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modern mathematical statistics with applications

Engineering Statistics 2nd Edition Douglas C. Montgomery, George C. Runger, Norma F. Hubele - Solutions

The mean breaking strength of yarn used in manufacturing drapery material is required to be at least 100 psi. Past experience has indicated that the standard deviation of breaking strength is 2 psi. A random sample of nine specimens is tested, and the average breaking strength is found to be 100.6
Medical researchers have developed a new artificial heart constructed primarily of titanium and plastic. The heart will last and operate almost indefinitely once it is implanted in the patient’s body, but the battery pack needs to be recharged about every 4 hours. A random sample of 50 battery
A random sample has been taken from a normal population and two confidence intervals constructed using exactly the same data. The two CIs are (38.02, 61.98) and (39.95, 60.05).(a) What is the value of the sample mean?(b) One of these intervals is a 90% CI and the other is a 95% CI. Which one is the
For a normal population with known variance 2, answer the following questions:(a) What is the confidence level for the CI\overline{x}-2.14\sigma\sqrt{n}\le\mu\le\overline{x}+2.14\sigma\sqrt{n}?(b) What is the confidence level for the CI
Consider the Minitab output below.(a) Fill in the missing values in the output. Can the null hypothesis be rejected at the 0.05 level of significance?Explain your answer.(b) Suppose that the alternative hypothesis had been What is the P-value in this situation? Can the null hypothesis be rejected
Consider the Minitab output below.(a) Fill in the missing values in the output. Can the null hypothesis be rejected at the 0.05 level of significance?Explain your answer.(b) Suppose that the alternative hypothesis had been What is the P-value in this situation? Can the null hypothesis be rejected
Consider the Minitab output shown below.(a) Fill in the missing values in the output. Can the null hypothesis be rejected at the 0.05 level? Why?(b) Is this a one-sided or a two-sided test?(c) If the hypotheses had been versus would you reject the null hypothesis at the 0.05 level? Can you answer
Consider the Minitab output shown below.(a) Fill in the missing values in the output. What conclusion would you draw?(b) Is this a one-sided or a two-sided test?(c) Use the output and the normal table to find a 99% CI on the mean.(d) What is the P-value if the alternative hypothesis is H1: 7 30?
Suppose that we are testing Ηο: μ = μο versusΗ₁: μ < μο. Calculate the P-value for the following observed values of the test statistic:(a) zo = -2.15(b) zo = -1.80(c) zo = -2.50(d) zo = -1.60(e) zo = 0.35
Suppose that we are testing Ηο: μ = μο versusΗ₁: μ ≠ μο. Calculate the P-value for the following observed values of the test statistic:(a) zo = 2.45(b) zo = -1.53(c) zo = 2.15(d) zo = 1.95(e) zo = -0.25
Suppose that we are testing Ηο: μ=μο versusΗ₁: μ > μο. Calculate the P-value for the following observed values of the test statistic:(a) zo= 2.35(b) zo= 1.53(c) zo= 2.00(d) zo= 1.85(e) zo = -0.15
Consider Exercise 4-25, and suppose that the process engineer wants the type I error probability for the test to be 0.05. Where should the critical region be located?
Rework Exercise 4-25 when n 16 batches and the boundaries of the critical region do not change.
A manufacturer is interested in the output voltage of a power supply used in a PC. Output voltage is assumed to be normally distributed, with standard deviation 0.25 V, and the manufacturer wishes to test H0: 9 V against using n 10 units.(a) The critical region is Find the value of .(b) Find
Consider Exercise 4-21, and suppose that the sample size is increased to n 16.(a) Where would the boundary of the critical region be placed if the type I error probability is 0.05?(b) Using n 16 and the critical region found in part (a), find the type II error probability if the true mean foam
Repeat Exercise 4-21 assuming that the sample size is n 16.
In Exercise 4-21, suppose that the sample data result in(a) What conclusion would you reach in a fixed-level test with 0.05?(b) How “unusual” is the sample value mm if the true mean is 175 mm? That is, what is the probability that you would observe a sample average as large as 190 mm (or
A consumer products company is formulating a new shampoo and is interested in foam height (in mm). Foam height is approximately normally distributed and has a standard deviation of 20 mm. The company wishes to test mm versus mm, using the results of n 10 samples.(a) Find the P-value if the sample
Repeat Exercise 4-19 using a sample size of n5 and the same critical region.
The heat evolved in calories per gram of a cement mixture is approximately normally distributed. The mean is thought to be 100 and the standard deviation is 2. We wish to test H0: 100 versus with a sample of n 9 specimens.(a) If the rejection region is defined as or find the type I error
In Exercise 4-16 with n 16:(a) Find the boundary of the critical region if the type I error probability is specified to be 0.05.(b) Find for the case when the true mean elongation force is 13.0 kg.(c) What is the power of the test from part (b)?
In Exercise 4-15 with n 5:(a) Find the boundary of the critical region if the type I error probability is specified to be 0.01.(b) Find for the case when the true mean elongation force is 13.5 kg.(c) What is the power of the test?
Repeat Exercise 4-15 using a sample size of n 16 and the same critical region.
A textile fiber manufacturer is investigating a new drapery yarn, which has a standard deviation of 0.3 kg. The company wishes to test the hypothesis H0: 14 against using a random sample of five specimens.(a) What is the P-value if the sample average is(b) Find for the case where the true mean
Let X₁, X₂, ..., Xₙ be a random sample of size n.(a) Show that $$X̄²$$ is a biased estimator for μ².(b) Find the amount of bias in this estimator.(c) What happens to the bias as the sample size n increases?
(a) Show that $$Σᵢ₌₁ⁿ(Xᵢ - X̄)²/n$$ is a biased estimator of σ²(b) Find the amount of bias in the estimator.(c) What happens to the bias as the sample size n increases?
Let three random samples of sizes n₁ = 20, n₂ = 10, and n₃ = 8 be taken from a population with mean μ and variance σ². Let S₁², S₂², and S₃² be the sample variances. Show that$$S² = (20S₁² + 10S₂² + 8S₃²)/38$$ is an unbiased estimator of σ².
Suppose that θ̂₁, θ̂₂, and θ̂₃ are estimators of θ. We know that E(θ̂₁) = E(θ̂₂) = θ, E(θ̂₃) ≠ θ, V(θ̂₁) = 16, V(θ̂₂) = 11, and E(θ̂₃ - θ)² = 6. Compare these three estimators. Which do you prefer? Why?
Suppose that and are estimators of the parameter. We know that E(®ˆ 1) , E(®ˆ 2) 2, V(®ˆ 1) 10,'V(θ̂₁) = 4. Which estimator is "better"? In what sense is it better?
Calculate the relative efficiency of the two estimators in Exercise 4-7.
Calculate the relative efficiency of the two estimators in Exercise 4-6.
Suppose that and are unbiased estimators of the parameter . We know that and Which estimator is better, and in what sense is it better?
Let X1, X2, . . . , X9 denote a random sample from a population having mean and variance 2. Consider the following estimators of :a) Is either estimator unbiased?(b) Which estimator is “better”? In what sense is it better? +2 9 3X - X6 + 2X4 2
Suppose we have a random sample of size 2n from a population denoted by X, and E(X ) and V(X ) 2.Letbe two estimators of . Which is the better estimator of ?Explain your choice. X 2n 2n X and X n
The Minitab output for a random sample of data is shown below. Some of the quantities are missing. Compute the values of the missing quantities. Variable N Mean SE Mean Variance Sum X 15 ? ? ? Sum of Squares Minimum Maximum 2977.70 592589.64 181.90 212.62
The Minitab output for a random sample of data is shown below. Some of the quantities are missing. Compute the values of the missing quantities. Sum of Variable N Mean Variance Sum Squares Minimum Maximum X 10 ?? 109.891 1258.899 6.451 13.878
The Minitab output for a random sample of data is shown below. Some of the quantities are missing. Compute the values of the missing quantities Variable N Mean SE Mean StDev Sum X 16 ? 0.159 ? 399.851
The Minitab output for a random sample of data is shown below. Some of the quantities are missing. Compute the values of the missing quantities Variable N Mean SE Mean StDev Variance Min. Max. X 9 19.96 ? 3.12 ? 15.94 27.16
Consider the data on weekly waste (percent) as reported for five suppliers of the Levi-Strauss clothing plant in Albuquerque and reported on the Web site http://lib.stat.cmu.edu/DASL/Stories/wasterunup.html.Test each of the data sets for conformance to a normal probability model using a normal
Computer software can be used to simulate data from a normal distribution. Use a package such as Minitab to simulate dimensions for parts A, B, and C in Fig. 3-42 of Exercise 3-181.(a) Simulate 500 assemblies from simulated data for parts A, B, and C and calculate the length of gap D for each.(b)
Using the data set that you found or collected in the first team exercise of Chapter 2, or another data set of interest, answer the following questions:(a) Is a continuous or discrete distribution model more appropriate to model your data? Explain.(b) You have studied the normal, exponential,
Suppose that time to prepare a bed at a hospital is modeled with an exponential distribution with 3 bedshour.Determine the following:(a) Probability that a bed is prepared in less than 10 minutes(b) Probability that the time to prepare a bed is more than 30 minutes(c) Probability that each of 10
Let X denote the number of major cracks in a mile of roadway with the following probabilities: P(X = 0) = 0.4, P(X = 1)=0.1, P(X = 2) = 0.1P(X> 2) = 0.4. Determine the following probabilities: (a) P(X 1) (b) At least one crack (c) Two or more cracks (d) More than zero but less than three cracks
Given the pdf f(x) = exp(-x) for 0 x, determine the following: (a) P(X < 1) (b) P(X > 2) (c) P(1 X < 2) (d) x such that P(X < x) = 0.95
Given the pdf f(x) = x for 0 x 3, determine the following: (a) P(X 1) (b) P(X = 2) (c) x such that P(X < x) = 0.95 (d) E(X) (e) V(X)
An article in Knee Surgery, Sports Traumatology, Arthroscopy, “Effect of Provider Volume on Resource Utilization for Surgical Procedures” (Vol. 13, 2005, pp. 273–279), showed a mean time of 129 minutes and a standard deviation of 14 minutes for ACL reconstruction surgery at high-volume
The lifetime of a mechanical assembly in a vibration test is exponentially distributed with a mean of 400 hours.(a) What is the probability that an assembly on test fails in less than 100 hours?(b) What is the probability that an assembly operates for more than 500 hours before failure?(c) If an
Arthroscopic meniscal repair was successful 70% of the time for tears greater than 25 millimeters (in 50 surgeries)and 76% of the time for shorter tears (in 100 surgeries).(a) Describe the random variable used in these probability statements.(b) Is the random variable continuous or discrete?(c)
Cholesterol is a fatty substance that is an important part of the outer lining (membrane) of cells in the body of animals. Suppose that the mean and standard deviation for a population of individuals are 180 mg/dl and 20 mg/dl, respectively.Samples are obtained from 25 individuals, and these are
Nonuniqueness of Probability Models. It is possible to fit more than one model to a set of data. Consider the life data given in Exercise 3-247.(a) Transform the data using logarithms; that is, let y* (new value) log y (old value). Perform a normal probability plot on the transformed data and
Consider the following data that represent the life of packaged magnetic disks exposed to corrosive gases (in hours):4, 86, 335, 746, 80, 1510, 195, 562, 137, 1574, 7600, 4394, 4, 98, 1196, 15, 934, 11(a) Perform a Weibull probability plot and determine the adequacy of the fit.(b) Using the
Consider the following data, which represent the life of roller bearings (in hours).7203 3917 7476 5410 7891 10,033 4484 12,539 2933 16,710 10,702 16,122 13,295 12,653 5610 6466 5263 2,504 9098 7759(a) Perform a Weibull probability plot and determine the adequacy of the fit.(b) Using the estimated
Consider the following data, which represent the number of hours of operation of a surveillance camera until failure:246,785 183,424 1060 23,310 921 35,659 127,015 10,649 13,859 53,731 10,763 1456 189,880 2414 21,414 411,884 29,644 1473(a) Perform a normal probability plot and comment on the
To illustrate the effect of a log transformation, consider the following data, which represent cycles to failure for a yarn product:675, 3650, 175, 1150, 290, 2000, 100, 375(a) Using a normal probability plot, comment on the adequacy of the fit.(b) Transform the data using logarithms, that is, let
Show that the gamma density function integrates to 1.
Consider the following system made up of functional components in parallel and series. The probability that each component functions is shown in Fig. 3-46.(a) What is the probability that the system operates?(b) What is the probability that the system fails due to the components in series? Assume
A cartridge company develops ink cartridges for a printer company and supplies both the ink and the cartridge.The following is the probability mass function of the number of cartridges over the life of the printer.x 5 6 7 8 9 f (x) 0.04 0.19 0.61 0.13 0.03(a) What is the expected number of
A keyboard for a personal computer is known to have a mean life of 5 years. The life of the keyboard can be modeled using an exponential distribution.(a) What is the probability that a keyboard will have a life between 2 and 4 years?(b) What is the probability that the keyboard will still function
Manufacturers need to determine that each medical linear accelerator works within proper parameters before shipping to hospitals. An individual machine is known to have a probability of failure during initial testing of 0.10. Eight accelerators are tested.(a) What is the probability that at most
Overheating is a major problem in microprocessor operation. After much testing, it has been determined that the operating temperature is normally distributed with a mean of 150 degrees and a standard deviation of 7 degrees. The processor will malfunction at 165 degrees.(a) What is the probability
The research and development team of a medical device manufacturer is designing a new diagnostic test strip to detect the breath alcohol level. The materials used to make the device are listed here together with their mean and standard deviation of their thickness.The materials are stacked as shown
The weights of bags filled by a machine are normally distributed with standard deviation 0.05 kilogram and mean that can be set by the operator. At what level should the mean be set if it is required that only 1% of the bags weigh less than 10 kilograms?
The thickness of glass sheets produced by a certain process are normally distributed with a mean of 3.00 millimeters and a standard deviation of 0.12 millimeters. What is the value of c for which there is a 99% probability that a glass sheet has a thickness within the interval [3.00c, 3.00
The weight of a certain type of brick has an expectation of 1.12 kilograms with a variance of 0.0009 kilogram. How many bricks would need to be selected so that their average weight has a standard deviation of no more than 0.005 kilogram?
Continuation of Exercise 2-2.(a) Plot the data on normal probability paper. Do these data appear to have a normal distribution?(b) Remove the largest observation from the data set. Does this improve the fit of the normal distribution to the data?
Continuation of Exercise 2-1.(a) Plot the data on normal probability paper. Does concentration appear to have a normal distribution?(b) Suppose it has been determined that the largest observation, 68.7, was suspected to be an outlier. Consequently, it can be removed from the data set. Does this
Continuation of Exercise 3-82. Recall that it was determined that a normal distribution adequately fit the dimensional measurements for parts from two different machines.Using this distribution, suppose that s1 2.28, and s27.58 are used to estimate the population parameters. Estimate the
Continuation of Exercise 3-81. Recall that it was determined that a normal distribution adequately fit the internal pressure strength data. Use this distribution and suppose that the sample mean of 206.04 and standard deviation of 11.57 are used to estimate the population parameters. Estimate the
A process is said to be of six-sigma quality if the process mean is at least six standard deviations from the nearest specification. Assume a normally distributed measurement.(a) If a process mean is centered between the upper and lower specifications at a distance of six standard deviations from
A bearing assembly contains 10 bearings. The bearing diameters are assumed to be independent and normally distributed with a mean of 1.5 millimeters and a standard deviation of 0.025 millimeter. What is the probability that the maximum diameter bearing in the assembly exceeds 1.6 millimeters?
A mechanical assembly used in an automobile engine contains four major components. The weights of the components are independent and normally distributed with the following means and standard deviations (in ounces).(a) What is the probability that the weight of an assembly exceeds 29.5 ounces?(b)
The time for an automated system in a warehouse to locate a part is normally distributed with mean 45 seconds and standard deviation 30 seconds. Suppose that independent requests are made for 10 parts.(a) What is the probability that the average time to locate 10 parts exceeds 60 seconds?(b) What
A disk drive assembly consists of one hard disk and spacers on each side, as shown in Fig. 3-45. The height of the top spacer, W, is normally distributed with mean 120 millimeters and standard deviation 0.5 millimeter; the height of the disk, X, is normally distributed with mean 20 millimeters and
The weight of adobe bricks for construction is normally distributed with a mean of 3 pounds and a standard de-viation of 0.25 pound. Assume that the weights of the bricks are independent and that a random sample of 25 bricks is chosen. What is the probability that the mean weight of the sample is
From contractual commitments and extensive past laboratory testing, we know that compressive strength measurements are normally distributed with the true mean compressive strength 5500 psi and standard deviation 100 psi. A random sample of structural elements is tested for compressive strength
A random sample of 36 observations has been drawn. Find the probability that the sample mean is in the interval 47 X 53 for each of the following population distributions and population parameter values.(a) Normal with mean 50 and standard deviation 12(b) Exponential with mean 50(c) Poisson
The lifetimes of six major components in a copier are independent exponential random variables with means of 8000, 10,000, 10,000, 20,000, 20,000, and 25,000 hours, respectively.(a) What is the probability that the lifetimes of all the components exceed 5000 hours?(b) What is the probability that
Polyelectrolytes are typically used to separate oil and water in industrial applications. The separation process is dependent on controlling the pH. Fifteen pH readings of wastewater following these processes were recorded. Is it reasonable to model these data using a normal distribution?6.2 6.5
Continuation of Exercise 3-218. Let Y be the random variable defined as the number of pages between errors.(a) What is the distribution of Y? What is the mean of Y?(b) What is the probability that there are fewer than 100 pages between errors?(c) What is the probability that there are no errors in
The number of errors in a textbook follows a Poisson distribution with mean of 0.01 error per page.(a) What is the probability that there are three or fewer errors in 100 pages?(b) What is the probability that there are four or more errors in 100 pages?(c) What is the probability that there are
Continuation of Exercise 3-216. Let Y be the random variable defined as the time between messages arriving to the computer bulletin board.(a) What is the distribution of Y? What is the mean of Y?(b) What is the probability that the time between messages exceeds 15 minutes?(c) What is the
The number of messages sent to a computer Web site is a Poisson random variable with a mean of 5 messages per hour.(a) What is the probability that 5 messages are received in 1 hour?(b) What is the probability that 10 messages are received in 1.5 hours?(c) What is the probability that fewer than 2
The probability that a call to an emergency help line is answered in less than 15 seconds is 0.85. Assume that all calls are independent.(a) What is the probability that exactly 7 of 10 calls are answered within 15 seconds?(b) What is the probability that at least 16 of 20 calls are answered in
The average life of a certain type of compressor is 10 years with a standard deviation of 1 year. The manufacturer replaces free all compressors that fail while under guarantee.The manufacturer is willing to replace 3% of all compressors sold. For how many years should the guarantee be in
A standard fluorescent tube has a life length that is normally distributed with a mean of 7000 hours and a standard deviation of 1000 hours. A competitor has developed a compact fluorescent lighting system that will fit into incandescent sockets.It claims that a new compact tube has a normally
A driveshaft will suffer fatigue failure with a mean time-to-failure of 40,000 hours of use. If it is known that the probability of failure before 36,000 hours is 0.04 and that the distribution governing time-to-failure is a normal distribution, what is the standard deviation of the time-to-failure
The random variable X has the following probability distribution.x 2 3 5 8 Probability 0.2 0.4 0.3 0.1 Determine the following.(a) P(X 3) (b) P(X 2.5)(c) P(2.7 X 5.1) (d) E(X )(e) V(X)
Suppose that f (x) e x2 for 0 x and f (x) 0 for x 0.(a) Determine x such that P(x X) 0.20.(b) Determine x such that P(X x) 0.75.
Suppose that f (x) e x for 0 x and f (x) 0 for x 0. Determine the following probabilities.(a) P(X 1.5) (b) P(X 1.5)(c) P(1.5 X 3) (d) P(X 3)(e) P(X 3)
The mean and standard deviation of the lifetime of a battery in a portable computer are 3.5 and 1.0 hours, respectively.(a) Approximate the probability that the mean lifetime of 25 batteries exceeds 3.25 hours.(b) Approximate the probability that the mean lifetime of 100 batteries exceeds 3.25
Suppose that the time to prepare a bed at a hospital is modeled with a random variable with a mean of 20 minutes and a variance of 16 minutes. Approximate the probabilities of the following events:(a) Mean time to prepare 100 beds is less than 21 minutes.(b) Total time to prepare 100 beds is less
A random sample of n 9 structural elements is tested for compressive strength. We know that the true mean compressive strength 5500 psi and the standard deviation is 100 psi. Find the probability that the sample mean compressive strength exceeds 4985 psi.
The viscosity of a fluid can be measured in an experiment by dropping a small ball into a calibrated tube containing the fluid and observing the random variable X, the time it takes for the ball to drop the measured distance.Assume that X is normally distributed with a mean of 20 seconds and a
Suppose that X has the following discrete distribution f (x) e 13, x 1, 2, 3 0, otherwise A random sample of n 36 is selected from this population.Approximate the probability that the sample mean is greater than 2.1 but less than 2.5.
The amount of time that a customer spends waiting at an airport check-in counter is a random variable with mean 8.2 minutes and standard deviation 1.5 minutes. Suppose that a random sample of n 49 customers is observed. Find the probability that the average time waiting in line for these
The compressive strength of concrete has a mean of 2500 psi and a standard deviation of 50 psi. Find the probability that a random sample of n 5 specimens will have a sample mean strength that falls in the interval from 2490 psi to 2510 psi.
A synthetic fiber used in manufacturing carpet has tensile strength that is normally distributed with mean 75.5 psi and standard deviation 3.5 psi. Find the probability that a random sample of n 6 fiber specimens will have sample mean tensile strength that exceeds 75.75 psi.
The time to complete a manual task in a manufacturing operation is considered a normally distributed random variable with mean of 0.50 minute and a standard deviation of 0.05 minute. Find the probability that the average time to complete the manual task, after 49 repetitions, is less than 0.465
The photoresist thickness in semiconductor manufacturing has a mean of 10 micrometers and a standard deviation of 1 micrometer. Assume that the thickness is normally distributed and that the thicknesses of different wafers are independent.(a) Determine the probability that the average thickness of
Intravenous fluid bags are filled by an automated filling machine. Assume that the fill volumes of the bags are independent, normal random variables with a standard deviation of 0.08 fluid ounce.(a) What is the standard deviation of the average fill volume of 20 bags?(b) If the mean fill volume of

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