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modern mathematical statistics with applications

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

Consider the following Minitab output.(a) Fill in the missing quantities.(b) At what level of significance can the null hypothesis be rejected? (c) If the hypotheses had been Ho: p = 44 versus H: p > 44, would the P-value be larger or smaller? (d) If the hypotheses had been Ho: = 44.5 versus H:
State the null and the alternative hypotheses, and indicate the type of critical region (either two-, lower-, or upper-tailed) to test the following claims.(a) A manufacturer of lightbulbs has a new type of lightbulb that is advertised to have a mean burning lifetime in excess of 5000 hours.(b) A
Consider the 20 observations collected on the number of errors in a string of 1000 bits of a communication channel given in Exercise 2-64 of Chapter 2.(a) Based on the description of the random variable and these 20 observations, is a binomial distribution an appropriate model? Perform a
Consider the 40 observations collected on the number of nonconforming coil springs in production batches of size 50 given in Exercise 2-63 of Chapter 2.(a) Based on the description of the random variable and these 40 observations, is a binomial distribution an appropriate model? Perform a
The proportion of residents in Phoenix favoring the building of toll roads to complete the freeway system is believed to be p 0.3. If a random sample of 20 residents shows that 2 or fewer favor this proposal, we will conclude that p 0.3.(a) Find the probability of type I error if the true
The proportion of adults who live in Tempe, Arizona, who are college graduates is estimated to be p 0.4. To test this hypothesis, a random sample of 15 Tempe adults is selected. If the number of college graduates is between 4 and 8, the hypothesis will be accepted; otherwise, we will conclude
A standardized test for graduating high school seniors is designed to be completed by 75% of the students within 40 minutes. A random sample of 100 graduates showed that 64 completed the test within 40 minutes.(a) Find a 90% two-sided traditional CI on the proportion of such graduates completing
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
A manufacturer of electronic calculators claims that less than 1% of its production output is defective. A random sample of 1200 calculators contains 8 defective units.(a) Confirm using a test procedure and an level of 0.01 that there is insufficient evidence to support the claim that the
A biotechnology company produces a therapeutic drug whose concentration has a standard deviation of 4 g/l. A new method of producing this drug has been proposed, although some additional cost is involved. Management will authorize a change in production technique only if the standard deviation of
A manufacturer of precision measuring instruments claims that the standard deviation in the use of the instruments is at most 0.00002 mm. An analyst, who is unaware of the claim, uses the instrument eight times and obtains a sample standard deviation of 0.00001 mm.(a) Confirm using a test procedure
Consider the fatty acid measurements for the diet margarine described in Exercise 4-58.(a) For this sample size n 6, using a two-sided alternative hypothesis and 0.01, test H0: 2 1.0.(b) Suppose that instead of n 6, the sample size were n 51. Using the estimate s2 from the original
The data below are the number of earthquakes per year of magnitude 7.0 and greater since 1900. (Source:U.S. Geological Survey, National Earthquake Information Center, Golden, CO).(a) Use computer software to summarize these data into a frequency distribution. Test the hypothesis that the number of
An article in the British Medical Journal (“Comparison of Treatment of Renal Calculi by Operative Surgery, Percutaneous Nephrolithotomy, and Extra-corporeal Shock Wave Lithotripsy,” Vol. 292, 1986, pp. 879–882) found that percutaneous nephrolithotomy (PN) had a success rate in removing kidney
An article in Medicine and Science in Sports and Exercise (“Maximal Leg-Strength Training Improves Cycling Economy in Previously Untrained Men,” Vol. 37, 2005, pp.131–136) reported the results of a study of cycling performance before and after eight weeks of leg-strength training. The sample
An article in Biological Trace Element Research[“Interaction of Dietary Calcium, Manganese, and Manganese Source (Mn Oxide or Mn Methionine Complex) on Chick Performance and Manganese Utilization,” Vol. 29, No. 3, 1991, pp. 217–228] showed the following results of tissue assay for liver
An article in Growth: A Journal Devoted to Problems of Normal and Abnormal Growth (“Comparison of Measured and Estimated Fat-Free Weight, Fat, Potassium and Nitrogen of Growing Guinea Pigs,” Vol. 46, No. 4, 1982, pp.306–321) reported the results of a study that measured the body weight
An article in Food Testing and Analysis (“Improving Reproducibility of Refractometry Measurements of Fruit Juices,” Vol. 4, No. 4, 1999, pp. 13–17) reported the results of H1: p 0.5.x 157.65 x 86.a study that measured the sugar concentration (Brix) in clear apple juice. All readings
Consider the biomedical device experiment described in Exercise 4-57.(a) For this sample size n 15, do the data support the claim that the standard deviation of life is less than 280 hours?(b) Suppose that instead of n 15, the sample size was 51.Repeat the analysis performed in part (a) using n
Suppose we are testing H0: p0.5 versus(a) Using 0.05, find the power of the test for n 100, 150, and 300, assuming the true value p 0.6. Comment on the effect of sample size on the power of the test.(b) Using 0.01, find the power of the test for n 100, 150, and 300, assuming the true
The cooling system in a nuclear submarine consists of an assembly of welded pipes through which a coolant is circulated.Specifications require that weld strength must meet or exceed 150 psi.(a) Suppose that the design engineers decide to test the hypothesis H0: 150 versus H1: 150. Explain
Suppose we wish to test the hypothesis H0: 85 versus the alternative H1: 85 where 16. Suppose that the true mean is 86 and that in the practical context of the problem this is not a departure from 0 85 that has practical significance.(a) For a test with 0.01, compute for the
The life in hours of a heating element used in a furnace is known to be approximately normally distributed. A random sample of 15 heating elements is selected and found to have an average life of 598.14 hours and a sample standard deviation of 16.93 hours.(a) At the 0.05 level of significance,
Suppose that in Exercise 4-113 the experimenter had believed that 14. For each of the following sample sizes and a fixed 0.05, find the probability of a type II error if the true mean is 205 ml/h.(a) n 20 (b) n 50 (c) n 100(d) Comparing your answers to those in Exercise 4-113, does
A quality control inspector of flow metering devices used to administer fluid intravenously will perform a hypothesis test to determine whether the mean flow rate is different from the flow rate setting of 200 ml/h. Based on prior information the standard deviation of the flow rate is assumed to be
Consider the description of Exercise 4-111. After collecting a sample, we are interested in computing the error in estimating the true value p. For each of the sample sizes and estimates of p, compute the error at the 95% confidence level.(a) n 60 and(b) n 70 and(c) n 100 and(d) Compare your
A manufacturer of semiconductor devices takes a random sample of size n of chips and tests them, classifying each chip as defective or nondefective. Let Xi 0 if the chip is nondefective and Xi 1 if the chip is defective. The sample fraction defective is What are the sampling distribution, the
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 8 milliseconds. A new version of the operating system is installed, and we wish to estimate the mean response time for the
The article “Mix Design for Optimal Strength Development of Fly Ash Concrete” (Cement and Concrete Research, Vol. 19, No. 4, 1989, pp. 634–640) investigates the compressive strength of concrete when mixed with fly ash(a mixture of silica, alumina, iron, magnesium oxide, and other
An article in the Journal of Sports Science (Vol. 5, 1987, 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 16.2 14.9 15.7 15.3 14.6 15.7 16.0 15.0 15.7 16.2 14.7
A normal population has known mean 50 and variance 2 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 71.(d) Compare
Formulate the appropriate null and alternative hypotheses to test the following claims.x z1 1nx z2 1n(a) A plastics production engineer claims that 99.95% of the plastic tube manufactured by her company meets the engineering specifications requiring the length to exceed 6.5 inches.(b) A
Consider the confidence interval for with known standard deviation :where 1 2 . Let 0.05 and find the interval for1 2 2 0.025. Now find the interval for the case1 0.01 and 2 0.04. Which interval is shorter? Is there any advantage to a “symmetric” CI?
Convert the OC curve in the previous problem into a plot of the power function of the test.
If we plot the probability of accepting H0: 0 versus various values of and connect the points with a smooth curve, we obtain the operating characteristic curve (or the OC curve) of the test procedure. These curves are used extensively in industrial applications of hypothesis testing to display
Define X as the number of underfilled bottles in a filling operation in a carton of 12 bottles. Eighty cartons are inspected, and the following observations on X are recorded(a) Based on these 80 observations, is a binomial distribution an appropriate model? Perform a goodness-of-fit procedure with
Consider the following frequency table of observations on the random variable X.(a) Based on these 50 observations, is a binomial distribution with n 6 and p 0.25 an appropriate model? Perform a goodness-of-fit procedure with 0.05.(b) Calculate the P-value for this test. Values 0 1 2 3 4
The number of cars passing eastbound through the intersection of Mill Avenue and University Avenue has been tabulated by a group of civil engineering students. They have obtained the following data:(a) Does the assumption of a Poisson distribution seem appropriate as a probability model for this
The number of calls arriving to a switchboard from noon to 1 P.M. during the business days Monday through Friday is monitored for 4 weeks (i.e., 30 days). Let X be defined as the number of calls during that 1-hour period. The observed frequency of calls was recorded and reported as follows:(a) Does
Let X denote the number of flaws observed on a large coil of galvanized steel. Seventy-five coils are inspected, and the following data were observed for the values of X.(a) Does the assumption of a Poisson distribution with a mean of 6.0 seem appropriate as a probability model for these data? Use
Consider the following frequency table of observations on the random variable X.(a) Based on these 100 observations, is a Poisson distribution with a mean of 2.4 an appropriate model? Perform a goodness-of-fit procedure with 0.05.(b) Calculate the P-value for this test. Values 0 1 2 3 4 5
Consider the metal rods described in Exercise 4-60.(a) Construct a 90% PI for the diameter of a single rod.(b) Find a tolerance interval for the diameter that includes 90% of the rods with 90% confidence.
Consider the breakdown voltage of diodes described in Exercise 4-59.(a) Construct a 99% PI for the breakdown voltage of a single diode.(b) Find a tolerance interval for the breakdown voltage that includes 99% of the diodes with 99% confidence.
Consider the fatty acid content of margarine described in Exercise 4-58.(a) Construct a 95% PI for the fatty acid content of a single package of margarine.(b) Find a tolerance interval for the fatty acid content that includes 95% of the margarine packages with 99% confidence.
Consider the life of biomedical devices described in Exercise 4-57.(a) Construct a 99% PI for the life of a single device.(b) Find a tolerance interval for the device life that includes 99% of the devices in the population with 90% confidence.
Consider the Izod impact strength problem described in Exercise 4-56.(a) Construct a 90% PI for the impact strength of a single specimen of PVC pipe.(b) Find a tolerance interval for the impact strength that includes 95% of the specimens in the population with 95%confidence.
Consider the tire life problem described in Exercise 4-55.(a) Construct a 95% PI on the life of a single tire.(b) Find a tolerance interval for the tire life that includes 90%of the tires in the population with 95% confidence.
Consider the knee surgery data given in Exercise 4-87. Calculate the 95% Agresti-Coull two-sided CI from equation 4-76 and compare it to the traditional CI in part (a) of the original exercise.
Consider the helmet data given in Exercise 4-76.Calculate the 95% Agresti-Coull two-sided CI from equation 4-76 and compare it to the traditional CI in the original exercise.
Consider the lung cancer data given in Exercise 4-74.Calculate the 95% Agresti-Coull two-sided CI from equation 4-76 and compare it to the traditional CI in the original exercise.
An article in Knee Surgury, Sports Traumatology, Arthroscopy (“Arthroscopic Meniscal Repair with an Absorbable Screw: Results and Surgical Technique,” 2005, Vol. 13, pp. 273–279) showed that 25 out of 37 tears located between 3 and 6 mm from the meniscus rim were healed.(a) Calculate a
The warranty for batteries for mobile phones is set at 400 operating hours, with proper charging procedures. A study of 2000 batteries is carried out and three stop operating prior to 400 hours. Do these experimental results support the claim that less than 0.2% of the company’s batteries will
A random sample of 500 registered voters in Phoenix is asked whether they favor the use of oxygenated fuels year round to reduce air pollution. If more than 315 voters respond positively, we will conclude that at least 60% of the voters favor the use of these fuels.(a) Find the probability of type
A researcher claims that at least 10% of all football helmets have manufacturing flaws that could potentially cause injury to the wearer. A sample of 200 helmets revealed that 24 helmets contained such defects.(a) Does this finding support the researcher’s claim? Use a fixed-level test with
A manufacturer of interocular lenses is qualifying a new grinding machine. She will qualify the machine if the percentage of polished lenses that contain surface defects does not exceed 4%. A random sample of 300 lenses contains 11 defective lenses.(a) Formulate and test an appropriate set of
An article in Fortune (September 21, 1992) claimed that one-half of all engineers continue academic studies beyond the B.S. degree, ultimately receiving either an M.S. or a Ph.D. degree. Data from an article in Engineering Horizons(Spring 1990) indicated that 117 of 484 new engineering graduates
Consider the defective circuit data and hypotheses in Exercise 4-80.(a) Suppose that the fraction defective is actually p 0.05.What is the -error for this test?(b) Suppose that the manufacturer is willing to accept a-error of 0.10 if the true value of p is 0.05. With 0.05, what sample size
The fraction of defective integrated circuits produced in a photolithography process is being studied. A random sample of 300 circuits is tested, revealing 18 defectives.(a) Use the data to test the hypothesis that the proportion is not 0.04. Use 0.05.(b) Find the P-value for the test.(c) Find
A study is to be conducted of the percentage of homeowners who have a high-speed Internet connection. How large a sample is required if we wish to be 95% confident that the error in estimating this quantity is less than 0.02?
A manufacturer of electronic calculators is interested in estimating the fraction of defective units produced. A random sample of 800 calculators contains 10 defectives.(a) Formulate and test an appropriate hypothesis to determine if the fraction defective exceeds 0.01 at the 0.05 level of
The Arizona Department of Transportation wishes to survey state residents to determine what proportion of the population would be in favor of building a citywide light-rail system. How many residents do they need to survey if they want to be at least 99% confident that the sample proportion is
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) Test the hypotheses H0: p 0.3 versus with 0.05.(b) Find the P-value for this test.(c) Find a 95% two-sided
Large passenger vans are thought to have a high propensity of rollover accidents when fully loaded. Thirty accidents of these vans were examined, and 11 vans had rolled over.(a) Test the claim that the proportion of rollovers exceeds 0.25 with 0.10.(b) Suppose that the true p 0.35 and
Of 1000 randomly selected cases of lung cancer, 823 resulted in death.(a) Test the hypotheses H0: p 0.85 versus with 0.05.(b) Construct a 95% two-sided traditional CI on the death rate from lung cancer.(c) How large a sample would be required to be at least 95%confident that the error in
Consider the following Minitab output.(a) Is this a one-sided or a two-sided test?(b) Was this test conducted using the normal approximation to the binomial? Was that appropriate?(c) Fill in the missing values. Test and CI for One Proportion Test of p = 0.65 vs p > 0.65 95% Lower Sample X N Sample
Consider the Minitab output shown below.(a) Is this a one-sided or a two-sided test?(b) Was this test conducted using the normal approximation to the binomial? Was that appropriate?(c) Can the null hypothesis be rejected at the 0.05 level?(d) Can the null hypothesis versus be rejected at the 0.05
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.(a) Test the hypothesis versus using 0.05. State any necessary assumptions about the underlying distribution of the data.(b) Find the P-value for
Consider the Izod impact test data in Exercise 4-56.(a) Test the hypothesis that against an alternative specifying that using 0.01, and draw a conclusion. State any necessary assumptions about the underlying distribution of the data.2 0.10,2 0.10(b) What is the P-value for this test?(c)
Consider the tire life data in Exercise 4-55.(a) Can you conclude, using 0.05, that the standard deviation of tire life exceeds 3000 km? State any necessary assumptions about the underlying distribution of the data.(b) Find the P-value for this test.(c) Find a 95% lower confidence bound for .(d)
The sugar content of the syrup in canned peaches is normally distributed, and the variance is thought to be 218 (mg)2.(a) Test the hypothesis that the variance is not 18 (mg)2 if a random sample of n 10 cans yields a sample standard deviation of s 4 mg, using a fixed-level test with 0.05.
A rivet is to be inserted into a hole. If the standard deviation of hole diameter exceeds 0.02 mm, there is an unacceptably high probability that the rivet will not fit. A random sample of n 15 parts is selected, and the hole diameter is measured. The sample standard deviation of the hole
Suppose that we are testing H: = versus H: > with a sample size of n = 15. Calculate bounds on the P-value for the following observed values of the test statistic: (a) x=22.35 (c) x = 25.00 (b) x = 23.50 (d) x = 28.55
Cloud seeding has been studied for many decades as a weather modification procedure (for an interesting study of this subject, see the article in Technometrics, “A Bayesian Analysis of a Multiplicative Treatment Effect in Weather Modification,” Vol. 17, 1975, pp. 161–166). The rainfall in
An article in the Journal of Composite Materials (Vol.23, 1989, 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 Hz):230.66, 233.05,
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 fluid ounces and a standard deviation of s 0.016 fluid ounces.(a) Do the data presented in
Measurements on the percentage of enrichment of 12 fuel rods used in a nuclear reactor were reported as follows:3.11 2.88 3.08 3.01 2.84 2.86 3.04 3.09 3.08 2.89 3.12 2.98(a) Use a normal probability plot to check the normality assumption.(b) Test the hypothesis H0: 2.95 versus and draw
The wall thickness of 25 glass 2-liter bottles was measured by a quality-control engineer. The sample mean was mm, and the sample standard deviation was s 0.081 mm.(a) Suppose that it is important to demonstrate that the wall thickness exceeds 4.0 mm. Formulate and test appropriate hypotheses
A machine produces metal rods used in an automobile suspension system. A random sample of 12 rods is selected, and the diameter is measured. The resulting data in millimeters are shown here.8.23 8.31 8.42 8.29 8.19 8.24 8.19 8.29 8.30 8.14 8.32 8.40(a) Check the assumption of normality for rod
In building electronic circuitry, the breakdown voltage of diodes is an important quality characteristic. The breakdown voltage of 12 diodes was recorded as follows: 9.099, 9.174, 9.327, 9.377, 8.471, 9.575, 9.514, 8.928, 8.800, 8.920, 9.913, and 8.306.(a) Check the normality assumption for the
A particular brand of diet margarine was analyzed to determine the level of polyunsaturated fatty acid (in percent).A sample of six packages resulted in the following data: 16.8, 17.2, 17.4, 16.9, 16.5, and 17.1.(a) It is important to determine if the mean is not 17.0. Test an appropriate
The life in hours of a biomedical device under development in the laboratory is known to be approximately normally distributed. A random sample of 15 devices is selected and found to have an average life of 5625.1 hours and a sample standard deviation of 226.1 hours.(a) Test the hypothesis that the
An Izod impact test was performed on 20 specimens of PVC pipe. The ASTM standard for this material requires that Izod impact strength must be greater than 1.0 ft-lb/in. The sample average and standard deviation obtained were and s 0.328, respectively. Test H0: 1.0 versus H1: 1.0 using
A research engineer for a tire manufacturer is investigating tire life for a new rubber compound. She has built 10 tires and tested them to end-of-life in a road test. The sample mean and standard deviation are 61,492 and 3035 kilometers, respectively.(a) The engineer would like to demonstrate that
An article in the ASCE Journal of Energy Engineering(Vol. 125, 1999, pp. 59–75) describes a study of the thermal inertia properties of autoclaved aerated concrete used as a building material. Five samples of the material were tested in a structure, and the average interior temperature reported
An article in Computers in Electrical Engineering(“Parallel Simulation of Cellular Neural Networks,” 1996, Vol. 22, pp. 61–84) considered the speed-up of cellular neural networks (CNN) for a parallel general-purpose computing architecture. The data follow.a) Is there sufficient evidence to
Consider the Minitab output below.(a) How many degrees of freedom are there on the t-test statistic?(b) Fill in the missing information. You may calculate bounds on the P-value. Test of mu = 50 vs not = 50 Variable N Mean StDev SE Mean C1 10 49.30 3.62 ? 95% CI (46.71, 51.89) T P ??
Consider the Minitab output shown below.(a) How many degrees of freedom are there on the t-test statistic?(b) Fill in the missing information. One-Sample T: X Test of mu 25 vs > 25 95% SE Lower Variable N Mean StDev Mean Bound T P X 12 25.6818 ? 0.3360 ? ? 0.034
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?(d) Use the
Suppose that we are testing Η: μ = μ₀ versus Η₁: μ < μ₀, with a sample size of n = 25. Calculate bounds on the P-value for the following observed values of the test statistic:(a) t₀ = -2.59(c) t₀ = -3.05(b) t₀ = -1.76(d) t₀ = -1.30
Suppose that we are testing Η: μ = μ₀ versus H₁: μ ≠ μ₀, with a sample size of n = 10. Calculate bounds on the P-value for the following observed values of the test statistic:(a) t₀ = 2.48(c) t₀ = 2.69(b) t₀ = -3.95(d) t₀ = 1.88(e) t₀ = -1.25
Suppose that we are testing Η:µ = μ₀ versus Η₁:μ > μ₀, with a sample size of n = 15. Calculate bounds on the P-value for the following observed values of the test statistic:(a) t₀ = 2.35(c) t₀ = 2.00(b) t₀ = 3.55(d) t₀ = 1.55
Consider the golf club testing problem from Example 4-7. If the mean coefficient of restitution differs from 0.82 by as much as 0.02, is the sample size n 15 adequate to ensure that H0: 0.82 will be rejected with probability at least 0.8?
Suppose that in Exercise 4-43 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?
Suppose that in Exercise 4-41 we wanted to be 95%confident that the error in estimating the mean distance is less than 0.01 cm. What sample size should we use?
Suppose that in Exercise 4-42 we wanted to be 95%confident that the error in estimating the mean life is less than 5 hours. What sample size should we use?
A civil engineer is analyzing the compressive strength of concrete. Compressive strength is approximately normally distributed with variance 21000 psi2. A random sample of 12 specimens has a mean compressive strength of psi.(a) Test the hypothesis that mean compressive strength is 3500 psi. Use a
The life in hours of a thermocouple used in a furnace is known to be approximately normally distributed, with standard deviation 20 hours. A random sample of 15 thermocouples resulted in the following data: 553, 552, 567, 579, 550, 541, 537, 553, 552, 546, 538, 553, 581, 539, 529.(a) Is there
In the production of airbag inflators for automotive safety systems, a company is interested in ensuring that the mean distance of the foil to the edge of the inflator is at least 2.00 cm. Measurements on 20 inflators yielded an average value of 2.02 cm. Assume a standard deviation of 0.05 on the
Benzene is a toxic chemical used in the manufacturing of medicinal chemicals, dyes, artificial leather, and linoleum. A manufacturer claims that its exit water meets the federal regulation with a mean of less than 7980 ppm of benzene.To assess the benzene content of the exit water, 10 independent
The yield of a chemical process is being studied.From previous experience with this process the standard deviation of yield is known to be 3. The past 5 days of plant operation have resulted in the following yields: 91.6, 88.75, 90.8, 89.95, and 91.3%. Use 0.05.(a) Is there evidence that the mean

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