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

9.7 With reference to Exercise 7.62, test the null hypothesis σ = 600 psi for the compressive strength of the given kind of steel against the alternative hypothesis σ > 600 psi. Use the 0.05 level of significance.
9.6 Use the value s obtained in Exercise 9.3 to construct a 98% confidence interval for σ, measuring the actual variability in the hardness of Alloy 1.
9.5 With reference to Exercise 7.63, construct a 99% confidence interval for the variance of the population sampled.
9.4 With reference to Exercise 7.56, construct a 95% confidence interval for the variance of the yield.
9.3 Use the data of part (a) of Exercise 8.13 to estimate σ for the Brinell hardness of Alloy 1 in terms of (a) the sample standard deviation; (b) the sample range. Compare the two estimates by expressing their difference as a percentage of the first.
9.2 With reference to Example 7, Chapter 8, use the range of the second sample to estimate σ for the resiliency modulus of recycled materials from the second location. Compare the result with the standard deviation of the second sample.
9.1 Use the data of Exercise 7.14 to estimate σ for the key performance indicator in terms of (a) the sample standard deviation; (b) the sample range. Compare the two estimates by expressing their difference as a percentage of the first.
8.37 Refer to Example 12 concerning the improvement in lost worker-hours. Obtain a 90% confidence interval for the mean of this paired difference.
8.36 Refer to Example 13 concerning an array of sites that smell toxic chemicals. When exposed to the common manufacturing chemical Arsine, a product of arsenic and acid, the change in the red component is measured six times. (Courtesy of the authors) 0.10 −0.33 −1.12 −1.95 −3.63 −1.48
8.35 Two samples in C1 and C2 can be analyzed using the MINITAB commands Dialog box: Stat > Basic Statistics > 2-Sample t Pull down Each sample in its own column Type C1 in Sample 1 and C2 in Sample 2. Click Options and then Assume equal variances. Click OK. Click OK.If you do not click Assume
8.34 With reference to part (a) of Exercise 8.33, how would you pair and then randomize for a paired test?
8.33 How would you randomize, for a two sample test, in each of the following cases? (a) Forty combustion engines are available for a speed test and you want to compare a modified exhaust valve with the regular valve. (b) A new cold storage freezer will be compared with the old. Twenty jugs of
8.32 With reference to Example 8, find a 90% confidence interval for the difference of mean strengths of the alloys (a) using the pooled procedure; (b) using the large samples procedure.
8.31 Random samples are taken from two normal popula- tions with = 9.6 and 02 = 13.2 to test the null - hypothesis 12 = 41.2 against the alternative hypothesis μ1 − μ2 > 41.2 at the level of significance α = 0.05. Determine the common sample size n = n1 − n2 that is required if the
8.30 With reference to the previous exercise, find a 90% confidence interval for the difference of the two means.
8.29 With reference to Example 2, Chapter 2, test that the mean copper content is the same for both heats.
8.28 Two adhesives for pasting plywood boards are to be compared. 10 tubes are prepared using Adhesive I and 8 tubes are prepared using Adhesive II. Then 18 different pairs of plywood boards are pasted together, one tube per pair of boards. (a) The response is the time in minutes for the boards to
8.27 With reference to the previous exercise, find a 90% confidence interval for the difference of the two means.
8.26 With reference to Exercise 2.64, test that the mean charge of the electron is the same for both tubes. Use α = 0.05.
8.25 How would you randomize, for a two sample test, if 50 cars are available for an emissions study and you want to compare a modified air pollution device with that used in current production?
8.24 It takes an average of 30 classes for an instructor to teach a civil engineering student probability. The instructor introduces a new software which they feel will lead to faster calculations. The instructor intends to teach 10 students with the new software and compare their calculation
8.23 An electrical engineer has developed a modified circuit board for elevators. Suppose 3 modified circuit boards and 6 elevators are available for a comparative test of the old versus the modified circuit boards. (a) Describe how you would select the 3 elevators in which to install the modified
8.22 An engineer wants to compare two busy hydraulic belts by recording the number of finished goods that are successfully transferred by the belts in a day. Describe how to select 3 of the next 6 working days to try Belt A. Belt B would then be tried on the other 3 days.
8.21 In a study of the effectiveness of physical exercise in weight reduction, a group of 16 persons engaged in a prescribed program of physical exercise for one month showed the following results: Weight before Weight after Weight before Weight after (pounds) (pounds) (pounds) (pounds) 209 196 170
8.20 Referring to Example 13, conduct a test to show that the mean change μD is different from 0. Take α = 0.05.
8.19 A shoe manufacturer wants potential customers to compare two types of shoes, one made of the current PVC material X and one made of a new PVC material Y. Shoes made of both are available. Each person, in a sample of 52, is asked to wear one pair of each type for a whole day. After a walk of 2
8.18 Refer to Example 14 concerning suspended solids in effluent from a treatment plant. Take the natural logarithm of each of the measurements and then take the difference. (a) Construct a 95% confidence interval for μD. (b) Conduct a level α = 0.05 level test of H0: μD = 0 against a two-sided
8.17 Refer to Example 14 concerning suspended solids in effluent from a treatment plant. Take the square root of each of the measurements and then take the difference. (a) Construct a 95% confidence interval for μD. (b) Conduct a level α = 0.05 level test of H0: μD = 0 against a two-sided
8.16 The following data were obtained in an experiment designed to check whether there is a systematic difference in the weights obtained with two different scales Weight in grams Scale I Scale II Rock Specimen 1 11.23 11.27 Rock Specimen 2 14.36 14.41 Rock Specimen 3 8.33 8.35 Rock Specimen 4
8.15 Refer to Exercise 8.14. Test with α = 0.01, that the mean difference is 0 versus a two-sided alternative.
8.14 A civil engineer wants to compare two machines for grinding cement and sand. A sample of a fixed quantity of cement and sand is taken and put in each machine. The machines are run and the fineness of each mixture is noted. This process is repeated five times. The results, in microns, are as
8.13 In each of the parts below, first decide whether or not to use the pooled estimator of variance. Assume that the populations are normal. (a) The following are the Brinell hardness values obtained for samples of two magnesium alloys before testing: Alloy 1: 66.3 63.5 64.9 61.8 64.3 64.7 65.1
8.12 With reference to Example 5 construct a 95% confidence interval for the true difference between the average resistance of the two kinds of wire.
8.11 The following are the number of hydraulic pumps which a sample of 10 industrial machines of Type A and a sample of 8 industrial machines of Type B manufactured over a certain fixed period of time: Type A: 8 6 7 9 4 11 8 10 6 9 Type B:43677 1 9 6 Assuming that the populations sampled can be
8.10 We know that silk fibers are very tough but in short supply. Breakthroughs by one research group result in the summary statistics for the stress (MPa) of synthetic silk fibers (Source: F. Teulé, et. al. (2012), Combining flagelliform and dragline spider silk motifs to produce tunable
8.9 Measuring specimens of nylon yarn taken from two spinning machines, it was found that 8 specimens from the first machine had a mean denier of 9.67 with a standard deviation of 1.81, while 10 specimens from the second machine had a mean denier of 7.43 with a standard deviation of 1.48.
8.8 Two methods for manufacturing a product are to be compared. Given 12 units, six are manufactured using method M and six are manufactured using method N. (a) How would you assign manufacturing methods to the 12 units? (b) The response is the percent of finished product that did not meet quality
8.7 Given the n1 = 3 and n2 = 2 observations from Population 1 and Population 2, respectively, Population 1 6 27 Population 2 14 10 (a) Calculate the three deviations x − x and two deviations y − y. (b) Use your results from part (a) to obtain the pooled variance.
8.6 Studying the flow of traffic at two busy intersections between 4 p.m. and 6 p.m. (to determine the possible need for turn signals), it was found that on 40 weekdays there were on the average 247.3 cars approaching the first intersection from the south that made left turns while on 30 weekdays
8.5 An investigation of two types of bulldozers showed that 50 failures of one type of bulldozer took on an average 6.8 hours to repair with a standard deviation of 0.85 hours, while 50 failures of the other type of bulldozer took on an average 7.3 hours to repair with a standard deviation of 1.2
8.4 Refer to Exercise 8.3 and obtain a 95% confidence interval for the difference in mean dynamic modulus.
8.3 The dynamic modulus of concrete is obtained for two different concrete mixes. For the first mix, n1 = 33, x = 115.1, and s1 = 0.47 psi. For the second mix, n2 = 31, y = 114.6, and s2 = 0.38. Test, with α = 0.05, the null hypothesis of equality of mean dynamic modulus versus the two-sided
8.2 Refer to Exercise 8.1 and obtain a 95% confidence interval for the difference in mean tensile strength.
8.1 Refer to Exercise 2.58, where n1 = 30 specimens of 2 × 4 lumber have x = 1,908.8 and s1 = 327.1 psi. A second sample of size n2 = 40 specimens of larger dimension, 2 × 6, lumber yielded y = 2,114.3 and s2 = 472.3. Test, with α = 0.05, the null hypothesis of equality of mean tensile
7.89 The compressive strength of parts made from a composite material are known to be nearly normally distributed. A scientist, using the testing device for the first time, obtains the tensile strength (psi) of 20 specimens 95 102 105 107 109 110 111 112 114 115 134 135 136 138 139 141 142 144 150
7.88 Refer to Exercise 7.87. (a) Perform a test with the intention of establishing that the mean time to return a call is greater than 1.5 hours. Use α = 0.05. (b) In light of your conclusion in part (a), what error could you have made? Explain in the context of this problem. (c) In a long series
7.87 An industrial engineer concerned with service at a large medical clinic recorded the duration of time from the time a patient called until a doctor or nurse returned the call. A sample of size 180 calls had a mean of 1.65 hours and a standard deviation of 0.82. (a) Obtain a 95% confidence
7.86 In a fatigue study, the time spent working by employees in a factory was observed. The ten readings (in hours) were 4.8 3.6 10.8 5.7 8.2 6.8 7.5 7.7 6.3 8.6 Assuming the population sampled is normal, construct a 90% confidence interval for the corresponding true mean.
7.80 It is desired to estimate the mean number of hours of continuous use until a printer overheats. If it can be assumed that σ = 4 hours, how large a sample is needed so that one will be able to assert with 95% confidence that the sample mean is off by at least 15 hours? 7.81 A sample of 15
7.79 With reference to the preceding exercise, construct a 95% confidence interval for the true average increase in the pulse rate of astronaut trainees performing the given task.
7.78 While performing a certain task under simulated weightlessness, the pulse rate of 32 astronaut trainees increased on the average by 26.4 beats per minute with a standard deviation of 4.28 beats per minute. What can one assert with 95% confidence about the maximum error if x = 26.4 is used as
7.77 With reference to Example 7 on page 29, find a 95% confidence interval for the mean strength of the aluminum alloy.
7.76 Specify the null hypothesis and the alternative hypothesis in each of the following cases. (a) An engineer hopes to establish that an additive will increase the viscosity of an oil. (b) An electrical engineer hopes to establish that a modified circuit board will give a computer a higher
7.75 MINITAB calculation of power or OC curve Refer to the steps in Exercise 7.72, but enter a range of values for the difference. Here 0:3/.1 goes in steps from 0 to 3 in steps of .1 for Example 22. Stat > Power and sample size > 1-Sample Z. Type 15 in Sample sizes, 0:3/.1 in differences and 3.6
7.74 MINITAB calculation of sample size Refer to Exercise 7.72, but this time leave Sample size blank but Type 0.90 in power to obtain the partial output concerning sample size Sample Target Difference Size Power Actual Power 1.8 35 0.9 0.905440 Refer to the example concerning sound intensity on
7.73 Use computer software to repeat Exercise 7.71.
7.72 MINITAB calculation of power These calculations pertain to normal populations with known variance and provide an accurate approxima- tion in the large sample case where is unknown. To calculate the power of the Z test at 1, you need to enter the difference 1-po- Although n = 15 is not large,
7.71 Refer to the example concerning average sound inten- sity on page 260. Calculate the power at = 77 when (a) the level is changed to a = 0.03. (b) = 0.05 but the alternative is changed to the two- sided H: #75.2.
7.70 Repeat Exercise 7.69 but replace the t test with the large sample Z test.
7.69 Refer to the labor time data in Exercise 7.3. Using the 90% confidence interval, based on the t distribution, for the mean labor time N Mean StDev SE Mean 90% CI 52 1.86462 1.24992 0.17333 (1.57423, 2.15500) (a) decide whether or not to reject H0 : μ = 1.6 in favor of H1 : μ = 1.6 at α =
7.68 Refer to the green gas data on page 241. Using the 95% confidence interval, based on the t distribution for the mean yield N Mean StDev SE Mean 95% CI 15 6.00933 1.07780 0.27829 (5.41247, 6.60620) (a) decide whether or not to reject H0 : μ = 5.5 gal in favor of H0 : μ = 5.5 at α = 0.05;
7.67 Repeat Exercise 7.66 but replace the t test with the large sample Z test.
7.66 Refer to the nanopillar height data on page 25. Using the 95% confidence interval, based on the t distribution, for the mean nanopillar height N Mean StDev SE Mean 95% CI 50 305.580 36.971 5.229 (295.073, 316.087) (a) decide whether or not to reject H0 : μ = 320 nm in favor of H1 : μ = 320
7.65 The statistical program MINITAB will calculate t tests. With the nanopillar height data in C1,Dialog box: Stat > Basic Statistics > 1-Sample t. Click on box and type C1. Click Perform hypothesis test and Type 300 in Hypothesized mean. Choose Options. Type 0.95 in Confidence level and choose
7.64 Suppose that in the preceding exercise the first measurement is recorded incorrectly as 16.0 instead of 14.5. Show that, even though the mean of the sample increases to x = 14.7, the null hypothesis H0: μ = 14.0 is not rejected at level α = 0.05. Explain the apparent paradox that even
7.63 A manufacturer claims that the average tar content of a certain kind of cigarette is μ = 14.0. In an attempt to show that it differs from this value, five measurements are made of the tar content (mg per cigarette): 14.5 14.2 14.4 14.3 14.6 Show that the difference between the mean of this
7.62 A random sample of 6 steel beams has a mean compressive strength of 58,392 psi (pounds per square inch) with a standard deviation of 648 psi. Use this information and the level of significance α = 0.05 to test whether the true average compressive strength of the steel from which this sample
7.61 With reference to the thickness measurements in Exercise 2.41, test the null hypothesis that μ = 30.0 versus a two-sided alternative. Take α = 0.05.
7.60 In 64 randomly selected hours of production, the mean and the standard deviation of the number of acceptable pieces produced by a automatic stamping machine are x = 1,038 and s = 146. At the 0.05 level of significance, does this enable us to reject the null hypothesis μ = 1,000 against the
7.59 Refer to Exercise 2.34, page 46, concerning the number of board failures for n = 32 integrated circuits. A computer calculation gives x = 7.6563 and s = 5.2216. At the 0.01 level of significance, conduct a test of hypotheses with the intent of showing that the mean is greater than 7.
7.58 Refer to Exercise 7.22, where, in n = 81 cases, the coffee machine needed to be refilled with beans after 225 cups with a standard deviation of 22 cups. (a) Conduct a test of hypotheses with the intent of showing that the mean number of cups is greater than 218 cups. Take α = 0.01. (b) Based
7.57 Refer to Exercise 7.14, where n = 9 measurements were made on a key performance indicator. 123 106 114 128 113 109 120 102 111 (a) Conduct a test of hypotheses with the intent of showing that the mean key performance indicator is different from 107. Take α = 0.05 and assume a normal
7.56 Refer to Exercise 7.12, where, in a pilot process, vertical spirals were cut to produce latex from n = 8 trees to yield (in liters) in a week. 26.8 32.5 29.7 24.6 31.5 39.8 26.5 19.9 (a) Conduct a test of hypotheses with the intent of showing that the mean production is less than 36.2. Take
7.55 Refer to Exercise 7.5, where the number of unremovable defects, for each of n = 45 displays, has x = 2.467 and s = 3.057 unremovable defects. (a) Conduct a test of hypotheses with the intent of showing that the mean number of unremovable defects is less than 3.6. Take α = 0.025. (b) Based on
7.54 Refer to data in Exercise 7.3 on the labor time required to produce an order of automobile mufflers using a heavy stamping machine. The times (hours) for n = 52 orders of different parts has x = 1.865 hours and s2 = 1.5623, so s = 1.250 hours. (a) Conduct a test of hypotheses with the intent
7.53 Refer to Exercise 7.1 where a construction engineer recorded the quantity of gravel (in metric tons) used in concrete mixes. The quantity of gravel for n = 24 sites has x = 5,818 tons and s2 = 7,273,809 so s = 2,697 tons. (a) Construct a test of hypotheses with the intent of showing that the
7.52 Specify the null and alternative hypotheses in each of the following cases. (a) A car manufacturer wants to establish the fact that in case of an accident, the installed safety gadgets saved the lives of the passengers in more than 90% of accidents. (b) An electrical engineer wants to
7.51 A producer of extruded plastic products finds that his mean daily inventory is 1,250 pieces. A new marketing policy has been put into effect and it is desired to test the null hypothesis that the mean daily inventory is still the same. What alternative hypothesis should be used if (a) it is
7.50 Several square inches of gold leaf are required in the manufacture of a high-end component. Suppose that, the population of the amount of gold leaf has σ = 8.4 square inches. We want to test the null hypothesis μ = 80.0 square inches against the alternative hypothesis μ < 80.0 square
7.49 It is desired to test the null hypothesis μ = 30 minutes against the alternative hypothesis μ < 30 minutes on the basis of the time taken by a newly developed oven for n = 50 cakes baked. The population has σ = 5 minutes. For what values of X must the null hypothesis be rejected if the
7.48 Suppose that in the electric car battery example on page 242, n is changed from 36 to 50 while everything else remains the same. Find (a) the probability of a Type I error; (b) the probability of a Type II error when μ = 1680 cycles.
7.47 If the criterion on page 242 is modified so that the manufacturer’s claim is accepted for X > 1640 cycles, find (a) the probability of a Type I error; (b) the probability of a Type II error when μ = 1680 cycles.
7.46 Suppose that we want to test the null hypothesis that an antipollution device for cars is effective. Explain under what conditions we would commit a Type I error and under what conditions we would commit a Type II error.
7.45 Suppose that an engineering firm is asked to check the safety of a dam. What type of error would it commit if it erroneously rejects the null hypothesis that the dam is safe? What type of error would it commit if it erroneously fails to reject the null hypothesis that the dam is safe? Would
7.44 Suppose you are scheduled to ride a space vehicle that will orbit the earth and return. A statistical test of hypotheses includes the step of setting a maximum for the probability of falsely rejecting the null hypothesis. Engineers need to make various measurements to decide if it is safe or
7.43 A statistical test of hypotheses includes the step of setting a maximum for the probability of falsely rejecting the null hypothesis. Engineers make many measurements on critical bridge components to decide if a bridge is safe or unsafe. (a) Explain how you would formulate the null hypothesis.
7.42 A manufacturer wants to establish that the mean life of a gear when used in a crusher is over 55 days. The data will consist of how long gears in 80 different crushers have lasted. (a) Formulate the null and alternative hypotheses. (b) If the true mean is 55 days, what error could be made?
7.41 An airline claims that the typical flying time between two cities is 56 minutes. (a) Formulate a test of hypotheses with the intent of establishing that the population mean flying time is different from the published time of 56 minutes. (b) If the true mean is 50 minutes, what error can be
7.40 A manufacturer of four-speed clutches for automobiles claims that the clutch will not fail until after 50,000 miles. (a) Interpreting this as a statement about the mean, formulate a null and alternative hypothesis for verifying the claim. (b) If the true mean is 55,000 miles, what error can
7.39 A civil engineer wants to establish that the average time to construct a new two-storey building is less than 6 months. (a) Formulate the null and alternative hypotheses. (b) What error could be made if μ = 6? Explain in the context of the problem. (c) What error could be made if μ = 5.5?
7.38 Let X have the negative binomial distribution f (x) = x − 1 r − 1 pr (1 − p) x−r for x = r,r + 1,... (a) Obtain the maximum likelihood estimator of p. (b) For one engineering application, it is best to use components with a superior finish. Suppose X = 27 identical components are
7.37 Let x1,..., xn be the observed values of a random sample of size n from the exponential distribution f (x; β) = β−1e−x/β for x > 0. (a) Find the maximum likelihood estimator of β. (b) Obtain the maximum likelihood estimator of the probability that the next observation is greater than 1.
7.36 Find the maximum likelihood estimator of p when f (x; p) = px(1 − p) 1−x for x = 0, 1
7.35 Refer to Exercise 7.14. (a) Obtain the maximum likelihood estimates of μ and σ. (b) Find the maximum likelihood of the coefficient of variation σ/μ.
7.34 Refer to Exercise 7.12. (a) Obtain the maximum likelihood estimates of μ and σ. (b) Find the maximum likelihood of the probability that the next run will have a production greater than 38 liters.
7.33 In one area along the interstate, the number of dropped wireless phone connections per call follows a Poisson distribution. From four calls, the number of dropped connections is 2031 (a) Find the maximum likelihood estimate of λ. (b) Obtain the maximum likelihood estimate that the next two
7.32 The daily number of accidental disconnects with a server follows a Poisson distribution. On five days 2 5 3 3 7 accidental disconnects are observed. (a) Obtain the maximum likelihood estimate of 2. (b) Find the maximum likelihood estimate of the probability that 3 or more accidental
7.31 Refer to Example 7, Chapter 10, where 48 of 60 transceivers passed inspection. (a) Obtain the maximum likelihood estimate of the probability that a transceiver will pass inspection. (b) Obtain the maximum likelihood estimate that the next two transceivers tested will pass inspection.
7.30 Refer to Example 13, Chapter 3, where 294 out of 300 ceramic insulators were able to survive a thermal shock. (a) Obtain the maximum likelihood estimate of the probability that a ceramic insulator will survive a thermal shock. (b) Suppose a device contains 3 ceramic insulators and all must
7.29 You can simulate the coverage of the small sample confidence intervals for μ by generating 20 samples of size 10 from a normal distribution with μ = 20 and σ = 5 and computing the 95% confidence intervals according to the formula on page 231. Using MINITAB:Calc > Random Data > Normal Type

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