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Miller And Freunds Probability And Statistics For Engineers 9th Edition Richard A. Johnson - Solutions

7.27 Suppose that we observe a random variable having the binomial distribution. Let X be the number of successes in n trials.a) Show that X n is an unbiased estimate of the binomial parameter p. (b) Show that X + 1 n + 2 is not an unbiased estimate of the binomial parameter p.7.28 The statistical
7.26 Instead of the large sample confidence interval formula for μ on page 230, we could have given the alternative formula x − zα/3 · σ √n
7.25 Modify the formula for E on page 216 so that it applies to large samples which constitute substantial portions of finite populations, and use the resulting formula for the following problems: (a) A sample of 50 scores on the admission test for a school of engineering is drawn at random from
7.24 In an air-pollution study performed at an experiment station, the following amount of suspended benzenesoluble organic matter (in micrograms per cubic meter) was obtained for eight different samples of air: 2.2 1.8 3.1 2.0 2.4 2.0 2.1 1.2 Assuming that the population sampled is normal,
7.23 Refer to Example 1 and the data on the resiliency modulus of recycled concrete. (a) Obtain a 95% confidence interval for the population mean resiliency modulus μ. (b) Is the population mean contained in your interval in part (a)? Explain. (c) What did you assume about the population in your
7.22 A café records that in n = 81 cases, the coffee beans for the coffee machine lasted an average of 225 cups with a standard deviation of 22 cups. (a) Obtain a 90% confidence interval for μ, the population mean number of cups before the coffee machine needs to be refilled with beans. (b) Does
7.21 The freshness of produce at a mega-store is rated a scale of 1 to 5, with 5 being very fresh. From a random sample of 36 customers, the average score was 3.5 with a standard deviation of 0.8. (a) Obtain a 90% confidence interval for the population mean, μ, or the mean score for all
7.20 Ten bearings made by a certain process have a mean diameter of 0.5060 cm and a standard deviation of 0.0040 cm. Assuming that the data may be looked upon as a random sample from a normal population, construct a 95% confidence interval for the actual average diameter of bearings made by this
7.19 With reference to the thickness measurements in Exercise 2.41, page 47, obtain a 95% confidence interval for the mean thickness.
7.18 Refer to the data on page 50, on the number of defects per board for Product B. Obtain a 95% confidence interval for the population mean number of defects per board.
7.17 Refer to the 2×4 lumber strength data in Exercise 2.58, page 48. According to the computer output, a sample of n = 30 specimens had x = 1908.8 and s = 327.1. Find a 95% confidence interval for the population mean strength.
7.16 Refer to Exercise 2.34, page 46, concerning the number for board failures for n = 32 integrated circuits (IC). A computer calculates x = 7.6563 and s = 5.2216. Obtain a 95% confidence interval for the mean IC board failures.
7.15 With reference to the previous exercise, assume that the key performance indicator has a normal distribution and obtain a 95% confidence interval for the true value of the indicator.
7.14 To monitor complex chemical processes, chemical engineers will consider key process indicators, which may be just yield but most often depend on several quantities. Before trying to improve a process, n = 9 measurements were made on a key performance indicator. 123 106 114 128 113 109 120 102
7.13 With reference to the previous exercise, assume that production has a normal distribution and obtain a 99% confidence interval for the true mean production of the pilot process.
7.12 An effective way to tap rubber is to cut a panel in the rubber tree’s bark in vertical spirals. In a pilot process, an engineer measures the output of latex from such cuts. Eight cuts on different trees produced latex (in liters) in a week 26.8 32.5 29.7 24.6 31.5 39.8 26.5 19.9 What can the
7.11 The dean of a college wants to use the mean of a random sample to estimate the average amount of time students take to get from one class to the next, and she wants to be able to assert with 99% confidence that the error is at most 0.25 minute. If it can be presumed from experience that σ =
7.10 Refer to Example 8. How large a sample will we need in order to assert with probability 0.95 that the sample mean will not differ from the true mean by more than 1.5. (replacing σ by s is reasonable here because the estimate is based on a sample of size eighteen.)
7.9 In a study of automobile collision insurance costs, a random sample of 80 body repair costs for a particular kind of damage had a mean of $472.36 and a standard deviation of $62.35. If x = $472.36 is used as a point estimate of the true average repair cost of this kind of damage, with what
7.8 With reference to the previous exercise, construct a 95% confidence interval for the true mean interrequest time.
7.7 With reference to the n = 50 interrequest time observations in Example 6, Chapter 2, which have mean 11,795 and standard deviation 14,056, what can one assert with 95% confidence about the maximum error if x = 11,795 is used as a point estimate of the true population mean interrequest time?
7.6 With reference to the previous exercise, construct a 98% confidence interval for the true population mean number of unremovable defects per display.
7.5 The manufacture of large liquid crystal displays (LCD’s) is difficult. Some defects are minor and can be removed; others are unremovable. The number of unremovable defects, for each of n = 45 displays (Courtesy of Shiyu Zhou) 105 307600468 509 108603200 0 6 0 10 0 6 0 0 1 0 0 0 015 105002 has
7.4 With reference to the previous exercise, construct a 95% confidence interval for the true population mean labor time.
7.3 An industrial engineer collected data on the labor time required to produce an order of automobile mufflers using a heavy stamping machine. The data on times (hours) for n = 52 orders of different parts 2.15 2.27 0.99 0.63 2.45 1.30 2.63 2.20 0.99 1.00 1.05 3.44 0.49 0.93 2.52 1.05 1.39 1.22
7.2 With reference to the previous exercise, construct a 90% confidence interval for the true population mean quantity of gravel in concrete mixes.
7.1 A construction engineer collected data from some construction sites on the quantity of gravel (in metric tons) used in mixing concrete. The quantity of gravel for n = 24 sites 4861 5158 8642 2896 7654 9891 8381 6215 1116 7918 2313 8114 3517 8852 5712 4312 8023 1215 3598 2429 8211 4613 9168
6.64 Several pickers are each asked to gather 30 ripe apples and put them in a bag. (a) Would you expect all of the bags to weigh the same? For one bag, let X1 be the weight of the first apple, X2 the weight of the second apple, and so on. Relate the weight of this bag, 30 i=1 Xi to the
6.63 Explain why the following may not lead to random samples from the desired population: (a) To determine the mix of animals in a forest, a forest officer records the animals observed after each interval of 2 minutes. (b) To determine the quality of print, an observer observes the quality of
6.62 A traffic engineer collects data on traffic flow at a busy intersection during the rush hour by recording the number of westbound cars that are waiting for a green light. The observations are made for each light change. Explain why this sampling technique will not lead to a random sample.
6.61 When we sample from an infinite population, what happens to the standard error of the mean if the sample size is (a) increased from 100 to 200; (b) increased from 200 to 300; (c) decreased from 360 to 90?
6.60 If 2 independent samples of sizes n1 = 26 and n2 = 8 are taken from a normal population, what is the probability that the variance of the second sample will be at least 2.4 times the variance of the first sample?
6.59 If 2 independent random samples of size n1 = 31 and n2 = 11 are taken from a normal population, what is the probability that the variance of the first sample will be at least 2.7 times as large as the variance of the second sample?
6.58 Adding graphite to iron can improve its ductile qualities. If measurements of the diameter of graphite spheres within an iron matrix can be modeled as a normal distribution having standard deviation 0.16, what is the probability that the mean of a sample of size 36 will differ from the
0.63?
6.57 If measurements of the elasticity of a fabric yarn can be looked upon as a sample from a normal population having a standard deviation of 1.8, what is the probability that the mean of a random sample of size 26 will be less elastic by
6.56 The number of pieces of mail that a department receives each day can be modeled by a distribution having mean 44 and standard deviation 8. For a random sample of 35 days, what can be said about the probability that the sample mean will be less than 40 or greater than 48 using (a)
6.55 The time to check out and process payment information at an office supplies Web site can be modeled as a random variable with mean μ = 63 seconds and variance σ2 = 81. If the sample mean X will be based on a random sample of n = 36 times, what can we assert about the probability of getting
6.54 Referring to Exercise 6.52, find the value of the finite population correction factor in the formula for σ2 X for part (a) and part (b).
6.53 With reference to Exercise 6.52, what is the probability of choosing each sample in part (a) and the probability of choosing each sample in part (b), if the samples are to be random?
6.52 How many different samples of size n = 2 can be chosen from a finite population of size (a) N = 12; (b) N = 20?
6.51 The panel for a national science fair wishes to select 10 states from which a student representative will be chosen at random from the students participating in the state science fair. (a) Use Table 7W or software to select the 10 states. (b) Does the total selection process give each student
6.50 Use the discrete convolution formula, Theorem 6.10, to obtain the probability distribution of X + Y when X and Y are independent and each has the uniform distribution on {0, 1, 2}.
6.49 Use the transformation method, Theorem 6.9, to obtain the distribution of the ratio Y/X when when X and Y are independent and each has the same gamma distribution.
6.48 Use the convolution formula, Theorem 6.9, to obtain the density of X + Y when X and Y are independent and each has the exponential distribution with β = 1.
6.47 Use the transformation method to obtain the distribution of − ln ( X ) when X has the uniform distribution on (0, 1).
6.46 Use the transformation method to obtain the density of X3 when X has density f ( x ) = 1.5 X for 0 < x < 4.
6.45 Use the distribution function method to obtain the density of ln (X) when X has the exponential distribution with β = 1.
6.44 Use the distribution function method to obtain the density of 1 − e−X when X has the exponential distribution with β = 1.
6.43 Use the distribution function method to obtain the density of Z3 when Z has a standard normal distribution.
6.42 Referring to Example 16, verify that g( y ) = 1 √2 π y−1/2 e−y/2
6.41 Refer to Exercise 6.40. Let X1, X2,..., Xn be n independent random variables each having a negative binomial distribution with success probability p but where Xi has parameter ri. (a) Show that the mgf MXi (t) = E( et(X1+X2+···+Xr) ) of the sum Xi is [pet / (1 − (1 − p) et )] n i =
6.40 Let X1, X2,..., Xr be r independent random variables each having the same geometric distribution. (a) Show that the moment generating function MXi (t) = E( et(X1+X2+···+Xr) ) of the sum is [pet /(1 − (1 − p) et )]r (b) Relate the sum to the total number of trials to obtain r
6.39 Refer to Exercise 6.38. (a) Show that 7X1 + X2 − 2X3 + 7 has a normal distribution. (b) Find the mean and variance of the random variable in part (a).
6.38 Let X1, X2, and X3 be independent normal variables with E(X1) = −4 and σ2 1 = 1 E(X2) = 0 and σ2 2 = 4 E(X3) = 3 and σ2 3 = 1(a) Show that 2 X1 − X2 + 5X3 has a normal distribution. (b) Find the mean and variance of the random variable in part (a).
6.37 Refer to Exercise 6.36. (a) Show that 2X1 − X2 − 4X3 − 12 has a normal distribution. (b) Find the mean and variance of the random variable in part (a).
6.36 Let X1, X2, and X3 be independent normal variables with E(X1) = 5 and σ2 1 = 9 E(X2) = −2 and σ2 2 = 2.25 E(X3) = 5 and σ2 3 = 4 (a) Show that 2X1 + 2 X2 + 5X3 has a normal distribution. (b) Find the mean and the variance of the random variable in part (a).
6.35 Let X1, X2,..., X5 be 5 independent random variables. Find the moment generating function MXi (t) = E( et(X1+X2+···+X5 ) ) of the sum when Xi has a gamma distribution with αi = 2 i and βi = 2.
6.34 Let X1, X2,..., X8 be 8 independent random variables. Find the moment generating function MXi (t) = E( et(X1+X2+···+X8 ) ) of the sum when Xi has a Poisson distribution with mean (a) λi = 0.5 (b) λi = 0.04
6.33 Let the chi square variables χ2 1 , with ν1 degrees of freedom, and χ2 2 , with ν2 degrees of freedom, be independent. Establish the result on page 211, that their sum is a chi square variable with ν1 + ν2 degrees of freedom.
6.32 Let Z1,..., Z7 be independent and let each have a standard normal distribution. (a) Specify the distribution of Z2 1 + Z2 2 + Z2 3 + Z2 4 . (b) Specify the distribution of Z2 5 + Z2 6 + Z2 7 . (c) Specify the distribution of the sum of variables in part (a) and part (b).
6.31 Let Z1,..., Z6 be independent and let each have a standard normal distribution. Specify the distribution of Z1 − Z2 Z2 3 + Z2 4 + Z2 5 + Z2 6 8
6.30 Let Z1,..., Z5 be independent and let each have a standard normal distribution. (a) Specify the distribution of Z2 2 + Z2 3 + Z2 4 + Z2 5 . (b) Specify the distribution of Z1 Z2 2 + Z2 3 + Z2 4 + Z2 5 4
6.29 The F distribution with 4 and 4 degrees of freedom is given by f (F) = 6F(1 + F) −4 F > 0 0 F ≤ 0 If random samples of size 5 are taken from two normal populations having the same variance, find the probability that the ratio of the larger to the smaller sample variance will exceed 3.
6.28 The t distribution with 1 degree of freedom is given by f (t) = 1 π (1 + t 2) −1 − ∞ < t < ∞ Verify the value given for t0.05 for ν = 1 in Table 4.
6.27 The chi square distribution with 4 degrees of freedom is given by f (x) = ⎧ ⎨ ⎩ 1 4 · x · e−x/2 x > 0 0 x ≤ 0 Find the probability that the variance of a random sample of size 5 from a normal population with σ = 15 will exceed 180.
6.26 Find the values of (a) F0.95 for 15 and 12 degrees of freedom; (b) F0.99 for 5 and 20 degrees of freedom.
6.25 If independent random samples of size n1 = n2 = 8 come from normal populations having the same variance, what is the probability that either sample variance will be at least 7 times as large as the other?
6.24 A random sample of 15 observations is taken from a normal population having variance σ2 = 90.25. Find the approximate probability of obtaining a sample standard deviation between 7.25 and 10.75.
6.23 Engine bearings depend on a film of oil to keep shaft and bearing surfaces separated. Samples are regularly taken from production lines and each bearing in a sample is tested to measure the thickness of the oil film. After many samples, it is concluded that the population is normal. The
6.22 The process of making concrete in a mixer is under control if the rotations per minute of the mixer has a mean of 22 rpm. What can we say about this process if a sample of 20 of these mixers has a mean rpm of 22.75 rpm and a standard deviation of 3 rpm?
6.21 The following is the time taken (in hours) for the delivery of 8 parcels within a city: 28, 32, 20, 26, 42, 40, 28, and 30. Use these figures to judge the reasonableness of delivery services when they say it takes 30 hours on average to deliver a parcel within the city.
6.20 The tensile strength (1,000 psi) of a new composite can be modeled as a normal distribution. A random sample of size 25 specimens has mean x = 45.3 and standard deviation s = 7.9. Does this information tend to support or refute the claim that the mean of the population is 40.5?
6.19 Prove that μX = μ for random samples from discrete (finite or countably infinite) populations.
6.18 If X is a continuous random variable and Y = X − μ, show that σ2 Y = σ2 X .
6.17 If the distribution of scores of all students in an examination has amean of 296 and a standard deviation of 14, what is the probability that the combined gross score of 49 randomly selected students is less than 14,250?
6.16 A wire-bonding process is said to be in control if the mean pull strength is 10 pounds. It is known that the pull-strength measurements are normally distributed with a standard deviation of 1.5 pounds. Periodic random samples of size 4 are taken from this process and the process is said to be
and standard deviation 0.0125 ml. The sample mean of insufficient lubrication will be obtained from a random sample of 60 bearings. What is the probability that X will be between 0.600 ml and 0.640 ml?
6.15 Engine bearings depend on a film of oil to keep shaft and bearing surfaces separated. Insufficient lubrication causes bearings to be overloaded. The insufficient lubrication can be modeled as a random variable having mean 0.6520 ml
6.14 The mean of a random sample of size n = 25 is used to estimate the mean of an infinite population that has standard deviation σ = 2.4. What can we assert about the probability that the error will be less than 1.2, if we use (a) Chebyshev’s theorem; (b) the central limit theorem?
6.13 For large sample size n, verify that there is a 50-50 chance that the mean of a random sample from an infinite population with the standard deviation σ will differ from μ by less than 0.6745 · σ/√n. It has been the custom to refer to this quantity as the probable error of the mean.
6.12 What is the value of the finite population correction factor in the formula for σ2 X when (a) n = 8 and N = 640? (b) n = 100 and N = 8,000? (c) n = 250 and N = 20,000?
6.11 When we sample from an infinite population, what happens to the standard error of the mean if the sample size is (a) increased from 40 to 1,000? (b) decreased from 256 to 65? (c) increased from 225 to 1,225? (d) decreased from 450 to 18?
6.10 Suppose that we convert the 50 samples referred to on page 197 into 25 samples of size n = 20 by combining the first two, the next two, and so on. Find the means of these samples and calculate their mean and their standard deviation. Compare this mean and this standard deviation with the
6.9 Given an infinite population whose distribution is given by x f(x) 1 0.20 2 0.20 3 0.20 4 0.20 5 0.20 list the 25 possible samples of size 2 and use this list to construct the distribution of X for random samples of size 2 from the given population. Verify that the mean and the variance of this
6.8 Repeat Exercise 6.7, but select each sample with replacement; that is, replace each slip of paper and reshuffle before the next one is drawn.
6.7 Take 30 slips of paper and label five each −4 and 4, four each −3 and 3, three each −2 and 2, and two each −1, 0 and 1. (a) If each slip of paper has the same probability of being drawn, find the probability of getting −4, −3, −2, −1, 0, 1, 2, 3, 4 and find the mean and the
6.6 With reference to Exercise 6.5, what is the probability of each sample in part (a) and the probability of each sample in part (b) if the samples are to be random?
6.5 How many different samples of size n = 4 can be chosen from a finite population of size (a) N = 15? (b) N = 35?
6.4 A market research organization wants to try a new product in 8 of 50 states. Use Table 7W or software to make this selection.
6.3 Explain why the following will not lead to random samples from the desired populations. (a) To determine what the average person spends on a vacation, a market researcher interviews passengers on a luxury cruise. (b) To determine the average income of its graduates 10 years after graduation,
6.2 Large maps are printed on a plotter and rolled up. The supervisor randomly selects 12 printed maps and unfolds a part of each map to verify the quality of the printing. List one condition under which this method of sampling might not yield a random sample.
6.1 An inspector examines every twentieth piece coming off an assembly line. List some of the conditions under which this method of sampling might not yield a random sample.
5.128 Refer to the heights of pillars in the example on page 25. The variation in the population of heights of pillars can be modeled as a normal distribution with mean 306.6 nm and standard deviation 37.0 nm. (a) For a pillar selected at random, what is the probability that its height is greater
5.127 Refer to Example 7 concerning scanners. The maximum attenuation has a normal distribution with mean 10.1 dB and standard deviation 2.7 dB. (a) What proportion of the products has maximum attenuation less than 6 dB? (b) What proportion of the products has maximum attenuation between 6 dB and
5.126 Let X1, X2,..., X50 be independent and let each have the same marginal distribution with mean −5 and variance 8. Find (a) E ( X1 + X2 +···+ X50 ) ; (b) Var ( X1 + X2 +···+ X50 ) .
5.125 If X1 has mean −5 and variance 3 while X2 has mean 1 and variance 4, and the two are independent, find (a) E ( 3X1 + 5X2 + 2); (b) Var ( 3X1 + 5X2 + 2).
5.124 Two random variables are independent and each has a binomial distribution with success probability 0.6 and 2 trials. (a) Find the joint probability distribution. (b) Find the probability that the second random variable is greater than the first.
5.122 A software engineer models the crashes encountered when executing a new software as a random variable having the Weibull distribution with α = 0.06 and β = 6.0. What is the probability that the software crashes after 6 minutes? 5.123 Let the times to breakdown for the processors of a
5.121 If n salespeople are employed in a door-to-door selling campaign, the gross sales volume in thousands of dollars may be regarded as a random variable having the gamma distribution with α = 100√n and β = 1 2 . If the sales costs are $5,000 per salesperson, how many salespeople should be

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