Questions and Answers of Applied Statistics and Probability

Reconsider the hardness measurements in Exercise 15-19. Suppose that the specifications are 45 to 60.(a) Estimate the process standard deviation.(b) Calculate PCR and PCRk for the process.Exercise
Reconsider the syringe lengths in Exercise 15-10. Suppose that the specifications are set at 4.90 and 5.00.(a) Estimate the process standard deviation.(b) Calculate PCR and PCRk for the
A study was performed to determine whether men and women differ in repeatability in assembling components on printed circuit boards. Random samples of 25 men and 21 women were selected, and each
Reconsider the pull-strength measurements in Exercise 15-9. Estimate the fallout level if the specifications are 16 ± 5. Estimate PCR and interpret this ratio.Exercise 15-9The pull strength of
Reconsider the copper-content measurements in Exercise 15-8. Given that the specifications are at 6.0 ± 1.0, estimate PCR and PCRkand interpret these ratios.Exercise 15-8The copper content of
Reconsider the diameter measurements in Exercise 15-7. Use the revised control limits and process estimates.(a) Estimate PCR and PCRk. Interpret these ratios.(b) What percentage of defectives is
Consider the hypothesis test H0: σ21 = σ22 against H1: σ21 ≠ σ22. Suppose that the sample sizes are n1 = 15 and n2 = 15, and the sample variances are s21 = 2.3 and s22 = 1.9. Use α = 0.05.(a)
Consider the hypothesis test H0: σ21 = σ22 against H1: σ21 > σ22. Suppose that the sample sizes are n1 = 20 and n2 = 8, and that s21 = 4.5 and s22 = 2.3. Use α = 0.01. Test the hypothesis and
Reconsider Exercise 15-4(a). Assuming that both charts exhibit statistical control and that the process specifications are at 20 ± 5, estimate PCR and PCRk and interpret these ratios.Data from
Consider the hypothesis test H0: σ21 = σ22 against H1: σ21 < σ22 respectively. Suppose that the sample sizes are n1 = 5 and n2 = 10, and that s21 = 23.2 and s22 = 28.8. Use α = 0.05. Test the
In a series of tests to study the efficacy of ginkgo biloba on memory, Solomon et al. first looked at differences in memory tests of people six weeks before and after joining the study [“Ginkgo for
Neuroscientists conducted research in a Canadian prison to see whether solitary confinement affects brain wave activity [€œChanges in EEG Alpha Frequency and Evoked Response Latency During
Repeat Exercise 10-59 using the Wilcoxon signe drank test. State carefully what hypothesis is being tested and how it differs from the one tested in Exercise 10-59.
Use the sign test on the blood cholesterol data in Exercise 10-53. Is there evidence that diet and exercise reduce the median cholesterol level?Exercise 10-53Fifteen adult males between the ages of
In Biometrics (1990, Vol. 46, pp. 673€“87), the authors analyzed the circumference of five orange trees (labeled as A€“E) measured on seven occasions (xi).(a) Compare the mean
An article in Neurology (1998, Vol. 50, pp. 1246€“1252) discussed that monozygotic twins share numerous physical, psychological, and pathological traits. The investigators measured an
Two different analytical tests can be used to determine the impurity level in steel alloys. Eight specimens are tested using both procedures, and the results are shown in the following tabulation.(a)
Ten individuals have participated in a diet-modification program to stimulate weight loss. Their weight both before and after participation in the program is shown in the following list.(a) Is there
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
The manager of a fleet of automobiles is testing two brands of radial tires and assigns one tire of each brand at random to the two rear wheels of eight cars and runs the cars until the tires wear
A computer scientist is investigating the usefulness of two different design languages in improving programming tasks. Twelve expert programmers who are familiar with both languages are asked to code
Consider the parking data in Example 10-12.(a) Use the paired t-test to investigate the claim that the two types of cars have different levels of difficulty to parallel park. Use α = 0.10.(b)
Consider the shear strength experiment described in Example 10-11.(a) Construct a 95% confidence interval on the difference in mean shear strength for the two methods. Is the result you obtained
Reconsider Exercise 15-3. Suppose that the variable is normally distributed with specifications at 220 ± 50. What is the proportion out of specifications? Estimate and interpret PCR and
Using the data from Exercise 10-37, test the hypotheses using Tukey’s quick test and see whether you reach the same conclusions as you did for Exercise 10-37.Exercise 10-37Olympic swimmers are
Reconsider Exercise 15-2 in which the specification limits are 14.50 ± 0.50.(a) What conclusions can you draw about the ability of the process to operate within these limits? Estimate the
Another nonparametric test known as Tukey’s quick test can be useful with two groups when one group has the minimum value overall (which we call the lower group) and the other (which we call the
Consider the distance traveled by a golf ball in Exercise 10-33.(a) Use the Wilcoxon rank-sum test to investigate if the means differ. Use α = 0.05.(b) Use the normal approximation for the Wilcoxon
Reconsider Exercise 15-1. Suppose that the quality characteristic is normally distributed with specification at 220 ± 40. What is the fallout level? Estimate PCR and PCRk and interpret these
Consider the pipe deflection data in Exercise 10-22.(a) Use the Wilcoxon rank-sum test for the pipe deflection temperature experiment. If α = 0.05, what are your conclusions?(b) Use the normal
Consider the chemical etch rate data in Exercise 10-23.(a) Use the Wilcoxon rank-sum test to investigate the claim that the mean etch rate is the same for both solutions. If α = 0.05, what are your
The manufacturer of a hot tub is interested in testing two different heating elements for its product. The element that produces the maximum heat gain after 15 minutes would be preferable. The
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
An article in Quality Engineering [2012, Vol. 24(1)] described an experiment on a grinding wheel. The following are some of the grinding force data (in N) from this experiment at two different
A normally distributed process uses 85% of the specification band. It is centered at the nominal dimension, located halfway between the upper and lower specification limits.(a) Estimate PCR and PCRk.
A normally distributed process uses 66.7% of the specification band. It is centered at the nominal dimension, located halfway between the upper and lower specification limits.(a) Estimate PCR and
An article in Polymer Degradation and Stability (2006, Vol. 91) presented data from a nine-year aging study on S537 foam. Foam samples were compressed to 50% of their original thickness and stored at
A paper in Quality Engineering [2013, Vol. 25(1)] presented data on cycles to failure of solder joints at different temperatures for different types of printed circuit boards (PCB). Failure data for
Suppose that a quality characteristic is normally distributed with specifications at 100 ± 20. The process standard deviation is 6.(a) Suppose that the process mean is 100. What are the natural
Suppose that a quality characteristic is normally distributed with specifications from 20 to 32 units.(a) What value is needed for σ to achieve a PCR of 1.5?(b) What value for the process mean
Suppose that a quality characteristic is normally distributed with specifications from 10 to 30 units. The process standard deviation is 2 units.(a) Calculate the natural tolerance limits, fraction
Twenty successive hardness measurements are made on a metal alloy, and the data are shown in the following table.(a) Using all the data, compute trial control limits for individual observations and
Olympic swimmers are seeded according to their previous 12-month performances with faster swimmers going into the later heats. The last 24 swimmers, however, are distributed among the last three
An article in Quality & Safety in Health Care [€œStatistical Process Control as a Tool for Research and Healthcare Improvement,€ (2003)Vol. 12, pp. 458€“464]
An X̅ control chart with 3-sigma control limits and subgroup size n = 4 has control limits UCL = 48.75 and LCL = 40.55.(a) Estimate the process standard deviation.(b) Does the response to part (a)
European scientists sampled rivers in various seasons for chemical composition and algae growth (http://archive.ics.uci. edu/ml/datasets/Coil+1999+Competition+Data). The following is a random sample
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 follow.(a) Check the assumption that
Consider the data in Exercise 15-10. Calculate the sample standard deviation of all 100 measurements and compare this result to the estimate of Ïƒ obtained from your revised XÌ… and R charts.
Reconsider the paper towel absorbency data from Exercise 10-13. Find a 95% confidence interval on the difference in the towels’ mean absorbency produced by the two processes. Assume the standard
Consider the data in Exercise 15-9. Calculate the sample standard deviation of all 60 measurements and compare this result to the estimate of Ïƒ obtained from your revised XÌ…
Web traffic can be measured to help highlight security problems or indicate a potential lack of bandwidth. Data on Webtraffic (in thousand hits) fromhttp://en.wikipedia.org/wiki/Web_traffic are given
Apply the Western Electric Rules to the following control chart. The warning limits are shown as dotted lines. Describe any rule violations. 32.5- UCL = 30.82 30.0- 28.21 27,5- 25.60 25.0- X = 22.98
Apply the Western Electric Rules to the following X control chart. The warning limits are shown as dotted lines. Describe any rule violations. UCL=16 14 12 X=10 6. LCL=4 4 10 12 14 16 18 20
The following data were considered in Quality Engineering [€œAn SPC Case Study on Stabilizing Syringe Lengths€ (1999€“2000, Vol. 12(1))]. The syringe length is measured during a
The pull strength of a wire-bonded lead for an integrated circuit is monitored. The following table provides data for 20 samples each of size 3.(a) Use all the data to determine trial control limits
The copper content of a plating bath is measured three times per day, and the results are reported in ppm. The xÌ… and r values for 25 days are shown in the following table:(a) Using all
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
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
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
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. In completely randomized experiment, 11 substrates
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 x̅1 = 420°F and s1 =
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
Consider the hypothesis test H0: μ1 = μ2 against H1: μ1 ≠ μ2. Suppose that sample sizes n1 = 10 and n2 = 10, that x̅1 = 7.8 and x̅2 = 5.6, and that s21 = 4 and s22 = 9. Assume that σ21 =
Consider the hypothesis test H0: μ1 = μ2 against H1: μ1 = μ2. Suppose that sample sizes n1 = 15 and n2 = 15, that x̅1 = 6.2 and x̅2 = 7.8, and that s21 = 4 and s22 = 6.25. Assume that σ21 =
Consider the hypothesis test H0: μ1 = μ2 against H1: μ1 ≠ μ2. Suppose that sample sizes are n1 = 15 and n2 = 15, that x̅1 = 4.7 and x̅2 = 7.8, and that s21 = 4 and s22 = 6.25. Assume that
Consider the computer output below.Difference = mu (1) €“ mu (2)Estimate for difference: €“3.9195% upper bound for difference: ?T-test of difference = 0(vs <): T-value =
Consider the following computer output.Difference = mu (1) €“ mu (2)Estimate for difference: €“1.21095% CI for difference: (€“2.560, 0.140)T-test of difference = 0 (vs
In their book Statistical Thinking (2nd ed.), Roger Hoerl and Ron Snee provide data on the absorbency of paper towels that were produced by two different manufacturing processes. From process 1, the
Reconsider the study described in Exercise 10-10. Suppose that you wanted to detect a true difference in mean force of 0.25 pounds on the hands for these two activities. What level of type II error
Reconsider the data from Exercise 10-10. Find a 95% confidence interval on the difference in mean force on the hands for the two activities. How would you interpret this CI? Is the value zero in the
An article in Industrial Engineer (September 2012) reported on a study of potential sources of injury to equine veterinarians conducted at a university veterinary hospital. Forces on the hand were
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
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
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 σ21 = 1.5, and for formulation, 2
Two types of plastic are suitable for an electronics component manufacturer to use. The breaking strength of this plastic is important. It is known that σ1 = σ2 = 1.0 psi. From a random sample of
Two machines are used for filling plastic bottles with a net volume of 16.0 ounces. The fill volume can be assumed to be normal with standard deviation Ïƒ1= 0.020 and Ïƒ2= 0.025
Consider the hypothesis test H0: μ1 = μ2 against H1: μ1 > μ2 with known variances σ1 = 10 and σ2 = 5. Suppose that sample sizes n1 = 10 and n2 = 15 and that x1 = 24.5 and x2 = 21.3. Use α =
Consider the hypothesis test H0: μ1 = μ2 against H1: μ1 ≠ μ2 with known variances σ1 = 10 and σ2 = 5. Suppose that sample sizes n1 = 10 and n2 = 15 and that x̅1 = 4.7 and x̅2 = 7.8. Use α
Let X1, X2, …, Xn be a sample from an exponential distribution with parameter λ. It can be shown that 2λ Σni=1 Xi has a chi-square distribution with 2n degrees of freedom. Use this fact to
When X1, X2, €¦, Xnis a random sample from a normal distribution and n is large, the sample standard deviation has approximately a normal distribution with mean Ïƒ and variance
When X1, X2, €¦, Xnare independent Poisson random variables, each with parameter Î», and n is large, the sample mean X has an approximate normal distribution with mean
Derive an expression for β for the test on the variance of a normal distribution. Assume that the two-sided alternative is specified.
Suppose that we wish to test H0: μ = μ0 versus H1: μ ≠ μ0 where the population is normal with known σ. Let 0 < e < α, and define the critical region so that we will reject H0 if z0 >
Suppose that eight sets of hypotheses of the form H0: μ = μ0 H1: μ ≠ μ0 have been tested and that the P-values for these tests are 0.15, 0.06. 0.67, 0.01, 0.04, 0.08, 0.78, and 0.13. Use
A manufacturer of a pharmaceutical product is developing a generic drug and must show its the equivalence to the current product. The variable of interest is the activity level of the active
An article in the Journal of Electronic Material [“Progress in CdZnTe Substrate Producibility and Critical Drive of IRFPA Yield Originating with CdZnTe Substrates” (1998, Vol. 27(6), pp.
An article in Experimental Brain Research [€œSynapses in the Granule Cell Layer of the Rat Dentate Gyrus: Serial- Sectionin Study€ (1996, Vol. 112(2), pp. 237€“243)]
An article in Biological Trace Element Research [€œInteraction of Dietary Calcium, Manganese, and Manganese Source (Mn Oxide or Mn Methionine Complex) or Chick Performance and Manganese
An article in Food Chemistry [€œA Study of Factors Affecting Extraction of Peanut (Arachis Hypgaea L.) Solids with Water€ (1991, Vol. 42(2), pp. 153€“165)] reported
Consider the computer output belowUsing the normal approximation:(a) Fill in the missing information.(b) What are your conclusions if Î± = 0.05?(c) The normal approximation to the
An article in Food Testing and Analysis [€œImproving Reproducibility of Refractometry Measurements of Fruit Juices€ (1999, Vol. 4(4), pp. 13€“17)] measured the sugar
Consider the situation of Exercise 9-144. After collecting a sample, we are interested in testing H0: p = 0.10 versus H1: p ≠ 0.10 with α = 0.05. For each of the following situations, compute the
An article in Fire Technology [“An Experimental Examination of Dead Air Space for Smoke Alarms” (2009, Vol. 45, pp. 97–115)] studied the performance of smoke detectors installed not less than
An article in Transfusion Science [“Early Total White Blood Cell Recovery Is a Predictor of Low Number of Apheresis and Good CD34+ Cell Yield” (2000, Vol. 23, pp. 91–100)] studied the white
Consider the following computer output.(a) How many degrees of freedom are there on the t-statistic?(b) Fill in the missing information. You may use bounds on the P-value.(c) What are your
Consider the following computer output.(a) How many degrees of freedom are there on the t-statistic?(b) Fill in the missing information. You may use bounds on the P-value.(c) What are your
Consider the following computer output.(a) Fill in the missing information.(b) Is this a one-sided or a two-sided test?(c) What are your conclusions if Î± = 0.05?(d) Find a 95% two-sided
The mean weight of a package of frozen fish must equal 22 oz. Five independent samples were selected, and the statistical hypotheses about the mean weight were tested. The P-values that resulted from
The standard deviation of fill volume of a container of a pharmaceutical product must be less than 0.2 oz to ensure that the container is accurately filled. Six independent samples were selected, and
Suppose that eight sets of hypotheses about a population proportion of the form H0: p = 0.3 H1: p > 0.3 have been tested and that the P-values for these tests are 0.15, 0.83, 0.103, 0.024, 0.03,

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