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inferential statistics
Probability And Statistics For Engineers And Scientists 9th Global Edition Ronald E. Walpole, Raymond Myers, Sharon L. Myers, Keying E. Ye - Solutions
8.42 The scores on a placement test given to college freshmen for the past five years are approximately normally distributed with a mean μ = 74 and a varianceσ2 = 8. Would you still consider σ2 = 8 to be a valid value of the variance if a random sample of 20 students who take the placement test
8.41 Assuming that the sample variances are continuous measurements, find the probability that a random sample of 30 observations, from a normal population with variance σ2 = 5, will have a sample variance of S2 that is(a) greater than 7.338;(b) between 2.766 and 7.883.
8.40 For a chi-square distribution, find χ2α such that(a) P(χ2 < χ2α) = 0.95, when v = 7;(b) P(4.601 > χ2 > χ2α) = 0.01, when v = 15;(c) P(χ2α < X2 < 32.852) = 0.95, when v = 19.
8.39 For a chi-square distribution, find χ2α such that(a) P(χ2 > χ2α) = 0.95, when v = 5;(b) P(χ2 > χ2α) = 0.99, when v = 13;(c) P(19.68 < χ2 < χ2α) = 0.04, when v = 11.
8.38 For a chi-square distribution, find(a) χ20.025, when v = 12;(b) χ20.01, when v = 14;(c) χ20.95, when v = 8.
8.37 For a chi-square distribution, find(a) χ20.005, when v = 10;(b) χ20.05, when v = 6;(c) χ20.01, when v = 16.
8.36 Let X1,X2, . . . , Xn be a random sample from a distribution that can take on only positive values. Use the Central Limit Theorem to produce an argument that if n is sufficiently large, then Y = X1X2 · · ·Xn has approximately a lognormal distribution.
8.35 Consider the situation described in Example 8.4 on page 254. Do these results prompt you to question the premise that μ = 800 hours? Give a probabilistic result that indicates how rare an event ¯X ≤ 775 is when μ = 800. On the other hand, how rare would it be if μ truly were, say, 760
8.34 Two alloys A and B are being used to manufacture a certain steel product. An experiment needs to be designed to compare the two in terms of maximum load capacity in tons (the maximum weight that can be tolerated without breaking). It is known that the two standard deviations in load capacity
8.33 The chemical benzene is highly toxic to humans.However, it is used in the manufacture of many medicine dyes, leather, and coverings. Government regulations dictate that for any production process involving benzene, the water in the output of the process must not exceed 7950 parts per million
8.32 Two different box-filling machines are used to fill cereal boxes on an assembly line. The critical measurement influenced by these machines is the weight of the product in the boxes. Engineers are quite certain that the variance of the weight of product is σ2 = 1 ounce.Experiments are
8.31 Consider Case Study 8.2 on page 258. Suppose 18 specimens were used for each type of paint in an experiment and ¯xA − ¯xB , the actual difference in mean drying time, turned out to be 1.0.(a) Does this seem to be a reasonable result if the two population mean drying times truly are
8.30 The mean score of the students from training centers for a particular competitive examination is 148, with a standard deviation of 24. Assuming that the means can be measured to any degree of accuracy, what is the probability that two groups selected at random, consisting of 42 and 64
8.29 The distribution of the weights of one-week-old chicks of a certain breed has a mean of 120 grams and a standard deviation of 12 grams. The distribution of weights of chicks of the same age of another breed is 92 grams with a standard deviation of 8 grams. Assuming that the sample mean can be
8.28 A sample of 30 people is randomly selected from city A, where the average height of adults is 160 centimeters with a standard deviation of 9 centimeters.A second random sample of size 42 is selected form city B, where the average height of adults is 158 centimeters with a standard deviation of
8.27 In a chemical process, the amount of a certain type of impurity in the output is difficult to control and is thus a random variable. Speculation is that the population mean amount of the impurity is 0.20 gram per gram of output. It is known that the standard deviation is 0.1 gram per gram. An
8.26 The amount of time that a vehicle spends in a petrol bunk is a random variable with the mean μ = 4.5 minutes and a standard deviation σ = 1.8 minutes. If a random sample of 24 vehicles is observed, find the probability that its mean time at the petrol bunk is(a) at most 3.6 minutes;(b) more
8.25 The average time taken to complete a project in a real estate company is 18 months, with a standard deviation of 3 months. Assuming that the project completion time approximately follows a normal distribution, find(a) the probability that the mean completion time of 4 such projects falls
8.24 If a certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard deviation of 2 ohms, what is the probability that a random sample of 36 of these resistors will have a combined resistance of more than 1458 ohms?
8.23 The random variable X, representing the number of cherries in a cherry puff, has the following probability distribution:x 4 5 6 7 P(X = x) 0.2 0.4 0.3 0.1(a) Find the mean μ and the variance σ2 of X.(b) Find the mean μX¯ and the variance σ2 X¯ of the mean¯X for random samples of 36
8.22 The heights of 1000 students are approximately normally distributed with a mean of 174.5 centimeters and a standard deviation of 6.9 centimeters. Suppose 200 random samples of size 25 are drawn from this population and the means recorded to the nearest tenth of a centimeter. Determine(a) the
8.21 A soft-drink machine is regulated so that the amount of drink dispensed averages 240 milliliters with a standard deviation of 15 milliliters. Periodically, the machine is checked by taking a sample of 40 drinks and computing the average content. If the mean of the 40 drinks is a value within
8.20 Given the probability mass function of the results of 72 independents tosses of an unbiased die f(x) =16, x= 1, 2, 3, 4, 5, and 6 0, elsewhere find the probability that it will yield a sample mean greater than 3.6 but less than 3.9. Assume the means are measured to the nearest tenth.
8.19 A certain type of automobile battery has a mean life of 1500 days and standard deviation of 250 days.How does the variance of the sample mean change when the sample size is(a) increased from 49 to 225?(b) decreased from 625 to 36?
8.18 If the standard deviation of the mean for the sampling distribution of random samples of size 49, from large or infinite population, is 3, how large must the sample size become if the standard deviation is to be reduced to 1.5?
8.17 If all possible samples of size 25 are drawn from a normal population with a mean of 60 and standard deviation of 9, what is the probability that a sample mean, ¯X , will fall in the interval between μX¯−0.96σ ¯ X and μX¯ +0.25σX¯ ? Assume that the sample means can be measured to
8.16 In the 2014–15 cricket season, the captain of a university cricket team scored the following runs in 12 different one-day matches: 84, 25, 74, 53, 40, 31, 64, 71, 18, 63, 88, and 49. Find(a) the mean runs;(b) the median runs.
8.15 Calculate the variance of the sample, 8, 10, 16, 18, 24, and 25. Use this answer, along with the results of Exercise 8.14, to find(a) the variance of the sample 16, 20, 32, 36, 48 and 50;(b) the variance of the sample 12, 14, 20, 22, 28, and 29.
8.14 (a) Show that the sample variance is unchanged if a constant c is added to or subtracted from each value in the sample.(b) Show that the sample variance becomes c2 times its original value if each observation in the sample is multiplied by c.
8.13 The heights, in meters, of 20 randomly selected college seniors are as follows:1.53 1.48 1.64 1.70 1.56 1.79 1.69 1.74, 1.79, 1.73 1.43 1.68 1.64 1.75 1.76 1.59 1.68 1.74, 1.73, 1.71 Calculate the standard deviation.
8.12 The average contents of saturated fat in eight bars of a certain brand of low-fat cereal selected at random are measured as follows: 0.65, 0.72, 0.45, 0.55, 0.58, 0.39, 0.68, and 0.52 grams. Calculate(a) the mean;(b) the variance.
8.11 For the data of Exercise 8.5, calculate the variance using the formula(a) of form (8.2.1);(b) in Theorem 8.1.
8.10 For the sample of reaction times in Exercise 8.3, calculate(a) the range;(b) the variance, using the formula of form (8.2.1).
8.9 Consider the data in Exercise 8.2, find(a) the range;(b) the standard deviation.
8.8 According to ecology writer Jacqueline Killeen, phosphates contained in household detergents pass right through our sewer systems, causing lakes to turn into swamps that eventually dry up into deserts. The following data show the amount of phosphates per load of laundry, in grams, for a random
8.7 A random sample of students from a city school scored the following marks for a paper in the annual examination: 45, 57, 68, 34, 50, 32, 89, 47, 97, 67, 79, 84, 43, 35, 68, 55, 72, 63, 68, and 49. Calculate(a) the mean;(b) the mode.
8.6 Find the mean, median, and mode for a sample whose observations, 5, 12, 3, 7, 11, 1, 85, 5, 9, and 5, represent the number of medical leaves claimed by 10 employees of a company in a year. Which of the values appears to be the best measure of the center of the data? State reasons for your
8.5 The numbers of incorrect answers on a true-false competency test for a random sample of 15 students were recorded as follows: 2, 1, 3, 0, 1, 3, 6, 0, 3, 3, 5, 2, 1, 4, and 2. Find(a) the mean;(b) the median;(c) the mode.
8.4 The number of tickets issued for traffic violations by 8 state troopers during the Memorial Day weekend are 5, 4, 7, 7, 6, 3, 8, and 6.(a) If these values represent the number of tickets issued by a random sample of 8 state troopers from Montgomery County in Virginia, define a suitable
8.3 The reaction times for a random sample of 9 subjects to a stimulant were recorded as 2.5, 3.6, 3.1, 4.3, 2.9. 2.3, 2.6, 4.1, and 3.4 seconds. Calculate(a) the mean;(b) the median.
8.2 The lengths of time, in minutes, that 10 patients waited in a doctor’s office before receiving treatment were recorded as follows: 5, 11, 9, 5, 10, 15, 6, 10, 5, and 10. Treating the data as a random sample, find(a) the mean;(b) the median;(c) the mode.
8.1 Define suitable populations from which the following samples are selected:(a) A car manufacturing company calls customers for feedback after servicing their vehicles at the company’s authorized service center.(b) Four out of 10 randomly selected college students are girls.(c) The total marks
7.24 By expanding etx in a Maclaurin series and integrating term by term, show that MX (t) =∞−∞etx f(x) dx= 1+μt + μ2 t2 2!+ · · · + μr tr r!+ · · · .
7.23 Both X and Y independently follow a geometric distribution with a probability mass function of P(X = r) = P(Y = r) = qr p, r = 0, 1, 2, . . ., where, p and q are positive numbers such that p + q = 1. Find(a) the probability distribution of U = X + Y ;(b) the conditional distribution of X/(X +
7.22 Using the moment-generating function of Exercise 7.21, show that the mean and variance of the chisquared distribution with v degrees of freedom are, respectively, v and 2v.
7.21 Show that the moment-generating function of the random variable X having a chi-squared distribution with v degrees of freedom is MX (t) = (1 − 2t)−v /2 .
7.20 The moment-generating function of a certain Poisson random variable X is given by MX (t) = e9(e t −1) .Find P(μ − σ ≤ X ≤ μ + σ).
7.19 A random variable X has the Poisson distribution p(x; μ) = e−μ μx /x! for x = 0, 1, 2, . . . . Show that the moment-generating function of X is MX (t) = eμ(e t −1) .Using MX (t), find the mean and variance of the Poisson distribution.
7.18 A random variable X has the geometric distribution g(x; p) = pqx−1 for x = 1, 2, 3, . . . . Show that the moment-generating function of X is MX (t) = pet 1 − qet, t
7.17 A random variable X has the discrete uniform distribution f(x; k) =1 k, x= 1, 2, . . . , k, 0, elsewhere.Show that the moment-generating function of X is MX (t) = et (1 − ek t )k(1 − et ) .
7.16 Show that the rth moment about the origin of the gamma distribution isμr = βr Γ(α + r)Γ(α) .[Hint: Substitute y = x/β in the integral defining μr and then use the gamma function to evaluate the integral.]
7.15 Let X have the probability distribution f(x) =2(x+1)9 , −1 < x < 2, 0, elsewhere.Find the probability distribution of the random variable Y = X2 .
7.14 Let X be a random variable with probability distribution f(x) =1+x 2 , −1 < x < 1, 0, elsewhere.Find the probability distribution of the random variable Y = X2 .
7.13 A current of I amperes flowing through a resistance of R ohms varies according to the probability distribution f(i) =6i(1 − i), 0 < i < 1, 0, elsewhere.If the resistance varies independently of the current according to the probability distribution g(r) =2r, 0 < r < 1, 0, elsewhere, find
7.12 Let X1 and X2 be independent random variables each having the probability distribution f(x) =e−x, x>0, 0, elsewhere.Show that the random variables Y1 and Y2 are independent when Y1 = X1 + X2 and Y2 = X1 /(X1 + X2 ).
7.11 The amount of kerosene, in thousands of liters, in a tank at the beginning of any day is a random amount Y from which a random amount X is sold during that day. Assume that the joint density function of these variables is given by f(x, y) =2, 0 < x < y, 0 < y < 1, 0, elsewhere.Find the
7.10 The random variables X and Y , representing the weights of creams and toffees, respectively, in 1-kilogram boxes of chocolates containing a mixture of creams, toffees, and cordials, have the joint density function f(x, y) =24xy, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, x + y ≤ 1, 0, elsewhere.(a) Find
7.9 The hospital period, in days, for patients following treatment for a certain type of kidney disorder is a random variable Y = X + 4, where X has the density function f(x) =32(x+4)3, x>0, 0, elsewhere.(a) Find the probability density function of the random variable Y .(b) Using the density
7.8 A dealer’s profit, in units of $5000, on a new automobile is given by Y = X2, where X is a random variable having the density function f(x) =2(1 − x), 0 < x < 1, 0, elsewhere.(a) Find the probability density function of the random variable Y .(b) Using the density function of Y , find the
7.7 The speed of a molecule in a uniform gas at equilibrium is a random variable V whose probability distribution is given by f(v) =kv2 e−bv 2, v>0, 0, elsewhere, where k is an appropriate constant and b depends on the absolute temperature and mass of the molecule.Find the probability
7.6 Given the random variable X with probability distribution f(x) =e−x , 0 < x < ∞0, elsewhere, find the probability distribution function of Y =−3X + 5.
7.5 If the variable X has the probability distribution of f(x) =1, 0 < x < 1 0 elsewhere, Show that the random variable Y = −2 loge X has an exponential distribution with λ =1 2.
7.4 Let X and Y be two discrete random variables with a joint probability distribution of f(x, y) =(x+2y )27 , x,y= 0, 1, 2.0, otherwise.Find the probability distribution of the random variable Z = X + Y .
7.3 Let X1 and X2 be discrete random variables with the joint multinomial distribution f(x1, x2 )=2 x1, x2 , 2 − x1 − x21 4x1 1 3x2 5 122−x1−x2 for x1 = 0, 1, 2; x2 = 0, 1, 2; x1 + x2 ≤ 2; and zero elsewhere. Find the joint probability distribution of Y1 = X1 + X2 and Y2 = X1 −
7.2 Let X be a binomial random variable with probability distribution f(x) =5Cx 13x 233−x, x= 0, 1, 2, . . . , 5 0, elsewhere.Find the probability distribution of the random variable Y = X3 + 2.
7.1 The probability distribution of the number X when an unbiased die is tossed is f(x) =16, x= 1, 2, 3, 4, 5, 6 0, elsewhere.Find the probability distribution of the random variable Y = 3X + 2.
6.87 Group Project: Have groups of students observe the number of people who enter a specific coffee shop or fast food restaurant over the course of an hour, beginning at the same time every day, for two weeks.The hour should be a time of peak traffic at the shop or restaurant. The data collected
6.86 The length of time, in seconds, that a computer user takes to read his or her e-mail is distributed as a lognormal random variable with μ = 1.8 and σ2 = 4.0.(a) What is the probability that a user reads e-mail for more than 20 seconds? More than a minute?(b) What is the probability that a
6.85 From the relationship between the chi-squared random variable and the gamma random variable, prove that the mean of the chi-squared random variable is v and the variance is 2v.
6.84 Explain why the nature of the scenario in Review Exercise 6.82 would likely not lend itself to the exponential distribution.
6.83 Derive the cdf for the Weibull distribution.[Hint: In the definition of a cdf, make the transformation z = yβ .]
6.82 The length of life, in hours, of a drill bit in a mechanical operation has a Weibull distribution withα = 2 and β = 50. Find the probability that the bit will fail before 10 hours of usage.
6.81 The length of time between breakdowns of an essential piece of equipment is important in the decision of the use of auxiliary equipment. An engineer thinks that the best model for time between breakdowns of a generator is the exponential distribution with a mean of 15 days.(a) If the generator
6.80 In a human factor experimental project, it has been determined that the reaction time of a pilot to a visual stimulus is normally distributed with a mean of 1/2 second and standard deviation of 2/5 second.(a) What is the probability that a reaction from the pilot takes more than 0.3 second?(b)
6.79 Consider Review Exercise 6.78. Given the assumption of the exponential distribution, what is the mean number of calls per hour? What is the variance in the number of calls per hour?
6.78 Consider now Review Exercise 3.74 on page 128.The density function of the time Z in minutes between calls to an electrical supply store is given by f(z) = 1 10 e−z /10 , 0 < z < ∞, 0, elsewhere.(a) What is the mean time between calls?(b) What is the variance in the time between calls?(c)
6.77 The beta distribution has considerable application in reliability problems in which the basic random variable is a proportion, as in the practical scenario illustrated in Exercise 6.50 on page 226. In that regard, consider Review Exercise 3.73 on page 128. Impurities in batches of product of a
6.76 In Exercise 6.54 on page 226, the lifetime of a transistor is assumed to have a gamma distribution with mean 10 weeks and standard deviation√50 weeks.Suppose that the gamma distribution assumption is incorrect.Assume that the distribution is normal.(a) What is the probability that a
6.75 For Review Exercise 6.74, what is the mean of the average water usage per hour in thousands of gallons?
6.74 The average rate of water usage (thousands of gallons per hour) by a certain community is known to involve the lognormal distribution with parametersμ = 5 and σ = 2. It is important for planning purposes to get a sense of periods of high usage. What is the probability that, for any given
6.73 For Review Exercise 6.72, what are the mean and variance of the time that elapses before 2 failures occur?
6.72 Consider the information in Review Exercise 6.66. What is the probability that less than 200 hours will elapse before 2 failures occur?
6.71 A technician plans to test a certain type of resin developed in the laboratory to determine the nature of the time required before bonding takes place. It is known that the mean time to bonding is 3 hours and the standard deviation is 0.5 hour. It will be considered an undesirable product if
6.70 A controlled satellite is known to have an error(distance from target) that is normally distributed with mean zero and standard deviation 4 feet. The manufacturer of the satellite defines a success as a firing in which the satellite comes within 10 feet of the target.Compute the probability
6.69 The time taken by all the contestants in a food festival to prepare a dish is normally distributed, with a mean of 25 minutes and standard deviation of 7 minutes.Find the probability that(a) it takes more than 30 minutes to prepare the item;(b) the item is prepared within 20 minutes;(c) the
6.68 The average consultation time for a patient in an ENT department of a hospital is 15 minutes.(a) Assuming that the consultation times for the patients are independent and exponential, what is the expected consultation time for two random patients?(b) What is the probability that the total
6.67 In a chemical processing plant, it is important that the yield of a certain type of batch product stay above 80%. If it stays below 80% for an extended period of time, the company loses money. Occasional defective batches are of little concern. But if several batches per day are defective, the
6.66 A certain type of device has an advertised failure rate of 0.01 per hour. The failure rate is constant and the exponential distribution applies.(a) What is the mean time to failure?(b) What is the probability that 200 hours will pass before a failure is observed?
6.65 A survey on gender status reveals that 56% of the families have more males than females. Assuming that this percentage is still valid, what is the probability that among 1000 randomly selected families, the male members exceed the female in between 570 and 650 (both included) of the families?
6.64 A manufacturer of a certain type of large machine wishes to buy rivets from one of two manufacturers.It is important that the breaking strength of each rivet exceed 10,000 psi. Two manufacturers (A and B) offer this type of rivet and both have rivets whose breaking strength is normally
6.63 When α is a positive integer n, the gamma distribution is also known as the Erlang distribution.Setting α = n in the gamma distribution on page 215, the Erlang distribution is f(x) =xn −1 e−x / ββ n (n−1)! , x>0, 0, elsewhere.It can be shown that if the times between successive
6.62 The exponential distribution is frequently applied to the waiting times between successes in a Poisson process. If the number of calls received per hour by a telephone answering service is a Poisson random variable with parameter λ = 6, we know that the time, in hours, between successive
6.61 A non-governmental agency working on traffic awareness conducted a survey in the city last year. It found that 42% of the population had a basic knowledge of the traffic rules. What is the probability that among any 500 randomly selected individuals of the city, between 220 and 250 (both
6.60 Show that the failure-rate function is given by Z(t) = αβtβ−1, t>0, if and only if the time to failure distribution is the Weibull distribution f(t) = αβtβ−1 e−αtβ, t>0.
6.59 Consider the information in Exercise 6.58.(a) What is the probability that more than 1 minute elapses between arrivals?(b) What is the mean number of minutes that elapse between arrivals?
6.58 The number of automobiles that arrive at a certain intersection per minute has a Poisson distribution with a mean of 5. Interest centers around the time that elapses before 10 automobiles appear at the intersection.(a) What is the probability that more than 10 automobiles appear at the
6.57 For Exercise 6.56, what is the mean power usage(average dB per hour)? What is the variance?
6.56 Rate data often follow a lognormal distribution.Average power usage (dB per hour) for a particular company is studied and is known to have a lognormal distribution with parameters μ = 4 and σ = 2. What is the probability that the company uses more than 270 dB during any particular hour?
6.55 According to a telephone operator, the average time for each call is 3.2 minutes. This time follows an exponential distribution.(a) What is the probability that the call time exceeds 5 minutes?(b) What is the probability that call completes in 2 minutes?
6.54 The lifetime, in weeks, of a certain type of transistor is known to follow a gamma distribution with mean 10 weeks and standard deviation√50 weeks.(a) What is the probability that a transistor of this type will last at most 50 weeks?(b) What is the probability that a transistor of this type
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