) P{3 ≤ X ≤ 9}.
When shooting at a target in a two-dimensional plane, suppose that the horizontal miss distance is normally distributed with mean 0 and variance 4 and is independent of the vertical miss distance, which is also normally distributed with mean 0 and variance 4. Let D denote the distance between the
Earthquakes occur in a given region in accordance with a Poisson process with rate 5 per year.(a) What is the probability there will be at least two earthquakes in the first half of 2010?(b) Assuming that the event in part (a) occurs, what is the probability that there will be no earthquakes during
Let X1, X2, . . . , Xn denote the first n interarrival times of a Poisson process and set Sn = ni=1 Xi .(a) What is the interpretation of Sn?(b) Argue that the two events {Sn ≤ t } and {N(t ) ≥ n} are identical.(c) Use part (b) to show that(d) By differentiating the distribution function of Sn
Jones figures that the total number of thousands of miles that a used auto can be driven before it would need to be junked is an exponential random variable with parameter 1 20 . Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the
The number of years a radio functions is exponentially distributed with parameterλ = 18. If Jones buys a used radio, what is the probability that it will be working after an additional 10 years?
The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter λ = 1.(a) What is the probability that a repair time exceeds 2 hours?(b) What is the conditional probability that a repair takes at least 3 hours, given that its duration exceeds 2 hours?
An IQ test produces scores that are normally distributed with mean value 100 and standard deviation 14.2. The top 1 percent of all scores are in what range?
The height of adult women in the United States is normally distributed with mean 64.5 inches and standard deviation 2.4 inches. Find the probability that a randomly chosen woman is(a) less than 63 inches tall;(b) less than 70 inches tall;(c) between 63 and 70 inches tall.(d) Alice is 72 inches
The annual rainfall in Cincinnati is normally distributed with mean 40.14 inches and standard deviation 8.7 inches. (a) What is the probability this year’s rainfall will exceed 42 inches?(b) What is the probability that the sum of the next 2 years’ rainfall will exceed 84 inches?(c) What is the
The lifetime of a television picture tube is a normal random variable with mean 8.2 years and standard deviation 1.4 years. What percentage of such tubes lasts(a) more than 10 years;(b) less than 5 years;(c) between 5 and 10 years?
In Problem 31, what is the probability that a batch of 100 chips will contain at least 4 whose lifetimes are less than 3.8 × 106 hours?
The lifetimes of interactive computer chips produced by a certain semiconductor manufacturer are normally distributed having mean 4.4×106 hours with a standard deviation of 3 × 105 hours. If a mainframe manufacturer requires that at least 90 percent of the chips from a large batch will have
A random variable X is said to have a lognormal distribution if log X is normally distributed. IfXis lognormal withE[log X] = μandVar(log X ) = σ2, determine the distribution function of X. That is, what is P{X ≤ x}?
Let I =∞−∞ e−x2/2 dx.(a) Show that for any μ and σis equivalent to I = √2π.(b) Show that I = √2π by writingand then evaluating the double integral by means of a change of variables to polar coordinates. (That is, let x = r cos θ, y = r sin θ, dx dy = r dr dθ.) 8 1 -(x-1) 120 dx =
A manufacturer produces bolts that are specified to be between 1.19 and 1.21 inches in diameter. If its production process results in a bolt’s diameter being normally distributed with mean 1.20 inches and standard deviation .005, what percentage of bolts will not meet specifications?
A certain type of lightbulb has an output that is normally distributed with mean 2,000 end foot candles and standard deviation 85 end foot candles. Determine a lower specification limit L so that only 5 percent of the lightbulbs produced will be defective. (That is, determine L so that P{X ≥ L} =
The weekly demand for a product approximately has a normal distribution with mean 1,000 and standard deviation 200. The current on hand inventory is 2,200 and no deliveries will be occurring in the next two weeks. Assuming that the demands in different weeks are independent,(a) what is the
The annual rainfall (in inches) in a certain region is normally distributed withμ = 40, σ = 4. What is the probability that in 2 of the next 4 years the rainfall will exceed 50 inches? Assume that the rainfalls in different years are independent.
The Scholastic Aptitude Test mathematics test scores across the population of high school seniors follow a normal distribution with mean 500 and standard deviation 100. If five seniors are randomly chosen, find the probability that(a) all scored below 600 and (b) exactly three of them scored above
If X is a normal random variable with parameters μ = 10, σ2 = 36, compute(a) P{X > 5};(b) P{4 < X < 16};(c) P{X < 8};(d) P{X < 20};(e) P{X > 16}.
You arrive at a bus stop at 10 o’clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. What is the probability that you will have to wait longer than 10 minutes? If at 10:15 the bus has not yet arrived, what is the probability that you will have to wait
If U is uniformly distributed on (0, 1), show that a + (b − a)U is uniform on (a, b).
Independent trials, each of which is a success with probability p, are successively performed. Let X denote the first trial resulting in a success. That is, X will equal k if the first k −1 trials are all failures and the kth a success. X is called a geometric random variable. Compute(a) P{X =
Let X denote a hypergeometric random variable with parameters n, m, and k.That is,(a) Derive a formula for P{X = i} in terms of P{X = i − 1}.(b) Use part (a) to compute P{X = i} for i = 0, 1, 2, 3, 4, 5 when n = m = 10, k = 5, by starting with P{X = 0}.(c) Based on the recursion in part (a),
A contractor purchases a shipment of 100 transistors. It is his policy to test 10 of these transistors and to keep the shipment only if at least 9 of the 10 are in working condition. If the shipment contains 20 defective transistors, what is the probability it will be kept?
If X is a Poisson random variable with mean λ, show that P{X = i } first increases and then decreases as i increases, reaching its maximum value when i is the largest integer less than or equal to λ.
The probability of error in the transmission of a binary digit over a communication channel is 1/103. Write an expression for the exact probability of more than 3 errors when transmitting a block of 103 bits. What is its approximate value?Assume independence.
The game of frustration solitaire is played by turning the cards of a randomly shuffled deck of 52 playing cards over one at a time. Before you turn over the first card, say ace; before you turn over the second card, say two, before you turn over the third card, say three. Continue in this manner
Approximately 80,000 marriages took place in the state of New York last year.Estimate the probability that for at least one of these couples(a) both partners were born on April 30;(b) both partners celebrated their birthday on the same day of the year.State your assumptions.
In the 1980s, an average of 121.95 workers died on the job each week. Give estimates of the following quantities:(a) the proportion of weeks having 130 deaths or more;(b) the proportion of weeks having 100 deaths or less.Explain your reasoning.
The number of times that an individual contracts a cold in a given year is a Poisson random variable with parameter λ = 3. Suppose a new wonder drug (based on large quantities of vitamin C) has just been marketed that reduces the Poisson parameter to λ = 2 for 75 percent of the population. For
If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is 1 100 , what is the (approximate) probability that you will win a prize (a)at least once, (b) exactly once, and (c) at least twice?
Compare the Poisson approximation with the correct binomial probability for the following cases:(a) P{X = 2} when n = 10, p = .1;(b) P{X = 0} when n = 10, p = .1;(c) P{X = 4} when n = 9, p = .2.
Derive the moment generating function of a binomial random variable and then use your result to verify the formulas for the mean and variance given in the text.
If X is a binomial random variable with parameters n and p, where 0 (b) As k goes from 0 to n, P{X = k} first increases and then decreases, reaching its largest value when k is the largest integer less than or equal to (n + 1)p. (a) P{X=k+1} = P n-k 1-pk+1 P{X=k}, k = 0, 1,...,n-1.
If X and Y are binomial random variables with respective parameters (n, p) and(n, 1 − p), verify and explain the following identities:(a) P{X ≤ i} = P{Y ≥ n − i};(a) P{X = k} = P{Y = n − k}.
Let X be a binomial random variable withFind (a) P{X = 4};(b) P{X >12}. E[X] 7 and Var(X) = 2.1
At least one-half of an airplane’s engines are required to function in order for it to operate. If each engine independently functions with probability p, for what values of p is a 4-engine plane more likely to operate than a 2-engine plane?
Suppose that a particular trait (such as eye color or left-handedness) of a person is classified on the basis of one pair of genes, and suppose that d represents a dominant gene and r a recessive gene. Thus, a person with dd genes is pure dominance, one with rr is pure recessive, and one with rd is
If each voter is for Proposition A with probability .7, what is the probability that exactly 7 of 10 voters are for this proposition?
A communications channel transmits the digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability .2. Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit 00000 instead of 0 and
A satellite system consists of 4 components and can function adequately if at least 2 of the 4 components are in working condition. If each component is, independently, in working condition with probability .6, what is the probability that the system functions adequately?
When we attempt to locate a target in two-dimensional space, suppose that the coordinate errors are independent normal random variables with mean 0 and standard deviation 2. Find the probability that the distance between the point chosen and the target exceeds 3.
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