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Applied Statistics And Probability For Engineers 6th Edition Douglas C. Montgomery, George C. Runger - Solutions

Construct a normal probability plot of the suspended solids concentration data in Exercise 6-24. Does it seem reasonable to assume that the concentration of suspended solids in water from this particular lake is normally distributed?
Construct two normal probability plots for the height data in Exercises 6-22 and 6-29. Plot the data for female and male students on the same axes. Does height seem to be normally distributed for either group of students? If both populations have the same variance, the two normal probability plots
It is possible to obtain a “quick and dirty” estimate of the mean of a normal distribution from the fiftieth percentile value on a normal probability plot. Provide an argument why this is so. It is also possible to obtain an estimate of the standard deviation of a normal distribution by
The concentration of a solution is measured six times by one operator using the same instrument. She obtains the following data: 63.2, 67.1, 65.8, 64.0, 65.1, and 65.3 (grams per liter).(a) Calculate the sample mean. Suppose that the desirable value for this solution has been specified to be 65.0
A sample of six resistors yielded the following resistances (ohms): and(a) Compute the sample variance and sample standard deviation.(b) Subtract 35 from each of the original resistance measurements and compute and s. Compare your results with those obtained in part (a) and explain your
Consider the following two samples:Sample 1: 10, 9, 8, 7, 8, 6, 10, 6Sample 2: 10, 6, 10, 6, 8, 10, 8, 6(a) Calculate the sample range for both samples. Would you conclude that both samples exhibit the same variability? Explain.(b) Calculate the sample standard deviations for both samples. Do these
An article in Quality Engineering (Vol. 4, 1992, pp. 487€“495) presents viscosity data from a batch chemical process. A sample of these data follows:(a) Reading down and left to right, draw a time series plot of all the data and comment on any features of the data that are revealed by
Reconsider the data from Exercise 6-76. Prepare comparative box plots for two groups of observations: the first 40 and the last 40. Comment on the information in the box plots.
The data shown in Table 6-7 are monthly champagne sales in France (1962-1969) in thousands of bottles.(a) Construct a time series plot of the data and comment on any features of the data that are revealed by this plot.(b) Speculate on how you would use a graphical procedure to forecast monthly
A manufacturer of coil springs is interested in implementing a quality control system to monitor his production process. As part of this quality system, it is decided to record the number of nonconforming coil springs in each production batch of size 50. During 40 days of production, 40 batches of
A communication channel is being monitored by recording the number of errors in a string of 1000 bits. Data for 20 of these strings follow: Read data across.(a) Construct a stem-and-leaf plot of the data.(b) Find the sample average and standard deviation.(c) Construct a time series plot of the
Reconsider the data in Exercise 6-76. Construct normal probability plots for two groups of the data: the first 40 and the last 40 observations. Construct both plots on the same axes. What tentative conclusions can you draw?
Construct a normal probability plot of the effluent discharge temperature data from Exercise 6-47. Based on the plot, what tentative conclusions can you draw?
Construct normal probability plots of the cold start ignition time data presented in Exercises 6-44 and 6-56. Construct a separate plot for each gasoline formulation, but arrange the plots on the same axes. What tentative conclusions can you draw?
For a normal population with known variance σ2, answer the following questions: (a) What is the confidence level for the interval x – 2.14σ/√n? (b) What is the confidence level for the interval x – 2.149σ/√n? (c) What is the confidence level for the interval x –
For a normal population with known variance σ2:
Consider the one-sided confidence interval expressions, Equations 8-9 and 8-10.(a) What value of za would result in a 90% CI?(b) What value of za would result in a 95% CI?(c) What value of za would result in a 99% CI?
A confidence interval estimate is desired for the gain in a circuit on a semiconductor device. Assume that gain is normally distributed with standard deviation σ = 20. (a) Find a 95% CI for μ when n = 10 and (b) Find a 95% CI for μ when n = 25 and (c) Find a 99% CI for μ when
Consider the gain estimation problem in Exercise 8-4. How large must n be if the length of the 95% CI is to be 40?
Following are two confidence interval estimates of the mean μ of the cycles to failure of an automotive door latch mechanism (the test was conducted at an elevated stress level to accelerate the failure). (a) What is the value of the sample mean cycles to failure? (b) The confidence level
n = 100 random samples of water from a fresh water lake were taken and the calcium concentration (milligrams per liter) measured. A 95% CI on the mean calcium concentration is 0.49 < μ < 0.82. (a) Would a 99% CI calculated from the same sample data been longer or shorter? (b) Consider the
The breaking strength of yarn used in manufacturing drapery material is required to be at least 100 psi. Past experience has indicated that breaking strength is normally distributed and that σ = 2 psi. A random sample of nine specimens is tested, and the average breaking strength is found to
The yield of a chemical process is being studied. From previous experience yield is known to be normally distributed and σ = 3. The past five days of plant operation have resulted in the following percent yields: 91.6, 88.75, 90.8, 89.95, and find a 95% two-sided confidence interval on the
The diameter of holes for cable harness is known to have a normal distribution with σ = 0.01 inch. A random sample of size 10 yields an average diameter of 1.5045 inch. Find a 99% two-sided confidence interval on the mean hole diameter.
A manufacturer produces piston rings for an automobile engine. It is known that ring diameter is normally distributed with σ = 0.001 millimeters. A random sample of 15 rings has a mean diameter of millimeters. (a) Construct a 99% two-sided confidence interval on the mean piston ring
The life in hours of a 75-watt light bulb is known to be normally distributed with σ = 25 hours. A random sample of 20 bulbs has a mean life of X = 1014 hours. (a) Construct a 95% two-sided confidence interval on the mean life. (b) Construct a 95% lower-confidence bound on the mean life.
A civil engineer is analyzing the compressive strength of concrete. Compressive strength is normally distributed with σ2 = 1000(psi)2. A random sample of 12 specimens has a mean compressive strength of x = 3250 psi. (a) Construct a 95% two-sided confidence interval on mean compressive
Suppose that in Exercise 8-12 we wanted to be 95% confident that the error in estimating the mean life is less than five hours. What sample size should be used?
Suppose that in Exercise 8-12 we wanted the total width of the two-sided confidence interval on mean life to be six hours at 95% confidence. What sample size should be used?
Suppose that in Exercise 8-13 it is desired to estimate the compressive strength with an error that is less than 15 psi at 99% confidence. What sample size is required?
By how much must the sample size n be increased if the length of the CI on μ in Equation 8-7 is to be halved?
If the sample size n is doubled, by how much is the length of the CI on μ in Equation 8-7 reduced? What happens to the length of the interval if the sample size is increased by a factor of four?
Find the values of the following percentiles: t0.025,15, t0.05,10, t0.10,20, t0.005,25, and t0.001,30.
Determine the t-percentile that is required to construct each of the following two-sided confidence intervals:(a) Confidence level = 95%, degrees of freedom = 12(b) Confidence level = 95%, degrees of freedom = 24(c) Confidence level = 99%, degrees of freedom = 13(d) Confidence level = 99.9%,
Determine the t-percentile that is required to construct each of the following one-sided confidence intervals:(a) Confidence level = 95%, degrees of freedom = 14(b) Confidence level = 99%, degrees of freedom = 19(c) Confidence level = 99.9%, degrees of freedom = 24
A research engineer for a tire manufacturer is investigating tire life for a new rubber compound and has built 16 tires and tested them to end-of-life in a road test. The sample mean and standard deviation are 60,139.7 and 3645.94 kilometers. Find a 95% confidence interval on mean tire life.
An Izod impact test was performed on 20 specimens of PVC pipe. The sample mean is X = 1.25 and the sample standard deviation is x = 0.25. Find a 99% lower confidence bound on Izod impact strength.
The brightness of a television picture tube can be evaluated by measuring the amount of current required to achieve a particular brightness level. A sample of 10 tubes results in x = 317.2 and s = 15.7. Find (in micro amps) a 99% confidence interval on mean current required. State any necessary
A particular brand of diet margarine was analyzed to determine the level of polyunsaturated fatty acid (in percentages). A sample of six packages resulted in the following data: 16.8, 17.2, 17.4, 16.9, 16.5, 17.1. (a) Is there evidence to support the assumption that the level of polyunsaturated
The compressive strength of concrete is being tested by a civil engineer. He tests 12 specimens and obtains the following data.(a) Is there evidence to support the assumption that compressive strength is normally distributed? Does this data set support your point of view? Include a graphical
A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected, and the diameter is measured. The resulting data (in millimeters) are as follows:(a) Check the assumption of normality for rod diameter.(b) Find a 95% two-sided confidence interval on mean
Rework Exercise 8-27 to compute a 95% lower confidence bound on rod diameter. Compare this bound with the lower limit of the two-sided confidence limit from Exercise 8-27. Discuss why they are different.
The wall thickness of 25 glass 2-liter bottles was measured by a quality-control engineer. The sample mean was x = 4.05 millimeters, and the sample standard deviation was s = 0.08 millimeter. Find a 95% lower confidence bound for mean wall thickness. Interpret the interval you have obtained.
An article in Nuclear Engineering International (February 1988, p. 33) describes several characteristics of fuel rods used in a reactor owned by an electric utility in Norway. Measurements on the percentage of enrichment of 12 rods were reported as follows:(a) Use a normal probability plot to check
A post mix beverage machine is adjusted to release a certain amount of syrup into a chamber where it is mixed with carbonated water. A random sample of 25 beverages was found to have a mean syrup content of x = 1.10 fluid ounces and a standard deviation of s = 0.015 fluid ounces. Find a 95% CI on
An article in the Journal of Composite Materials (December 1989, Vol 23, p. 1200) describes the effect of delamination on the natural frequency of beams made from composite laminates. Five such delaminated beams were subjected to loads, and the resulting frequencies were as follows (in hertz):
Determine the values of the following percentiles: X20.05,10, X20.025,15, X20.095,20, X20.099,18, X20.995,16, and X20.005,25,
Determine the 2 percentile that is required to construct each of the following CIs:(a) Confidence level = 95%, degrees of freedom = 24, one-sided (upper)(b) Confidence level = 99%, degrees of freedom = 9, one-sided (lower)(c) Confidence level = 90%, degrees of freedom = 19, two-sided.
A rivet is to be inserted into a hole. A random sample of n = 15 parts is selected, and the hole diameter is measured. The sample standard deviation of the hole diameter measurements is s = 0.008 millimeters. Construct a 99% lower confidence bound for σ2.
The sugar content of the syrup in canned peaches is normally distributed. A random sample of n = 10 cans yields a sample standard deviation of s = 4.8 milligrams. Find a 95% two-sided confidence interval for σ.
Consider the tire life data in Exercise 8-22. Find a 95% lower confidence bound for =2.
Consider the Izod impact test data in Exercise 8-23. Find a 99% two-sided confidence interval for σ2.
The percentage of titanium in an alloy used in aerospace castings is measured in 51 randomly selected parts. The sample standard deviation is s = 0.37. Construct a 95% two-sided confidence interval for σ.
Consider the hole diameter data in Exercise 8-35. Construct a 99% two-sided confidence interval for σ.
Consider the sugar content data in Exercise 8-37. Find a 90% lower confidence bound for σ.
Of 1000 randomly selected cases of lung cancer, 823 resulted in death within 10 years. Construct a 95% two-sided confidence interval on the death rate from lung cancer.
How large a sample would be required in Exercise 8-42 to be at least 95% confident that the error in estimating the 10-year death rate from lung cancer is less than 0.03?
A random sample of 50 suspension helmets used by motorcycle riders and automobile race-car drivers was subjected to an impact test, and on 18 of these helmets some damage was observed.(a) Find a 95% two-sided confidence interval on the true proportion of helmets of this type that would show damage
The Arizona Department of Transportation wishes to survey state residents to determine what proportion of the population would like to increase statewide highway speed limits to 75 mph from 65 mph. How many residents do they need to survey if they want to be at least 99% confident that the sample
A manufacturer of electronic calculators is interested in estimating the fraction of defective units produced. A random sample of 800 calculators contains 10 defectives. Compute a 99% upper-confidence bound on the fraction defective.
A study is to be conducted of the percentage of homeowners who own at least two television sets. How large a sample is required if we wish to be 99% confident that the error in estimating this quantity is less than 0.017?
The fraction of defective integrated circuits produced in a photolithography process is being studied. A random sample of 300 circuits is tested, revealing 13 defectives. Find a 95% two-sided CI on the fraction of defective circuits produced by this particular tool.
Consider the tire-testing data described in Exercise 8-22. Compute a 95% prediction interval on the life of the next tire of this type tested under conditions that are similar to those employed in the original test. Compare the length of the prediction interval with the length of the 95% CI on the
Consider the Izod impact test described in Exercise 8-23. Compute a 99% prediction interval on the impact strength of the next specimen of PVC pipe tested. Compare the length of the prediction interval with the length of the 99% CI on the population mean.
Consider the television tube brightness test described in Exercise 8-24. Compute a 99% prediction interval on the brightness of the next tube tested. Compare the length of the prediction interval with the length of the 99% CI on the population mean.
Consider the margarine test described in Exercise 8-25. Compute a 99% prediction interval on the polyunsaturated fatty acid in the next package of margarine that is tested. Compare the length of the prediction interval with the length of the 99% CI on the population mean.
Consider the test on the compressive strength of concrete described in Exercise 8-26. Compute a 90% prediction interval on the next specimen of concrete tested.
Consider the suspension rod diameter measurements described in Exercise 8-27. Compute a 95% prediction interval on the diameter of the next rod tested. Compare the length of the prediction interval with the length of the 95% CI on the population mean.
Consider the bottle wall thickness measurements described in Exercise 8-29. Compute a 90% prediction interval on the wall thickness of the next bottle tested.
How would you obtain a one-sided prediction bound on a future observation? Apply this procedure to obtain a 95% one-sided prediction bound on the wall thickness of the next bottle for the situation described in Exercise 8-29.
Consider the fuel rod enrichment data described in Exercise 8-30. Compute a 99% prediction interval on the enrichment of the next rod tested. Compare the length of the prediction interval with the length of the 95% CI on the population mean.
Consider the syrup dispensing measurements described in Exercise 8-31. Compute a 95% prediction interval on the syrup volume in the next beverage dispensed. Compare the length of the prediction interval with the length of the 95% CI on the population mean.
Consider the natural frequency of beams described in Exercise 8-32. Compute a 90% prediction interval on the diameter of the natural frequency of the next beam of this type that will be tested. Compare the length of the prediction interval with the length of the 95% CI on the population mean.
Compute a 95% tolerance interval on the life of the tires described in Exercise 8-22, that has confidence level 95%. Compare the length of the tolerance interval with the length of the 95% CI on the population mean. Which interval is shorter? Discuss the difference in interpretation of these two
Consider the Izod impact test described in Exercise 8-23. Compute a 99% tolerance interval on the impact strength of PVC pipe that has confidence level 90%. Compare the length of the tolerance interval with the length of the 99% CI on the population mean. Which interval is shorter? Discuss the
Compute a 99% tolerance interval on the brightness of the television tubes in Exercise 8-24 that has confidence level 95%. Compare the length of the prediction interval with the length of the 99% CI on the population mean. Which interval is shorter? Discuss the difference in interpretation of these
Consider the margarine test described in Exercise 8-25. Compute a 99% tolerance interval on the polyunsaturated fatty acid in this particular type of margarine that has confidence level 95%. Compare the length of the prediction interval with the length of the 99% CI on the population mean. Which
Compute a 90% tolerance interval on the compressive strength of the concrete described in Exercise 8-26 that has 90% confidence.
Compute a 95% tolerance interval on the diameter of the rods described in Exercise 8-27 that has 90% confidence. Compare the length of the prediction interval with the length of the 95% CI on the population mean. Which interval is shorter? Discuss the difference in interpretation of these two
Consider the bottle wall thickness measurements described in Exercise 8-29. Compute a 90% tolerance interval on bottle wall thickness that has confidence level 90%.
Consider the bottle wall thickness measurements described in Exercise 8-29. Compute a 90% lower tolerance bound on bottle wall thickness that has confidence level 90%. Why would a lower tolerance bound likely be of interest here?
Consider the fuel rod enrichment data described in Exercise 8-30. Compute a 99% tolerance interval on rod enrichment that has confidence level 95%. Compare the length of the prediction interval with the length of the 95% CI on the population mean.
Compute a 95% tolerance interval on the syrup volume described in Exercise 8-31 that has confidence level 90%. Compare the length of the prediction interval with the length of the 95% CI on the population mean.
Consider the confidence interval for μ with known standard deviationWhere a1 = a2 = a. Let a = 0.05 and find the interval for a1 = a2 = a/2 = 0.025. Now find the interval for the case a1 = 0.01 and a2 = 0.04. Which interval is shorter? Is there any advantage to a
A normal population has a known mean 50 and unknown variance.(a) A random sample of n = 16 is selected from this population, and the sample results are x = 52 and s = 8. How unusual are these results? That is, what is the probability of observing a sample average as large as 52 (or larger) if the
A normal population has known mean μ = 50 and variance σ2 = 5. What is the approximate probability that the sample variance is greater than or equal to 7.44? less than or equal to 2.56? (a) For a random sample of n = 16. (b) For a random sample of n = 30. (c) For a random sample of n
An article in the Journal of Sports Science (1987, Vol. 5, pp. 261€“271) presents the results of an investigation of the hemoglobin level of Canadian Olympic ice hockey players. The data reported are as follows (in g/dl):15.3 16.0 14.4 16.2 16.214.9 15.7 15.3 14.6 15.716.0 15.0
The article €œMix Design for Optimal Strength Development of Fly Ash Concrete€ (Cement and Concrete Research, 1989, Vol. 19, No. 4, pp. 634€“640) investigates the compressive strength of concrete when mixed with fly ash (a mixture of silica, alumina, iron, magnesium
An operating system for a personal computer has been studied extensively, and it is known that the standard deviation of the response time following a particular command is σ = 8 milliseconds. A new version of the operating system is installed, and we wish to estimate the mean response time
Consider the hemoglobin data in Exercise 8-73. Find the following:(a) An interval that contains 95% of the hemoglobin values with 90% confidence.(b) An interval that contains 99% of the hemoglobin values with 90% confidence.
Consider the compressive strength of concrete data from Exercise 8-74. Find a 95% prediction interval on the next sample that will be tested.
The maker of a shampoo knows that customers like this product to have a lot of foam. Ten sample bottles of the product are selected at random and the foam heights observed are as follows (in millimeters): 210, 215, 194, 195, 211, 201, 198, 204, 208, and 196.(a) Is there evidence to support the
During the 1999 and 2000 baseball seasons, there was much speculation that the unusually large number of home runs that were hit was due at least in part to a livelier ball. One way to test the €œliveliness€ of a baseball is to launch the ball at a vertical surface with a known
Consider the baseball coefficient of restitution data in Exercise 8-79. Suppose that any baseball that has a coefficient of restitution that exceeds 0.635 is considered too lively. Based on the available data, what proportion of the baseballs in the sampled population are too lively? Find a 95%
An article in the ASCE Journal of Energy Engineering (“Overview of Reservoir Release Improvements at 20 TVA Dams,” Vol. 125, April 1999, pp. 1–17) presents data on dissolved oxygen concentrations in streams below 20 dams in the Tennessee Valley Authority system. The observations are (in
The tar content in 30 samples of cigar tobacco follows:(a) Is there evidence to support the assumption that the tar content is normally distributed?(b) Find a 99% CI on the mean tar content.(c) Find a 99% prediction interval on the tar content for the next observation that will be taken on this
A manufacturer of electronic calculators takes a random sample of 1200 calculators and finds that there are eight defective units.(a) Construct a 95% confidence interval on the population proportion.(b) Is there evidence to support a claim that the fraction of defective units produced is 1% or less?
An article in The Engineer (“Redesign for Suspect Wiring,” June 1990) reported the results of an investigation into wiring errors on commercial transport aircraft that may produce faulty information to the flight crew. Such a wiring error may have been responsible for the crash of a British
An article in Engineering Horizons (Spring 1990, p. 26) reported that 117 of 484 new engineering graduates were planning to continue studying for an advanced degree. Consider this as a random sample of the 1990 graduating class.(a) Find a 90% confidence interval on the proportion of such graduates
An electrical component has a time-to-failure (or lifetime) distribution that is exponential with parameter λ, so the mean lifetime is μ = 1/λ. Suppose that a sample of n of these components is put on test, and let Xi be the observed lifetime of component i. The test continues only

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