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.
> 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 206. In that regard, consider Review Exercise 3.73 on page 10
> In Exercise 6.54 on page 206, 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. (
> Consider the situation in Review Exercise 8.74. Suppose a considerable effort is conducted to “tighten” the variability in the system. Following the effort, a random sample of size 40 is taken from the new assembly line and the sample variance is s2 = 0.
> Consider the situation of Case Study 9.1 on page 281 with a larger sample of metal pieces. The diameters are as follows: 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 1.01, 1.03, 0.99, 1.00, 1.00, 0.99, 0.98, 1.01, 1.02, 0.99 centimeters. Once again the normality
> Suppose a filling machine is used to fill cartons with a liquid product. The specification that is strictly enforced for the filling machine is 9 ± 1.5 oz. If any carton is produced with weight outside these bounds, it is considered by the supplier to be
> In Chapter 9, the concept of parameter estimation will be discussed at length. Suppose X is a random variable with mean μ and variance σ2 = 1.0. Suppose also that a random sample of size n is to be taken and ¯x is to be used as an estimate of μ. When the
> Given a normal random variable X with mean 20 and variance 9, and a random sample of size n taken from the distribution, what sample size n is necessary in order that P(19.9 ≤ ¯X ≤ 20.1) = 0.95?
> From the information in Review Exercise 8.70, compute (assuming μB = 65%) P(¯XB ≥ 70).
> A random sample of employees from a local manufacturing plant pledged the following donations, in dollars, to the United Fund: 100, 40, 75, 15, 20, 100, 75, 50, 30, 10, 55, 75, 25, 50, 90, 80, 15, 25, 45, and 100. Calculate (a) the mean; (b) the mode.
> Two distinct solid fuel propellants, type A and type B, are being considered for a space program activity. Burning rates of the propellant are crucial. Random samples of 20 specimens of the two propellants are taken with sample means 20.5 cm/sec for prop
> Consider the situation of Review Exercise 8.62. If the population from which the sample was taken has population mean μ = 53, 000 kilometers, does the sample information here seem to support that claim? In your answer, compute and determi
> The breaking strength X of a certain rivet used in a machine engine has a mean 5000 psi and standard deviation 400 psi. A random sample of 36 rivets is taken. Consider the distribution of ¯X, the sample mean breaking strength. (a) What is the probability
> Consider Review Exercise 8.56. Comment on any outliers in the data.
> Consider the data in Exercise 9.13. Suppose the manufacturer of the shearing pins insists that the Rockwell hardness of the product be less than or equal to 44.0 only 5% of the time. What is your reaction? Use a tolerance limit calculation as the basis f
> Consider Example 1.5 on page 25. Comment on any outliers.
> If S21 and S22 represent the variances of independent random samples of size n1 = 25 and n2 = 31, taken from normal populations with variances σ21 = 10 and σ22 = 15, respectively, find P(S21/S22 > 1.26).
> Consider the data of Exercise 1.19 on page 31. Construct a box-and-whisker plot. Comment. Compute the sample mean and sample standard deviation. Exercise 1.19 The following data represent the length of life in years, measured to the nearest tenth, of 30
> A taxi company tests a random sample of 10 steel-belted radial tires of a certain brand and records the following tread wear: 48,000, 53,000, 45,000, 61,000, 59,000, 56,000, 63,000, 49,000, 53,000, and 54,000 kilometers. Use the results of Exercise 8.14
> If the number of hurricanes that hit a certain area of the eastern United States per year is a random variable having a Poisson distribution with μ = 6, find the probability that this area will be hit by (a) exactly 15 hurricanes in 2 years; (b) at most
> A random sample of 5 bank presidents indicated annual salaries of $395,000, $521,000, $483,000, $479,000, and $510,000. Find the variance of this set.
> Find the mean, median, and mode for the sample whose observations, 15, 7, 8, 95, 19, 12, 8, 22, and 14, represent the number of sick days claimed on 9 federal income tax returns. Which value appears to be the best measure of the center of these data? Sta
> If S21 and S22 represent the variances of independent random samples of size n1 = 8 and n2 = 12, taken from normal populations with equal variances, find P(S21/S22 < 4.89).
> In testing for carbon monoxide in a certain brand of cigarette, the data, in milligrams per cigarette, were coded by subtracting 12 from each observation. Use the results of Exercise 8.14 on page 231 to find the standard deviation for the carbon monoxide
> If X1,X2, . . . , Xn are independent random variables having identical exponential distributions with parameter θ, show that the density function of the random variable Y = X1+X2+· · ·+Xn is that of a gamma distribution with parameters α = n and β = θ.
> Consider the drying time measurements in Exercise 9.14. Suppose the 15 observations in the data set are supplemented by a 16th value of 6.9 hours. In the context of the original 15 observations, is the 16th value an outlier? Show work. Exercise 9.14: Th
> Consider the data displayed in Exercise 1.20 on page 31. Construct a box-and-whisker plot and comment on the nature of the sample. Compute the sample mean and sample standard deviation.
> Computer response time is an important application of the gamma and exponential distributions. Suppose that a study of a certain computer system reveals that the response time, in seconds, has an exponential distribution with a mean of 3 seconds. (a) Wha
> Construct a quantile plot of these data, which represent the lifetimes, in hours, of fifty 40-watt, 110- volt internally frosted incandescent lamps taken from forced life tests: 919 1196 785 1126 936 918 1156 920 948 1067 1092 1162 1170 929 950 905
> Consider the following measurements of the heat-producing capacity of the coal produced by two mines (in millions of calories per ton): Mine 1: 8260 8130 8350 8070 8340 Mine 2: 7950 7890 7900 8140 7920 7840 Can it be concluded that the two population v
> 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.
> 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.
> 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). Se-², x > 0, f(x) = 10, elsewhere.
> 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
