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modern mathematical statistics with applications
Engineering Statistics 2nd Edition Douglas C. Montgomery, George C. Runger, Norma F. Hubele - Solutions
An article in the Materials Research Bulletin (Vol.26, No. 11, 1991) reported a study of four different methods of preparing the superconducting compound PbMo6S8. The authors contend that the presence of oxygen during the preparation process affects the material’s superconducting transition
Two different gauges can be used to measure the depth of bath material in a Hall cell used in smelting aluminum. Each gauge is used once in 15 cells by the same operator. Depth measurements from both gauges for 15 cells are shown below.(a) State any assumptions necessary to test the claim that both
A Rockwell hardness-testing machine presses a tip into a test coupon and uses the depth of the resulting depression to indicate hardness. Two different tips are being compared to determine whether they provide the same Rockwell C-scale hardness readings. Nine coupons are tested, with both tips
A materials engineer performs an experiment to investigate whether there is a difference among five types of foam pads used under carpeting. A mechanical device is constructed to simulate “walkers” on the pad, and four samples of each pad are randomly tested on the simulator. After a certain
A manufacturer of heart pacemakers is investigating changing the casing material to decrease the weight of the device.Three different alloys are being considered. Eight prototype parts are made from each alloy material and weighed, in grams. The data are compiled into the following partially
Consider the fire-fighting foam expanding agents investigated in Exercise 5-20, in which five observations of each agent were recorded. Suppose that, if agent 1 produces a mean expansion that differs from the mean expansion of agent 2 by 1.5, we would like to reject the null hypothesis with
An article in the Journal of the Environmental Engineering Division (“Distribution of Toxic Substances in Rivers,” Vol. 108, 1982, pp. 639–649) describes a study of the concentration of several hydrophobic organic substances in the Wolf River in Tennessee. Measurements of
An article in Neurology (Vol. 50, 1998, pp. 1246–1252)describes a finding that monozygotic twins share numerous physical, psychological, and pathological traits. The investigators measured an intelligence score of 10 pairs of twins, and the data are as follows:(a) Is the assumption that the
A fuel-economy study was conducted for two German automobiles, Mercedes and Volkswagen. One vehicle of each brand was selected, and the mileage performance was observed for 10 tanks of fuel in each car. The data are as follows (in mpg):(a) Construct a normal probability plot of each of the data
Suppose that we are testing H0: 1 2 versus H1:1 2, and we plan to use equal sample sizes from the two populations. Both populations are assumed to be normal with unknown but equal variances. If we use 0.05 and if the true mean 12 , what sample size must be used for the power of this
A manufacturer of a new pain relief tablet would like to demonstrate that its product works twice as fast as the competior’s product. Specifically, it would like to testwhere 1 is the mean absorption time of the competitive product and 2 is the mean absorption time of the new product.Assuming
Consider Example 5-12 in the text.(a) Redefine the parameters of interest to be the proportion of lenses that are unsatisfactory following tumble polishing with polishing fluids 1 or 2. Test the hypothesis that the two polishing solutions give different results using 0.01.(b) Compare your answer
Consider the previous exercise, which summarized data collected from drivers about their seat belt usage.(a) Do you think there is a reason not to believe these data?Explain your answer.(b) Is it reasonable to use the hypothesis testing results from the previous problem to draw an inference about
In a random sample of 200 Phoenix residents who drive a domestic car, 165 reported wearing their seat belt regularly, whereas another sample of 250 Phoenix residents who drive a foreign car revealed 198 who regularly wore their seat belt.(a) Perform a hypothesis testing procedure to determine
A study was carried out to determine the accuracy of Medicaid claims. In a sample of 1095 physician-filed claims, 942 claims exactly matched information in medical records. In a sample of 1042 hospital-filed claims, the corresponding number was 850.(a) Is there a difference in the accuracy between
The Salk polio vaccine experiment in 1954 focused on the effectiveness of the vaccine in combating paralytic polio. Because it was felt that without a control group of children there would be no sound basis for evaluating the efficacy of the Salk vaccine, the vaccine was administered to one group,
Consider the previous exercise. Suppose that prior to collecting the data, you decide that you want the error in estimating12 by to be less than 1.5 psi. Specify the sample size for the following percentage confidence:(a) 90%(b) 98%(c) Comment on the effect of increasing the percentage
The breaking strength of yarn supplied by two manufacturers is being investigated. We know from experience with the manufacturers’ processes that 1 5 and 2 4 psi. A random sample of 20 test specimens from each manufacturer results in and psi, respectively.(a) Using a 90% CI on the
A liquid dietary product implies in its advertising that use of the product for 1 month results in an average weight loss of at least 3 pounds. Eight subjects use the product for 1 month, and the resulting weight loss data are reported here.Use hypothesis testing procedures to answer the following
A manufacturing company uses a screen printing process to deposit ink on thin plastic substrates. The thickness of the deposit is a critical quality characteristic. A new automated ink depositing system has been added to reduce the variability in the thickness of the deposit. Weight measurements in
An article in the Journal of Materials Engineering(Vol 11, No. 4, 1989, pp. 275–282) reported the results of an experiment to determine failure mechanisms for plasma-sprayed thermal barrier coatings. The failure stress for one particular coating(NiCrAlZr) under two different test conditions is as
A procurement specialist has purchased 25 resistors from vendor 1 and 35 resistors from vendor 2. Each resistor’s resistance is measured with the following results.(a) What distributional assumption is needed to test the claim that the variance of resistance of product from vendor 1 is not
An experiment was conducted to investigate leaking current in a near-micrometer SOS MOSFETS device. The purpose of the experiment was to investigate how leakage current varies as the channel length changes. Four channel lengths were selected. For each channel length, five different widths were also
An article in the Food Technology Journal (Vol. 10, 1956, pp. 39–42) describes a study on the protopectin content of tomatoes during storage. Four storage times were selected, and samples from nine lots of tomatoes were analyzed. The protopectin content (expressed as hydrochloric acid-soluble
An article in the American Industrial Hygiene Association Journal (Vol. 37, 1976, pp. 418–422) describes a field test for detecting the presence of arsenic in urine samples. The test has been proposed for use among forestry workers because of the increasing use of organic arsenics in that
In Design and Analysis of Experiments, 7th edition(John Wiley & Sons, 2009), D. C. Montgomery describes an experiment that determined the effect of four different types of tips in a hardness tester on the observed hardness of a metal alloy. Four specimens of the alloy were obtained, and each
In “The Effect of Nozzle Design on the Stability and Performance of Turbulent Water Jets” (Fire Safety Journal, Vol. 4, August 1981), C. Theobald describes an experiment inwhich a shape measurement was determined for several different nozzle types at different levels of jet efflux
Consider the Minitab ANOVA output from the completely randomized single-factor experiment shown in Exercise 5-75. Suppose that this experiment had been conducted in a randomized complete block design and that the sum of squares for blocks was 80.00. Modify the ANOVA table to show the correct
An electronics engineer is interested in the effect on tube conductivity of five different types of coating for cathode ray tubes in a telecommunications system display device. The following conductivity data are obtained. If 0.05, can you isolate any differences in mean conductivity due to the
A study was carried out to determine if curing temperature significantly affects the tensile strength of silicone rubber.An axially controlled automated hydraulic force applicator was used to measure the tensile strength (in megapascals, MPa) of each of the specimens. The results are given
An experiment was run to determine whether four specific firing temperatures affect the density of a certain type of brick. The experiment led to the following data.(a) Does the firing temperature affect the density of the bricks?(b) Find the P-value for the F-statistic computed in part (a).
In Design and Analysis of Experiments, 7th edition(John Wiley & Sons, 2009), D. C. Montgomery describes an experiment in which the tensile strength of a synthetic fiber is of interest to the manufacturer. It is suspected that strength is related to the percentage of cotton in the fiber. Five
In “Orthogonal Design for Process Optimization and Its Application to Plasma Etching” (Solid State Technology, May 1987), G. Z. Yin and D. W. Jillie describe an experiment to determine the effect of C2F6 flow rate on the uniformity of the etch on a silicon wafer used in integrated circuit
The rollover rate of sport utility vehicles is a transportation safety issue. Safety advocates claim that manufacturer A’s vehicle has a higher rollover rate than that of manufacturer B.One hundred crashes for each of these vehicles were examined.The rollover rates were pA 0.35 and pB
Consider the situation described in Exercise 5-66.Suppose that p1 0.05 and p2 0.02.(a) With the sample sizes given here, what is the power of the test for this two-sided alternative?(b) Determine the sample size needed to detect this difference with a probability of at least 0.9. Use 0.05.
Consider the situation described in Exercise 5-66.Suppose that p1 0.05 and p2 0.01.(a) With the sample sizes given here, what is the power of the test for this two-sided alternative?(b) Determine the sample size needed to detect this difference with a probability of at least 0.9. Use 0.05.
Two different types of injection-molding machines are used to form plastic parts. A part is considered defective if it has excessive shrinkage or is discolored. Two random samples, each of size 300, are selected, and 15 defective parts are found in the sample from machine 1 whereas 8 defective
Fill in the blanks in the Minitab output shown below.(a) Is this a one-sided or a two-sided test?(b) Can the null hypothesis be rejected at the 0.05 level?(c) Can the null hypothesis versus be rejected at the 0.05 level? How can you do this without performing any additional calculations?(d)
Fill in the blanks in the Minitab output shown below(a) Is this a one-sided or a two-sided test?(b) Can the null hypothesis be rejected at the 0.05 level?(c) Can the null hypothesis versus be rejected at the 0.05 level? How can you do this without performing any additional calculations?(d) Can the
A study was performed to determine whether men and women differ in their repeatability in assembling components on printed circuit boards. Two samples of 25 men and 21 women were selected, and each subject assembled the units.The two sample standard deviations of assembly time were smen 0.914 min
Consider the foam data in Exercise 5-20. Construct the following:(a) A 90% two-sided CI on 1 22 2.(b) A 95% two-sided CI on 1 22 2. Comment on the comparison of the width of this interval with the width of the interval in part (a).(c) A 90% lower-confidence bound on 12.
Eleven resilient modulus observations of a ceramic mixture of type A are measured and found to have a sample average of 18.42 psi and sample standard deviation of 2.77 psi.Ten resilient modulus observations of a ceramic mixture of type B are measured and found to have a sample average of 19.28 psi
An experiment was conducted to compare the variances of two independent normal populations. The null hypothesis was versus The sample sizes from both populations were 16, and the computed value of the F-statistic was Find a bound on the P-value for this test statistic.
An experiment was conducted to compare the variances of two independent normal populations. The sample sizes from both populations was 10 and the computed value of the F-statistic was $f_o = 4.45$. Find a bound on the P-value for this test statistic.
A consumer products company uses direct-mail marketing to promote its products. To evaluate the effectiveness of two versions of a marketing brochure the company decides to test the two designs by mailing 5000 samples of each to randomly selected potential customers in four different regions of the
An article in the journal Biometrics (1990, Vol. 46, pp. 673–687) analyzed the circumference of orange trees. The measurements were made at two different points in time. Data for five trees follow.(a) Test the hypothesis that the mean increase in circumference between the two time periods exceeds
The Federal Aviation Administration requires material used to make evacuation systems retain their strength over the life of the aircraft. In an accelerated life test, the principal material, polymer-coated nylon weave, is aged by exposing it to 158 F for 168 hours. The tensile strength of the
An article in the Journal of Aircraft (Vol. 23, 1986, pp. 859–864) describes a new equivalent plate analysis method formulation that is capable of modeling aircraft structures such as cranked wing boxes and that produces results similar to the more computationally intensive finite element
Fifteen adult males between the ages of 35 and 50 participated in a study to evaluate the effect of diet and exercise on blood cholesterol levels. The total cholesterol was measured in each subject initially, and then 3 months after participating in an aerobic exercise program and switching to a
A computer scientist is investigating the usefulness of two different design languages in improving programming tasks. Twelve expert programmers, familiar with both languages, are asked to code a standard function in both languages, and the time in minutes is recorded. The data are shown here(a)
The manager of a fleet of automobiles is testing two brands of radial tires. She assigns one tire of each brand at random to the two rear wheels of eight cars and runs the cars until the tires wear out. The data are shown here (in kilometers). Find a 99% CI on the difference in mean life. Which
Reconsider the parking data in Example 5-9. Investigate the assumption that the differences in parking times are normally distributed.
Consider the parking data in Example 5-9. Use the paired t-test to investigate the claim that the two types of cars have different levels of difficulty to parallel park. Use 0.10.Compare your results with the confidence interval constructed in Example 5-9 and comment on why they are the same or
Reconsider the shear strength experiment described in Example 5-8. Does each of the individual shear strengths have to be normally distributed for the paired t-test to be appropriate, or is it only the difference in shear strengths that must be normal? Use a normal probability plot to investigate
Consider the shear strength experiment described in Example 5-8. Construct a 95% CI on the difference in mean shear strength for the two methods. Is the result you obtained consistent with the findings in Example 5-8? Explain why.
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) Use the output and the t table to find a 99% two-sided CI on the difference in means.(d) What is the
Consider the Minitab output shown on the right.(a) Fill in the missing values in the output, including a bound on the P-value. Can the null hypothesis be rejected at the 0.05 level? Why?(b) Is this a one-sided or a two-sided test?(c) Use the output and the t table to find a 99% two-sided CI on the
A pharmaceutical company is investigating the bioactivity of a new drug. Two dosage levels are used and the bioactivity measured in random samples as shown:(a) Is there evidence to support the claim that higher dosage levels lead to greater bioactivity?(b) Construct a 95% one-sided lower confidence
A regional opera company has tried two different approaches for soliciting donations from 16 potential sponsors.The 16 potential sponsors were selected at random from a population of potential sponsors and randomly divided into two groups of eight each. Then one of the approaches was used for each
Reconsider the coating thickness experiment in Exercise 5-29. Answer the question posed regarding the effect of temperature on coating thickness by using a CI. Explain your answer.
Reconsider the abrasive wear test in Exercise 5-28.Construct a CI that will address the questions in parts (a) and(b) in that exercise.
The thickness of a plastic film (in mils) on a substrate material is thought to be influenced by the temperature at which the coating is applied. A completely randomized experiment is carried out. Eleven substrates are coated at 125F, resulting in a sample mean coating thickness of and a sample
Two companies manufacture a rubber material intended for use in an automotive application. The part will be subjected to abrasive wear in the field application, so we decide to compare the material produced by each company in a test. Twenty-five samples of material from each company are tested in
Referring to the melting point experiment in Exercise 5-26, suppose that the true mean difference in melting points is 3F. How large a sample would be required to detect this difference using an 0.05 level test with probability at least 0.9? Use 1 2 4 as an initial estimate of the common
The melting points of two alloys used in formulating solder were investigated by melting 21 samples of each material.The sample mean and standard deviation for alloy 1 was and and for alloy 2 they were and(a) Do the sample data support the claim that both alloys have the same melting point? Use a
A photoconductor film is manufactured at a nominal thickness of 25 mils. The product engineer wishes to decrease the energy absorption of the film, and he believes this can be achieved by reducing the thickness of the film to 20 mils.Eight samples of each film thickness are manufactured in a pilot
Two suppliers manufacture a plastic gear used in a laser printer. The impact strength of these gears measured in foot-pounds is an important characteristic. A random sample of 10 gears from supplier 1 results in and and another random sample of 16 gears from the second supplier results in and(a) Is
In semiconductor manufacturing, wet chemical etching is often used to remove silicon from the backs of wafers prior to metalization. The etch rate is an important characteristic in this process and known to follow a normal distribution.Two different etching solutions have been compared, using two
The deflection temperature under load for two different types of plastic pipe is being investigated. Two random samples of 15 pipe specimens are tested, and the deflection temperatures observed are reported here (in F):(a) Construct box plots and normal probability plots for the two samples. Do
A consumer organization collected data on two types of automobile batteries, A and B. The summary statistics for 12 observations of each type are Assume that the data are normally distributed with A B.(a) Is there evidence to support the claim that type A battery mean life exceeds that of type
An article in Fire Technology describes the investigation of two different foam expanding agents that can be used in the nozzles of firefighting spray equipment. A random sample of five observations with an aqueous film-forming foam(AFFF) had a sample mean of 4.340 and a standard deviation of
The diameter of steel rods manufactured on two different extrusion machines is being investigated. Two random samples of sizes n115 and n217 are selected, and the sample means and sample variances are s1 2 0.35, and s2 2 0.40, respectively. Assume that 2 1 2 2and that the data are drawn
An article in IEEE International Symposium on Electromagnetic Compatibility (Vol. 2, 2002, pp. 667–670)describes the quantification of the absorption of electromagnetic energy and the resulting thermal effect from cellular phones. The experimental results were obtained from in vivo experiments
An article in Electronic Components and Technology Conference (Vol. 52, 2001, pp. 1167–1171) describes a study comparing single versus dual spindle saw processes for copper metallized wafers. A total of 15 devices of each type were measured for the width of the backside chipouts, and(a) Do the
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) Use the output and the t table to find a 99% one-sided upper confidence bound on the difference in
Consider the Minitab output shown below.(a) 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 Hou - = 2 versus H+2, would you reject the null hypothesis at the 0.05 level? (d) If the hypotheses had been Ho - P = 2
For the laundry detergent problem in Exercise 5-12, test the hypothesis that the mean active concentrations are the same for both types of catalyst. What is the P-value for this test? What are your conclusions? Compare your answer to that found in part (b) of Exercise 5-12, and comment on why they
Consider the polymer batch viscosity data in Exercise 5-11. If the difference in mean batch viscosity is 10 or less, the manufacturer would like to detect it with a high probability.(a) Formulate and test an appropriate hypothesis using the P-value approach. What are your conclusions?(b) Compare
The concentration of active ingredient in a liquid laundry detergent is thought to be affected by the type of catalyst used in the process. The standard deviation of active concentration is known to be 3 grams per liter, regardless of the catalyst type. Ten observations on concentration are taken
A polymer is manufactured in a batch chemical process. Viscosity measurements are normally made on each batch, and long experience with the process has indicated that the variability in the process is fairly stable with 20.Fifteen batch viscosity measurements are given as follows:724, 718, 776,
Consider the road octane test situation described in Exercise 5-8. What sample size would be required in each population if we wanted to be 95% confident that the error in estimating the difference in mean road octane number is less than 1?
Consider the situation described in Exercise 5-5. What sample size would be required in each population if we wanted the error in estimating the difference in mean burning rates to be less than 4 cm/s with 99% confidence?
Two different formulations of an oxygenated motor fuel are being tested to study their road octane numbers. The variance of road octane number for formulation 1 is 2 1 1.5, and for formulation 2 it is 2 2 1.2. Two random samples of size n1 15 and n2 20 are tested, and the mean road octane
Reconsider the situation described in Exercise 5-6.(a) Test the hypothesis that both machines fill to the same mean volume. Use the P-value approach.(b) If 0.05 and the -error of the test when the true difference in fill volume is 0.2 fluid ounces should not exceed 0.1, what sample sizes must be
Two machines are used to fill plastic bottles with dishwashing detergent. The standard deviations of fill volume are known to be 1 0.10 and 2 0.15 fluid ounces for the two machines, respectively. Two random samples of n1 12 bottles from machine 1 and n2 10 bottles from machine 2 are
The burning rates of two different solid-fuel propellants used in aircrew escape systems are being studied. It is known that both propellants have approximately the same standard deviation of burning rate; that is, 1 2 3 cm/s.Two random samples of n1 20 and n2 20 specimens are tested; the
Two types of plastic are suitable for use by an electronics component manufacturer. The breaking strength of this plastic is important. It is known that 12 1.0 psi. From a random sample of size n1 10 and n2 12, we obtain and The company will not adopt plastic 1 unless its mean breaking
Two machines are used for filling plastic bottles with a net volume of 16.0 ounces. The fill volume can be assumed normal, with standard deviation 1 0.020 and 2 0.025 ounces. A member of the quality engineering staff suspects that both machines fill to the same mean net volume, whether or not
A computer program has produced the following output for a hypothesis testing problem:Difference in sample means: 11.5 Standard error of the difference in sample means: ?Test statistic: z01.88 P-value: 0.0601(a) What is the missing value for the standard error?(b) Is this a two-sided or a
A computer program has produced the following output for a hypothesis testing problem:Difference in sample means: 2.35 Standard error of the difference in sample means: ?Test statistic: z0 2.01 P-value: 0.0222(a) What is the missing value for the standard error?(b) Is this a two-sided or a
Repeat Exercise 4-146 for sample sizes n 10, 15, and 25. How does the coverage of both CIs change with sample size?
This is a simulation exercise designed to illustrate the robustness of the t-test to the assumption of normality.(a) Generate 100,000 samples of size n 5 from a standard normal distribution ( ). For each sample find the 95% two-sided t CI on the mean. For each sample determine if the CI includes
Consider the experimental data collected to verify that the “true” speed of light is 710.5 (299,710.5 kilometers per second) in 1879 and in 1882 by the physicist A. A. Michelson. The data are shown in Exercise 2-68.Read the story associated with the data and reported on the Web site
Identify an example in which a standard is specified or claim is made about a population. For example,“This type of car gets an average of 30 miles per gallon in urban driving.” The standard or claim may be expressed as a mean (average), variance, standard deviation, or proportion.Collect an
A detection device is used to monitor the level of CO in the air. The following data, in ppm, were collected from a single location:7.28 6.98 8.50 6.33 5.56 7.34 3.18 5.56 4.03 4.69(a) Assess the assumption that the data are normally distributed.(b) There is concern that the device has significant
The following are recorded times, in hours, until failure of medical linear accelerators:953 1037 1068 1032 988 1014 1063 1000 983 942 945 921 915 921 1090 974 997 993 997 984(a) Assess the assumption that the data are normally distributed.(b) Assess the claim that the mean time until failure is
The thickness measurements, in millimeters, of a wall of plastic tubing used for intravenous bags were recorded as follows:1.9976 2.0008 2.0021 1.9995 2.0004 1.9972 1.9974 1.9989 2.0017 2.0030 1.9979 2.0035 1.9997 2.0014 2.0017 2.0018(a) Assess the assumption that the data are normally
Reconsider the data of Exercise 4-131. Find a 90%two-sided Agresti-Coull CI and compare it to the traditional CI in Exercise 4-131 part (a).
Reconsider the data in Exercise 4-130. Find a 99%two-sided Agresti-Coull CI and compare it to the traditional CI in Exercise 4-130.
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) Can the null hypothesis be rejected at the 0.05 level? At the 0.01 level?(d) Can the null hypothesis versus be
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