Question: Let X1 and X2 be independent random
Let X1 and X2 be independent random variables each having the probability distribution
Show that the random variables Y1 and Y2 are independent when Y1 = X1 +X2 and Y2 = X1/(X1 +X2).
Transcribed Image Text:
Se-², x > 0, f(x) = 10, elsewhere.
> Pull-strength tests on 10 soldered leads for a semiconductor device yield the following results, in pounds of force required to rupture the bond: Another set of 8 leads was tested after encapsulation to determine whether the pull strength had been incr
> For an F-distribution, find (a) f0.05 with v1 = 7 and v2 = 15; (b) f0.05 with v1 = 15 and v2 = 7: (c) f0.01 with v1 = 24 and v2 = 19; (d) f0.95 with v1 = 19 and v2 = 24; (e) f0.99 with v1 = 28 and v2 = 12.
> A maker of a certain brand of low-fat cereal bars claims that the average saturated fat content is 0.5 gram. In a random sample of 8 cereal bars of this brand, the saturated fat content was 0.6, 0.7, 0.7, 0.3, 0.4, 0.5, 0.4, and 0.2. Would you agree with
> 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.
> A normal population with unknown variance has a mean of 20. Is one likely to obtain a random sample of size 9 from this population with a mean of 24 and a standard deviation of 4.1? If not, what conclusion would you draw?
> A manufacturing firm claims that the batteries used in their electronic games will last an average of 30 hours. To maintain this average, 16 batteries are tested each month. If the computed t-value falls between −t0.025 and t0.025, the firm is satisfied
> Refer to Exercise 9.22 again. Suppose that specifications by a buyer of the thread are that the tensile strength of the material must be at least 62 kilograms. The manufacturer is satisfied if at most 5% of the manufactured pieces have tensile strength l
> Given a random sample of size 24 from a normal distribution, find k such that (a) P(−2.069 < T < k) = 0.965; (b) P(k < T < 2.807) = 0.095; (c) P(−k < T < k) = 0.90.
> (a) Find P(−t0.005 < T < t0.01) for v = 20. (b) Find P(T > −t0.025).
> (a) Find P(T < 2.365) when v = 7. (b) Find P(T > 1.318) when v = 24. (c) Find P(−1.356 < T < 2.179) when v = 12. (d) Find P(T > −2.567) when v = 17.
> (a) Find t0.025 when v = 14. (b) Find −t0.10 when v = 10. (c) Find t0.995 when v = 7.
> Show that the variance of S2 for random samples of size n from a normal population decreases as n becomes large.
> 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 variance σ2 = 8. Would you still consider σ2 = 8 to be a valid value of the variance if a random sample of 20
> Assume the sample variances to be continuous measurements. Find the probability that a random sample of 25 observations, from a normal population with variance σ2 = 6, will have a sample variance S2 (a) greater than 9.1; (b) between 3.462 and 10.745.
> For a chi-squared distribution, find χ2α such that (a) P(X2 > χ2α) = 0.01 when v = 21; (b) P(X2 < χ2α) = 0.95 when v = 6; (c) P(χ2α < X2 < 23.209) = 0.015 when v = 10.
> 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 Coun
> For a chi-squared distribution, find χ2 α such that (a) P(X2 > χ2α) = 0.99 when v = 4; (b) P(X2 > χ2α) = 0.025 when v = 19; (c) P(37.652 < X2 < χ2α) = 0.045 when v = 25.
> Refer to Exercise 9.22. Why are the quantities requested in the exercise likely to be more important to the manufacturer of the thread than, say, a confidence interval on the mean tensile strength? Exercise 9.22: A type of thread is being studied for i
> For a chi-squared distribution, find (a) χ20.005 when v = 5; (b) χ20.05 when v = 19; (c) χ20.01 when v = 12.
> For a chi-squared distribution, find (a) χ20.025 when v = 15; (b) χ20.01 when v = 7; (c) χ20.05 when v = 24.
> 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.
> Consider the situation described in Example 8.4 on page 234. 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
> 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 tw
> 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
> 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
> Consider Case Study 8.2 on page 238. 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 8.5 Sampling Di
> The mean score for freshmen on an aptitude test at a certain college is 540, with a standard deviation of 50. Assume the means to be measured to any degree of accuracy. What is the probability that two groups selected at random, consisting of 32 and 50 s
> 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.
> A type of thread is being studied for its tensile strength properties. Fifty pieces were tested under similar conditions, and the results showed an average tensile strength of 78.3 kilograms and a standard deviation of 5.6 kilograms. Assuming a normal di
> The distribution of heights of a certain breed of terrier has a mean of 72 centimeters and a standard deviation of 10 centimeters, whereas the distribution of heights of a certain breed of poodle has a mean of 28 centimeters with a standard deviation of
> A random sample of size 25 is taken from a normal population having a mean of 80 and a standard deviation of 5. A second random sample of size 36 is taken from a different normal population having a mean of 75 and a standard deviation of 3. Find the prob
> 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 stan
> The amount of time that a drive-through bank teller spends on a customer is a random variable with a mean μ = 3.2 minutes and a standard deviation σ = 1.6 minutes. If a random sample of 64 customers is observed, find the probability that their mean time
> The average life of a bread-making machine is 7 years, with a standard deviation of 1 year. Assuming that the lives of these machines follow approximately a normal distribution, find (a) the probability that the mean life of a random sample of 9 such mac
> 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?
> The random variable X, representing the number of cherries in a cherry puff, has the following probability distribution: (a) Find the mean μ and the variance σ2 of X. b) Find the mean μ ¯X and the vari
> 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 t
> 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 m
> Given the discrete uniform population find the probability that a random sample of size 54, selected with replacement, will yield a sample mean greater than 4.1 but less than 4.4. Assume the means are measured to the nearest tenth. x = 2, 4, 6, 10,
> In a study conducted by the Department of Zoology at Virginia Tech, fifteen samples of water were collected from a certain station in the James River in order to gain some insight regarding the amount of orthophosphorus in the river. The concentration of
> 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 mod
> A certain type of thread is manufactured with a mean tensile strength of 78.3 kilograms and a standard deviation of 5.6 kilograms. How is the variance of the sample mean changed when the sample size is (a) increased from 64 to 196? (b) decreased from 784
> If the standard deviation of the mean for the sampling distribution of random samples of size 36 from a large or infinite population is 2, how large must the sample size become if the standard deviation is to be reduced to 1.2?
> If all possible samples of size 16 are drawn from a normal population with mean equal to 50 and standard deviation equal to 5, what is the probability that a sample mean ¯X will fall in the interval from μ ¯X −1.9σ ¯X to μ ¯X −0.4σ ¯X ? Assume that the s
> In the 2004-05 football season, University of Southern California had the following score differences for the 13 games it played. Find (a) the mean score difference; (b) the median score differences. 11 49 32 3 6 38 38 30 8 40 31 5 36
> Verify that the variance of the sample 4, 9, 3, 6, 4, and 7 is 5.1, and using this fact, along with the results of Exercise 8.14, find (a) the variance of the sample 12, 27, 9, 18, 12, and 21; (b) the variance of the sample 9, 14, 8, 11, 9, and 12.
> (a) Show that the sample variance is unchanged if a constant c is added to or subtracted from each 232 Chapter 8 Fundamental Sampling Distributions and Data Descriptions value in the sample. (b) Show that the sample variance becomes c2 times its original
> The grade-point averages of 20 college seniors selected at random from a graduating class are as follows: Calculate the standard deviation. 3.2 1.9 2.7 2.4 2.8 2.9 3.8 3.0 2.5 3.3 1.8 2.5 3.7 2.8 2.0 3.2 2.3 2.1 2.5 1.9
> The tar contents of 8 brands of cigarettes selected at random from the latest list released by the Federal Trade Commission are as follows: 7.3, 8.6, 10.4, 16.1, 12.2, 15.1, 14.5, and 9.3 milligrams. Calculate (a) the mean; (b) the variance.
> Consider the situation of Exercise 9.11. Estimation of the mean diameter, while important, is not nearly as important as trying to pin down the location of the majority of the distribution of diameters. Find the 95% tolerance limits that contain 95% of t
> An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If a sample of 30 bulbs has an average life of 780 hours, find a 96% confidence interval for the popu
> 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 Find the probability
> 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 (a) Find the probability density func
> Let X be a random variable with probability Find the probability distribution of the random variable Y = 2X − 1. T = 1, 2, 3, 10, elsewhere. f(x) =
> For the data of Exercise 8.5, calculate the variance using the formula (a) of form (8.2.1); (b) in Theorem 8.1. Exercise 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
> Given the normally distributed variable X with mean 18 and standard deviation 2.5, find (a) P(X
> Given a normal distribution with μ = 30 and σ = 6, find (a) the normal curve area to the right of x = 17; (b) the normal curve area to the left of x = 22; (c) the normal curve area between x = 32 and x = 41; (d) the value of x that has 80% of the normal
> Given a standard normal distribution, find the value of k such that (a) P(Z > k) = 0.2946; (b) P(Z < k) = 0.0427; (c) P(−0.93 < Z < k) = 0.7235.
> Find the value of z if the area under a standard normal curve (a) to the right of z is 0.3622; (b) to the left of z is 0.1131; (c) between 0 and z, with z > 0, is 0.4838; (d) between −z and z, with z > 0, is 0.9500.
> Given a standard normal distribution, find the area under the curve that lies (a) to the left of z = −1.39; (b) to the right of z = 1.96; (c) between z = −2.16 and z = −0.65; (d) to the left of z = 1.43; (e) to the right of z = −0.89; (f) between z = −0.
> A bus arrives every 10 minutes at a bus stop. It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution. (a) What is the probability that the individual waits more than 7 minutes? (b) What
> A random sample of 25 tablets of buffered aspirin contains, on average, 325.05 mg of aspirin per tablet, with a standard deviation of 0.5 mg. Find the 95% tolerance limits that will contain 90% of the tablet contents for this brand of buffered aspirin. A
> The daily amount of coffee, in liters, dispensed by a machine located in an airport lobby is a random variable X having a continuous uniform distribution with A = 7 and B = 10. Find the probability that on a given day the amount of coffee dispensed by th
> The IQs of 600 applicants to a certain college are approximately normally distributed with a mean of 115 and a standard deviation of 12. If the college requires an IQ of at least 95, how many of these students will be rejected on this basis of IQ, regard
> If a set of observations is normally distributed, what percent of these differ from the mean by (a) more than 1.3σ? (b) less than 0.52σ?
> The tensile strength of a certain metal component is normally distributed with a mean of 10,000 kilograms per square centimeter and a standard deviation of 100 kilograms per square centimeter. Measurements are recorded to the nearest 50 kilograms per squ
> 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). Exercise 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
> Define suitable populations from which the following samples are selected: (a) Persons in 200 homes in the city of Richmond are called on the phone and asked to name the candidate they favor for election to the school board. (b) A coin is tossed 100 time
> 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 (a) Find the probability density function of the random variable Y. (b) Using the dens
> 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 (a) Find the probability density function of the random variable Y. (b) Using the density function
> For Review Exercise 6.74, what is the mean of the average water usage per hour in thousands of gallons? Exercise 6.74: The average rate of water usage (thousands of gallons per hour) by a certain community is known to involve the lognormal distribution
> The average rate of water usage (thousands of gallons per hour) by a certain community is known to involve the lognormal distribution with parameters μ = 5 and σ = 2. It is important for planning purposes to get a sense of periods of high usage. What is
> Referring to Exercise 9.13, construct a 95% tolerance interval containing 90% of the measurements. Exercise 9.13 A random sample of 12 shearing pins is taken in a study of the Rockwell hardness of the pin head. Measurements on the Rockwell hardness are
> For Review Exercise 6.72, what are the mean and variance of the time that elapses before 2 failures occur? Exercise 6.72: Consider the information in Review Exercise 6.66. What is the probability that less than 200 hours will elapse before 2 failures oc
> Consider the information in Review Exercise 6.66. What is the probability that less than 200 hours will elapse before 2 failures occur? Exercise 6.66: A certain type of device has an advertised failure rate of 0.01 per hour. The failure rate is constant
> 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 b
> 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
> The speed of a molecule in a uniform gas at equilibrium is a random variable V whose probability distribution is given by where k is an appropriate constant and b depends on the absolute temperature and mass of the molecule. Find the probability distri
> The elongation of a steel bar under a particular load has been established to be normally distributed with a mean of 0.05 inch and σ = 0.01 inch. Find the probability that the elongation is (a) above 0.1 inch; (b) below 0.04 inch; (c) between 0.025 and 0
> For an electrical component with a failure rate of once every 5 hours, it is important to consider the time that it takes for 2 components to fail. (a) Assuming that the gamma distribution applies, what is the mean time that it takes for 2 components to
> 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 se
> 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 i
> According to a recent census, almost 65% of all households in the United States were composed of only one or two persons. Assuming that this percentage is still valid today, what is the probability that between 590 and 625, inclusive, of the next 1000 ra
> Consider Exercise 9.9. Compute a 95% prediction interval for the sugar content of the next single serving of Alpha-Bits. Exercise 9.9 Regular consumption of presweetened cereals contributes to tooth decay, heart disease, and other degenerative diseases,
> 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 wh
> 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 195, the Erlang distribution is It can be shown that if the times between succe
> 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
> According to a study published by a group of sociologists at the University of Massachusetts, approximately 49% of the Valium users in the state of Massachusetts are white-collar workers. What is the probability that between 482 and 510, inclusive, of th
> Show that the failure-rate function is given by Z(t) = αβ tβ−1, t>0, if and only if the time to failure distribution is the Weibull distribution f(t) = αβ tβ−1 e–αtβ, t>0.
> Given the random variable X with probability distribution find the probability distribution of Y = 8X3. S2.x, 0<x < 1, f(x) = 10, elsewhere,
> 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? Exercise 6.58: The number of automobiles that arrive at a certa
> 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
> For Exercise 6.56, what is the mean power usage (average dB per hour)? What is the variance? Exercise 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 log
> 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
> Consider Exercise 9.10. Compute the 95% prediction interval for the next observed number of words per minute typed by a graduate of the secretarial school. Exercise 9.10: A random sample of 12 graduates of a certain secretarial school typed an average o
> 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 pr
> In a biomedical research study, it was determined that the survival time, in weeks, of an animal subjected to a certain exposure of gamma radiation has a gamma distribution with α = 5 and β = 10. (a) What is the mean survival time of a randomly selected
> Derive the mean and variance of the Weibull distribution.
> The lives of a certain automobile seal have the Weibull distribution with failure rate Z(t) =1/√t. Find the probability that such a seal is still intact after 4 years.
> If the proportion of a brand of television set requiring service during the first year of operation is a random variable having a beta distribution with α = 3 and β = 2, what is the probability that at least 80% of the new models of this brand sold this
