Question: Pull-strength tests on 10 soldered leads

> A lawyer commutes daily from his suburban home to his midtown office. The average time for a one-way trip is 24 minutes, with a standard deviation of 3.8 minutes. Assume the distribution of trip times to be normally distributed. (a) What is the probabili

> Let us define Show that and hence S’2 is a biased estimator for σ2. * = Ë(X, - X)*/n. S'2 i=1 E(S") = [(n – 1)/n]o°, [(n – 1)/n]o², ||

> The finished inside diameter of a piston ring is normally distributed with a mean of 10 centimeters and a standard deviation of 0.03 centimeter. (a) What proportion of rings will have inside diameters exceeding 10.075 centimeters? (b) What is the probabi

> A research scientist reports that mice will live an average of 40 months when their diets are sharply restricted and then enriched with vitamins and proteins. Assuming that the lifetimes of such mice are normally distributed with a standard deviation of

> The loaves of rye bread distributed to local stores by a certain bakery have an average length of 30 centimeters and a standard deviation of 2 centimeters. Assuming that the lengths are normally distributed, what percentage of the loaves are (a) longer t

> Consider the data in Exercise 8.2, find (a) the range; (b) the standard deviation. Exercise 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, 1

> A soft-drink machine is regulated so that it discharges an average of 200 milliliters per cup. If the amount of drink is normally distributed with a standard deviation equal to 15 milliliters, (a) what fraction of the cups will contain more than 224 mill

> According to Chebyshev’s theorem, the probability that any random variable assumes a value within 3 standard deviations of the mean is at least 8/9. If it is known that the probability distribution of a random variable X is normal with mean μ and varianc

> Given a continuous uniform distribution, show that (a) μ =(A+B)/2 and (b) σ2 = ((B-A)2)/12

> 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 minut

> 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.

> Explain why the nature of the scenario in Review Exercise 6.82 would likely not lend itself to the exponential distribution. Exercise 6.82: The length of life, in hours, of a drill bit in a mechanical operation has a Weibull distribution with α = 2 and

> In Section 9.3, we emphasized the notion of “most efficient estimator” by comparing the variance of two unbiased estimators ˆΘ1 and ˆΘ2. However, this does not take into ac

> Derive the cdf for the Weibull distribution.

> The length of life, in hours, of a drill bit in a mechanical operation has a Weibull distribution with α = 2 and β = 50. Find the probability that the bit will fail before 10 hours of usage.

> 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 wit

> 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

> 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

> 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? Exercise 6.78: Consider now Review Exercise 3.74 on page 108. The densi

> Consider now Review Exercise 3.74 on page 108. The density function of the time Z in minutes between calls to an electrical supply store is given by (a) What is the mean time between calls? (b) What is the variance in the time between calls? (c) What i

> 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

> 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 &Icirc;&frac14; and the variance &Iuml;&#131;2 of X. b) Find the mean &Icirc;&frac14; &Acirc;&macr;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