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applied statistics and multivariate
Applied Statistics And Probability For Engineers 5th Edition Douglas C. Montgomery, George C. Runger - Solutions
Consider the film speed data in Exercise 10-22.(a) Test H0: 2 1 2 2 versus using 0.02.(b) Suppose that one population standard deviation is 50%larger than the other. Is the sample size n1 n2 8 adequate to detect this difference with high probability?Use 0.01 in answering this
Consider the weight of paper data from Technometrics in Exercise 10-28.Is there evidence that the variance of the weight measurement differs between the sheets of paper? Use. Explain how this test can be conducted with a confidence interval.
Reconsider the overall distance data for golf balls in Exercise 10-29.Is there evidence to support the claim that the standard deviation of overall distance is the same for both brands of balls (use 0.05)? Explain how this question can be answered with a 95% confidence interval on . 1 2
Exercise 10-24 presented measurements of plastic coating thickness at two different application temperatures.Test H0: 2 1 2 2 against using 0.01.
Consider the melting-point data in Exercise 10-21.Do the sample data support a claim that both alloys have the same variance of melting point? Use 0.05 in reaching your conclusion.
Consider the gear impact strength data in Exercise 10-20.Is there sufficient evidence to conclude that the variance of impact strength is different for the two suppliers? Use 0.05.
Consider the foam data in Exercise 10-16.Construct the following:(a) A 90% two-sided confidence interval on 2 12 2.(b) A 95% two-sided confidence interval on 2 12 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
A study was performed to determine whether men and women differ in their repeatability in assembling components on printed circuit boards. Random 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 0.98
Consider the hypothesis test H0 :2 1 2 2 against H0 :2 1 2 2.Suppose that the sample sizes are and, and that and . Use . Test the hypothesis and explain how the test could be conducted with a confidence interval on 12.
Consider the hypothesis test H0 :2 1 2 2 against H1 :2 1 2 2.Suppose that the sample sizes are and, and that and . Use . Test the hypothesis and explain how the test could be conducted with a confidence interval on 12.
Repeat Exercise 10-47 using the Wilcoxon signedrank test. State carefully what hypothesis is being tested and how it differs from the one tested in Exercise 10-47.
Use the sign test on the blood cholesterol data in Exercise 10-41.Is there evidence that diet and exercise reduce the median cholesterol level?
The manager of a fleet of automobiles is testing two brands of radial tires. He 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 (in kilometers) follow.Find a 99% confidence interval on the difference in mean
Consider the shear strength experiment described in Example 10-10.(a) Construct a 95% confidence interval on the difference in mean shear strength for the two methods. Is the result you obtained consistent with the findings in Example 10-10?Explain why.(b) Do each of the individual shear strengths
Consider the distance traveled by a golf ball in Exercise 10-29.(a) Use the Wilcoxon rank-sum test to investigate if the means differ. Use 0.05.(b) Use the normal approximation for the Wilcoxon rank-sum test with 0.05. Find the approximate P-value for this test.
Consider the pipe deflection data in Exercise 10-18.(a) Use the Wilcoxon rank-sum test for the pipe deflection temperature experiment. If 0.05, what are your conclusions?(b) Use the normal approximation for the Wilcoxon rank-sum test. Assume that 0.05. Find the approximate P-value for this
The manufacturer of a hot tub is interested in testing two different heating elements for his product. The element that produces the maximum heat gain after 15 minutes would be preferable. He obtains 10 samples of each heating unit and tests each one. The heat gain after 15 minutes (in F)
An electrical engineer must design a circuit to deliver the maximum amount of current to a display tube to achieve sufficient image brightness. Within her allowable design constraints, she has developed two candidate circuits and tests prototypes of each. The resulting data (in microamperes) are as
The overall distance traveled by a golf ball is tested by hitting the ball with Iron Byron, a mechanical golfer with a swing that is said to emulate the legendary champion, Byron Nelson. Ten randomly selected balls of two different brands are tested and the overall distance measured. The data
An article in Technometrics (1999, Vol. 41, pp.202–211) studied the capability of a gauge by measuring the weights of two sheets of paper. The data are shown below.Paper Observations 1 3.481 3.448 3.485 3.475 3.472 3.477 3.472 3.464 3.472 3.470 3.470 3.470 3.477 3.473 3.474 2 3.258 3.254 3.256
An article in Radio Engineering and Electronic Physics (1984, Vol. 29, No. 3, pp. 63–66) investigated the behavior of a stochastic generator in the presence of external noise. The number of periods was measured in a sample of 100 trains for each of two different levels of noise voltage, 100 and
An article in IEEE International Symposium on Electromagnetic Compatibility (2002, Vol. 2, pp. 667–670)quantified the absorption of electromagnetic energy and the resulting thermal effect from cellular phones. The experimental results were obtained from in vivo experiments conducted on rats. The
An article in Electronic Components and Technology Conference (2001, Vol. 52, pp. 1167–1171) compared 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 sample data support
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 an
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 , while for alloy 2 they were and .(a) Do the sample data support the claim that both alloys have the same melting point?
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 s1 12, while another random sample of 16 gears from the second supplier results
In semiconductor manufacturing, wet chemical etching is often used to remove silicon from the backs of wafers prior to metallization. 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 as follows (in F):Type 1: 206, 188, 205, 187, 194, 193, 207, 185, 189, 213, 192, 210, 194, 178, 205
Two catalysts may be used in a batch chemical process. Twelve batches were prepared using catalyst 1, resulting in an average yield of 86 and a sample standard deviation of 3.Fifteen batches were prepared using catalyst 2, and they resulted in an average yield of 89 with a standard deviation of
An article in Fire Technology investigated two different foam expanding agents that can be used in the nozzles of fire-fighting spray equipment. A random sample of five observations with an aqueous film-forming foam (AFFF) had a sample mean of 4.7 and a standard deviation of 0.6. A random sample of
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,
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 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
Explain and use the relationship between confidence intervals and hypothesis tests
Compute power, type II error probability, and make sample size decisions for two-sample tests on means, variances, and proportions
Use the P-value approach for making decisions in hypotheses tests
Test hypotheses and construct confidence intervals on the difference in two population proportions
Test hypotheses and construct confidence intervals on the ratio of the variances or standard deviations of two normal distributions
Test hypotheses and construct confidence intervals on the difference in means of two normal distributions
Structure comparative experiments involving two samples as hypothesis tests
Consider the television picture tube brightness experiment described in Exercise 8-35.(a) For the sample size n 10, do the data support the claim that the standard deviation of current is less than 20 microamps?(b) Suppose that instead of n 10, the sample size was 51.Repeat the analysis
Consider the fatty acid measurements for the diet margarine described in Exercise 8-36.(a) For the 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 was n 51.Repeat the analysis performed in part (a) using
Consider the dissolved oxygen data in Exercise 8-94.Water quality engineers are interested in knowing whether these data support a claim that mean dissolved oxygen concentration is 2.5 milligrams per liter.(a) Formulate an appropriate hypothesis testing procedure to investigate this claim.(b) Test
Consider the golf ball overall distance data in Exercise 6-33.(a) Do these data support the claim that the mean overall distance for this brand of ball does not exceed 270 yards?Use 0.05.(b) What is the P-value for the test in part (a)?(c) Do these data appear to be well modeled by a normal
Consider the 20 observations collected on the number of errors in a string of 1000 bits of a communication channel given in Exercise 6-94.(a) Based on the description of the random variable and these 20 observations, is a binomial distribution an appropriate model? Perform a goodness of fit
Consider the 40 observations collected on the number of nonconforming coil springs in production batches of size 50 given in Exercise 6-93.(a) Based on the description of the random variable and these 40 observations, is a binomial distribution an appropriate model? Perform a goodness of fit
A biotechnology company produces a therapeutic drug whose concentration has a standard deviation of 4 grams per liter. 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
A manufacturer of precision measuring instruments claims that the standard deviation in the use of the instruments is at most 0.00002 millimeter. An analyst, who is unaware of the claim, uses the instrument eight times and obtains a sample standard deviation of 0.00001 millimeter.(a) Confirm using
The mean pull-off force of an adhesive used in manufacturing a connector for an automotive engine application should be at least 75 pounds. This adhesive will be used unless there is strong evidence that the pull-off force does not meet this requirement. A test of an appropriate hypothesis is to be
The marketers of shampoo products know that customers like their product to have a lot of foam. A manufacturer of shampoo claims that the foam height of his product exceeds 200 millimeters. It is known from prior experience that the standard deviation of foam height is 8 millimeters. For each of
Suppose that in Exercise 9-126, 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 milliliters per hour.(a) n 20(b) n 50(c) n 100(d) Comparing your answers to those in Exercise
An 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 milliliters per hour. Based on prior information, the standard deviation of the flow rate is assumed to
Consider the situation of Exercise 9-124.After collecting a sample, we are interested in testing H0: p 0.10 versus with 0.05. For each of the following situations, compute the p-value for this test:(a) n 50, ˆp 0.095(b) n 100, ˆp 0.095(c) n 500, ˆp 0.095(d) n 1000, ˆp
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
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 sample
An article in Fire Technology [“An Experimental Examination of Dead Air Space for Smoke Alarms” (Vol. 45, 2009, pp. 97–115)] studied the performance of smoke detectors installed not less than 100 mm from any adjoining wall if mounted on a flat ceiling, and not closer than 100 mm and not
The diameter of a ball bearing was measured by an inspector using a new type of caliper. The results were as follows (in mm): 0.265, 0.263, 0.266, 0.267, 0.267, 0.265, 0.267, 0.267, 0.265, 0.268, 0.268, and 0.263.(a) Use the Wilcoxon signed-rank test to evaluate the claim that the mean ball
Consider the compressive strength data in Exercise 8-37.(a) Use the sign test to investigate the claim that the median strength is at least 2250 psi. Use 0.05.(b) Use the normal approximation to test the same hypothesis that you formulated in part (a). What is the P-value for this test?
Consider the margarine fat content data in Exercise 8-36.Use the sign test to test versus with 0.05.(a) Find the P-value for the test statistic and use this quantity to make your decision.(b) Use the normal approximation to test the same hypothesis that you formulated in part (a). What is the
The impurity level (in ppm) is routinely measured in an intermediate chemical product. The following data were observed in a recent test:2.4, 2.5, 1.7, 1.6, 1.9, 2.6, 1.3, 1.9, 2.0, 2.5, 2.6, 2.3, 2.0, 1.8, 1.3, 1.7, 2.0, 1.9, 2.3, 1.9, 2.4, 1.6 Can you claim that the median impurity level is less
The titanium content in an aircraft-grade alloy is an important determinant of strength. A sample of 20 test coupons reveals the following titanium content (in percent):8.32, 8.05, 8.93, 8.65, 8.25, 8.46, 8.52, 8.35, 8.36, 8.41, 8.42, 8.30, 8.71, 8.75, 8.60, 8.83, 8.50, 8.38, 8.29, 8.46 The median
Ten samples were taken from a plating bath used in an electronics manufacturing process, and the bath pH was determined. The sample pH values are 7.91, 7.85, 6.82, 8.01, 7.46, 6.95, 7.05, 7.35, 7.25, and 7.42. Manufacturing engineering believes that pH has a median value of 7.0.(a) Do the sample
Reconsider Exercise 6-71.The data were the number of earthquakes per year of magnitude 7.0 and greater since 1900.(a) Use computer software to summarize these data into a frequency distribution. Test the hypothesis that the number of earthquakes of magnitude 7.0 or greater each year follows a
The number of cars passing eastbound through the intersection of Mill and University Avenues has been tabulated by a group of civil engineering students. They have obtained the data in the adjacent table:(a) Does the assumption of a Poisson distribution seem appropriate as a probability model for
Define X as the number of underfilled bottles from a filling operation in a carton of 24 bottles. Seventy-five cartons are inspected and the following observations on X are recorded:Values 0 1 2 3 Frequency 39 23 12 1(a) Based on these 75 observations, is a binomial distribution an appropriate
Let X denote the number of flaws observed on a large coil of galvanized steel. Seventy-five coils are in spected and the following data were observed for the values of X:Values 1 2 3 4 5 6 7 8 Observed Frequency 1 11 8 13 11 12 10 9 (a) Does the assumption of the Poisson distribution seem
Exercise 6-33 describes testing golf balls for an overall distance standard.(a) Can you support a claim that mean distance achieved by this particular golf ball exceeds 280 yards? Use 0.05.Find the P-value.(b) Check the normality assumption.(c) Compute the power of the test if the true mean
Exercise 6-30 gave data on the heights of female engineering students at ASU.(a) Can you support a claim that the mean height of female engineering students at ASU is at least 65 inches? Use 0.05. Find the P-value.(b) Check the normality assumption.(c) Compute the power of the test if the true
Reconsider the television tube brightness experiment in Exercise 8-35.Suppose that the design engineer claims that this tube will require at least 300 microamps of current to produce the desired brightness level. Formulate and test an appropriate hypothesis to confirm this claim using 0.05.
Reconsider the tire testing experiment described in Exercise 8-27.(a) The engineer would like to demonstrate that the mean life of this new tire is in excess of 60,000 kilometers. Formulate and test appropriate hypotheses, and draw conclusions using 0.05.(b) Suppose that if the mean life is as
Consider the dissolved oxygen concentration at TVA dams first presented in Exercise 8-94.(a) Test the hypothesis H0: 4 versus . Use 0.01. Find the P-value.(b) Check the normality assumption.(c) Compute the power of the test if the true mean dissolved oxygen concentration is as low as 3.(d)
Consider the baseball coefficient of restitution data first presented in Exercise 8-92.(a) Do the data support the claim that the mean coefficient of restitution of baseballs exceeds 0.635? Use 0.05. Find the P-value.(b) Check the normality assumption.(c) Compute the power of the test if the
The sodium content of twenty 300-gram boxes of organic cornflakes was determined. The data (in milligrams) are as follows: 131.15, 130.69, 130.91, 129.54, 129.64, 128.77, 130.72, 128.33, 128.24, 129.65, 130.14, 129.29, 128.71, 129.00, 129.39, 130.42, 129.53, 130.12, 129.78, 130.92.(a) Can you
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, pp. 161–166). The rainfall in acre-feet
A 1992 article in the Journal of the American Medical Association (“A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich”) reported body temperature, gender, and heart rate for a number of subjects. The
An article in the ASCE Journal of Energy Engineering (1999, Vol. 125, 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 temperatures (°C)
Medical researchers have developed a new artificial heart constructed primarily of titanium and plastic. The heart will last and operate almost indefinitely once it is implanted in the patient’s body, but the battery pack needs to be recharged about every four hours. A random sample of 50 battery
A bearing used in an automotive application is supposed to have a nominal inside diameter of 1.5 inches. A random sample of 25 bearings is selected and the average inside diameter of these bearings is 1.4975 inches. Bearing diameter is known to be normally distributed with standard deviation 0.01
A melting point test of n 10 samples of a binder used in manufacturing a rocket propellant resulted in Assume that the melting point is normally distributed with .(a) Test H0: 155 versus H1: 155 using 0.01.(b) What is the P-value for this test?(c) What is the -error if the true mean is
The mean water temperature downstream from a power plant cooling tower discharge pipe should be no more than 100°F. Past experience has indicated that the standard deviation of temperature is 2°F. The water temperature is measured on nine randomly chosen days, and the average temperature is found
A hypothesis will be used to test that a population mean equals 10 against the alternative that the population mean is greater than 10 with known variance . What is the critical value for the test statistic Z0 for the following significance levels?(a) 0.01 (b) 0.05 (c) 0.10
State the null and alternative hypothesis in each case.(a) A hypothesis test will be used to potentially provide evidence that the population mean is greater than 10.(b) A hypothesis test will be used to potentially provide evidence that the population mean is not equal to 7.(c) A hypothesis 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 the
A random sample of 500 registered voters in Phoenix is asked if they favor the use of oxygenated fuels year-round to reduce air pollution. If more than 400 voters respond positively, we will conclude that more than 60% of the voters favor the use of these fuels.(a) Find the probability of type I
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 10 residents shows that 1 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 living 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 that
In Exercise 9-20, find the boundary of the critical region if the type I error probability is(a) 0.01 and n 8 (c) 0.01 and n 16(b) 0.05 and n 8 (d) 0.05 and n 16
A manufacturer is interested in the output voltage of a power supply used in a PC. Output voltage is assumed to be normally distributed, with standard deviation 0.25 volt, and the manufacturer wishes to test H0: 5 volts against H1: volts, using n 8 units.(a) The acceptance region is Find the
In Exercise 9-15, calculate the P-value if the observed statistic is(a) = 180 (b) = 190 (c) = 170
Repeat Exercise 9-15 assuming that the sample size is n 16 and the boundary of the critical region is the same.
A consumer products company is formulating a new shampoo and is interested in foam height (in millimeters).Foam height is approximately normally distributed and has a standard deviation of 20 millimeters. The company wishes to test millimeters versus millimeters, using the results of samples.(a)
In Exercise 9-10, calculate the P-value if the observed statistic is(a) = 98 (b) = 101 (c) = 102
In Exercise 9-10, find the boundary of the critical region if the type I error probability is(a) 0.01 and n 9 (c) 0.01 and n 5(b) 0.05 and n 9 (d) 0.05 and n 5
Compute power and type II error probability, and make sample size selection decisions for tests on means, variances, and proportions
Use the P-value approach for making decisions in hypotheses tests
Test hypotheses on a population proportion
Test hypotheses on the variance or standard deviation of a normal distribution
Test hypotheses on the mean of a normal distribution using either a Z-test or a t-test procedure
Structure engineering decision-making problems as hypothesis tests
Suppose that X X X is a random sample from a continuous probability distribution with median. (a) Show that P{min(X) < < max{X}} Hiar: The complement of the event [min(X)
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