Questions and Answers of Applied Statistics and Probability

An experiment was run in a semiconductor fabrication plant in an effort to increase yield. Five factors, each at two levels, were studied. The factors (and levels) were A = aperture setting (small,
An article in Oikos: A Journal of Ecology [€œRegulation of Root Vole Population Dynamics by Food Supply and Predation: A Two-Factor Experiment€ (2005, Vol. 109, pp.
An article in IEEE Transactions on Semiconductor Manufacturing (1992, Vol. 5, pp. 214€“222) described an experiment to investigate the surface charge on a silicon wafer. The factors thought
Four factors are thought to influence the taste of a soft-drink beverage: type of sweetener (A), ratio of syrup to water (B), carbonation level (C), and temperature (D). Each factor can be run at two
An engineer is interested in the effect of cutting speed (A), metal hardness (B), and cutting angle (C) on the life of a cutting tool. Two levels of each factor are chosen, and two replicates of a
The percentage of hardwood concentration in raw pulp, the freeness, and the cooking time of the pulp are being investigated for their effects on the strength of paper. The data from a three-factor
The quality control department of a fabric finishing plant is studying the effects of several factors on dyeing for a blended cotton/synthetic cloth used to manufacture shirts. Three operators, three
An article in Bioresource Technology [€œQuantitative Response of Cell Growth and Tuber Polysaccharides Biosynthesis by Medicinal Mushroom Chinese Truffle Tuber Sinense to Metal Ion in
An article in Journal of Chemical Technology and Biotechnology [“A Study of Antifungal Antibiotic Production by Thermomonospora sp MTCC 3340 Using Full Factorial Design” (2003, Vol. 78, pp.
Consider a two-factor factorial experiment. Develop a formula for finding a 100(1 − α)% confidence interval on the difference between any two means for either a row or column factor. Apply this
An article in the Journal of Testing and Evaluation (1988, Vol. 16, pp. 508€“515) investigated the effects of cyclic loading frequency and environment conditions on fatigue crack growth at
An article in the IEEE Transactions on Electron Devices (November 1986, p. 1754) described a study on the effects of two variables€”polysilicon doping and anneal conditions (time and
Johnson and Leone (Statistics and Experimental Design in Engineering and the Physical Sciences, John Wiley, 1977) described an experiment conducted to investigate warping of copper plates. The two
An article in Technometrics [€œExact Analysis of Means with Unequal Variances€ (2002, Vol. 44, pp. 152€“160)] described the technique of the analysis of means (ANOM)
An experiment was conducted to determine whether either firing temperature or furnace position affects the baked density of a carbon anode. The data are as follows:(a) State the hypotheses of
In the book Design and Analysis of Experiments, 8th edition (2012, John Wiley & Sons), the results of an experiment involving a storage battery used in the launching mechanism of a shoulder-fired
An engineer suspects that the surface finish of metal parts is influenced by the type of paint used and the drying time. He selected three drying times€”20, 25, and 30
An article in Industrial Quality Control (1956) describes an experiment to investigate the effect of two factors (glass type and phosphor type) on the brightness of a television tube. The response
Sample Size Determination. In the single-factor completely randomized design, the accuracy of a100(1 ˆ’ Î±)% confidence interval on the difference in any two treatment means is
Consider the fixed-effects model of the completely randomized single-factor design. The model parameters are restricted by the constraint Î£ai=1Ï„i= 0. (Actually, other
Consider a random-effects model for the singlefactor completely randomized design.(a) Show that a 100(1 ˆ’ Î±)% confidence interval on the ratio Ïƒ2Ï„ /
Consider the random-effects model for the single factor completely randomized design. Show that a 100(1 ˆ’ Î±)% confidence interval on the ratio of variance components
Consider the single-factor completely randomized design. Show that a 100(1 ˆ’ Î±) percent confidence interval for Ïƒ2iswhere N is the total number of observations in
Consider the single-factor completely randomized design with a treatments and n replicates. Show that if the difference between any two treatment means is as large as D, the minimum value that the OC
In a fixed-effects model, suppose that there are n observations for each of four treatments. Let Q21 Q22, and Q23 be single-degree-of-freedom sums of squares for orthogonal contrasts. A contrast
Show that the variance of the linear combination Σςχ. is σΣn i=1 i=1
Consider the ANOVA with a = 2 treatments. Show that the MSE in this analysis is equal to the pooled variance estimate used in the two-sample t-test.
Consider testing the equality of the means of two normal populations for which the variances are unknown but are assumed to be equal. The appropriate test procedure is the two-sample t-test. Show
Show that in the fixed-effects model analysis of variance E (MSE ) = σ2. How would your development change if the random-effects model had been specified?
An article in Journal of Hazardous Materials [€œToxicity Assessment from Electro-Coagulation Treated-Textile Dye Waste Waters by Bioassays,€ 2009, Vol. 172(1)] discussed a study
Reconsider Exercise 13-9 in which the effect of different coating types on conductivity was investigated. Suppose that the five coating types were selected at random from a large number of types.(a)
An article in Bioresource Technology [€œPreliminary Tests on Nisin and Pediocin Production Using Waste Protein Sources: Factorial and Kinetic Studies€ (2006, Vol. 97(4))]
An article in Marine Biology [€œAllozymes an Morphometric Characters of Three Species of Mytilus in the Northern and Southern Hemispheres€ (1991, Vol. 111)] discussed the ratio
An article in Nature Genetics [(2003, Vol. 34(1)], €œTreatment-Specific Changes in Gene Expression Discriminate in vivo Drug Response in Human Leukemia Cells,€ reported the
Consider the following computer output.(a) How many levels of the factor were used in this experiment?(b) How many blocks were used?(c) Fill in the missing information. Use bounds for the P-value.(d)
Consider the following computer output.(a) How many levels of the factor were used in this experiment?(b) How many replicates were used?(c) Fill in the missing information. Use bounds for the
An article in Quality Engineering [€œDesigned Experiment to Stabilize Blood Glucose Levels€ (1999€“2000,)] described an experiment to minimize variations in blood
Reconsider the experiment of Exercise 13-5. Suppose that the experiment was conducted as a RCBD with blocks formed by days (denoted as columns in the data table). In the experiment, the primary
Exercise 13-4 introduced you to an experiment to investigate the potential effect of consuming chocolate on cardiovascular health. The experiment was conducted as a completely randomized design, and
Consider the following computer output from a RCBD. There are four levels of the factor and five blocks.(a) Fill in the missing information. Use bounds for the P-value.(b) What conclusions would you
Consider the following computer output from a RCBD.(a) How many levels of the factor were used in this experiment?(b) How many blocks were used in this experiment?(c) Fill in the missing information.
Reconsider Exercise 13-15 in which the effect of different diets on the protein content of cow€™s milk was investigated. Suppose that the three diets reported were selected at random from a
Reconsider Exercise 13-8 in which the effect of different circuits on the response time was investigated. Suppose that the three circuits were selected at random from a large number of circuits.(a)
Consider the cloth experiment described in Exercise 13-35.(a) Estimate the total variability in the output response.(b) How much of the total variability in the output response is due to the
In the book Bayesian Inference in Statistical Analysis (1973, John Wiley and Sons) by Box and Tiao, the total product yield for five samples was determined randomly selected from each of six randomly
Suppose that five normal populations with common variance σ2 are to be compared with a sample size of seven observations from each. Suppose that τ1 = ⋅ ⋅ ⋅ = τ4 = 0. What is the smallest
Suppose that four normal populations with common variance σ2 are to be compared with a sample size of eight observations from each population. Determine the smallest value for Σ4i=1 τ2i/
Diets in Exercise 13-15. Use Î± = 0.01.(a) Apply Fisher€™s LSD method with Î± = 0.05 and determine which levels of the factor differ.(b) Use the graphical method
Cross-linker Exercise 13-14. Use Î± = 0.05.(a) Apply Fisher€™s LSD method with Î± = 0.05 and determine which levels of the factor differ.(b) Use the graphical
Air voids in Exercise 13-13. Use Î± = 0.05.(a) Apply Fisher€™s LSD method with Î± = 0.05 and determine which levels of the factor differ.(b) Use the graphical
Preparation method in Exercise 13-12. Use Î± = 0.05.(a) Apply Fisher€™s LSD method with Î± = 0.05 and determine which levels of the factor differ.(b) Use the
Coating type in Exercise 13-9. Use Î± = 0.01.(a) Apply Fisher€™s LSD method with Î± = 0.05 and determine which levels of the factor differ.(b) Use the graphical
Circuit type in Exercise 13-8. Use Î± = 0.01.(a) Apply Fisher€™s LSD method with Î± = 0.05 and determine which levels of the factor differ.(b) Use the graphical
Mixing technique in Exercise 13-7. Use Î± = 0.05.(a) Apply Fisher€™s LSD method with Î± = 0.05 and determine which levels of the factor differ.(b) Use the
Flow rate in Exercise 13-6. Use Î± = 0.01.(a) Apply Fisher€™s LSD method with Î± = 0.05 and determine which levels of the factor differ.(b) Use the graphical
Cotton percentage in Exercise 13-5. Use Î± = 0.05.(a) Apply Fisher€™s LSD method with Î± = 0.05 and determine which levels of the factor differ.(b) Use the
Chocolate type in Exercise 13-4. Use α = 0.05.(a) Apply Fisher’s LSD method with α = 0.05 and determine which levels of the factor differ.(b) Use the graphical method to compare means described
An article in Fuel Processing Technology (€œApplication of the Factorial Design of Experiments to Biodiesel Production from Lard,€ 2009, Vol. 90, pp. 1447€“1451)
An article in Scientia Iranica [€œTuning the Parameters of an Artificial Neural Network (ANN) Using Central Composite Design and Genetic Algorithm€ (2011, Vol. 18(6))], described a series of
An article in Scientia Iranica [€œTuning the Parameters of an Artificial Neural Network (ANN) Using Central Composite Design and Genetic Algorithm€ (2011, Vol. 18(6), pp.
An article in Journal of Food Science [2001, Vol. 66(3), pp. 472€“477] reported on a study of potato spoilage based on different conditions of acidified oxine (AO), which is a mixture of
In the book Analysis of Longitudinal Data, 2nd ed. (2002, Oxford University Press), by Diggle, Heagerty, Liang, and Zeger, the authors analyzed the effects of three diets on the protein content of
An article in Quality Engineering [€œEstimating Sources of Variation: A Case Study from Polyurethane Product Research€ (1999€“2000, Vol.)] reported a study on the effects of additives on
An article in Quality Engineering [€œEstimating Sources of Variation: A Case Study from Polyurethane Product Research€ (1999€“2000,)] reported a study on the effects of
An article in Nature describes an experiment to investigate the effect on consuming chocolate on cardiovascular health (“Plasma Antioxidants from Chocolate,” 2003, Vol. 424,). The experiment
Consider the following computer output for an experiment.(a) How many replicates did the experimenter use?(b) Fill in the missing information in the ANOVA table. Use bounds for the P-value.(c) What
Consider the following computer output for an experiment.The factor was tested over four levels.(a) How many replicates did the experimenter use?(b) Fill in the missing information in the ANOVA
Consider the following computer output.(a) How many levels of the factor were used in this experiment?(b) How many replicates did the experimenter use?(c) Fill in the missing information in the ANOVA
Using the data from Exercise 6-22 on cloud seeding,(a) Find the median and quartiles for the unseeded cloud data.(b) Find the median and quartiles for the seeded cloud data.(c) Make two side-by-side
Using the data from Exercise 6-24 on swim times,(a) Find the median and quartiles for the data.(b) Make a box plot of the data.(c) Repeat (a) and (b) for the data without the extreme outlier and
The following data are the temperatures of effluent at discharge from a sewage treatment facility on consecutive days:(a) Calculate the sample mean, sample median, sample variance, and sample
The National Oceanic and Atmospheric Administration provided the monthly absolute estimates of global (land and ocean combined) temperature index (degrees C) from 2000. Read January to December from
It is possible to obtain a “quick-and-dirty” estimate of the mean of a normal distribution from the 50th percentile value on a normal probability plot. Provide an argument why this is so. It is
Construct a normal probability plot of the suspended solids concentration data in Exercise 6-40. Does it seem reasonable to assume that the concentration of suspended solids in water from this
Construct a normal probability plot of the cycles to failure data in Exercise 6-31. Does it seem reasonable to assume that cycles to failure is normally distributed?Data from 6-31 1115 865 1015 885
Construct a normal probability plot of the octane rating data in Exercise 6-30. Does it seem reasonable to assume that octane rating is normally distributed?Data from 6-30 88.5 98.8 89.6 92.2 92.7
Construct a normal probability plot of the O-ring joint temperature data in Exercise 6-19. Does it seem reasonable to assume that O-ring joint temperature is normally distributed? Discuss any
Construct a normal probability plot of the solar intensity data in Exercise 6-12. Does it seem reasonable to assume that solar intensity is normally distributed?Data from 6-12The following data are
Construct a normal probability plot of the visual accommodation data in Exercise 6-11. Does it seem reasonable to assume that visual accommodation is normally distributed?Data from 6-11An article in
Construct a normal probability plot of the insulating fluid breakdown time data in Exercise 6-8. Does it seem reasonable to assume that breakdown time is normally distributed?Data from 6-8In Applied
Construct a normal probability plot of the piston ring diameter data in Exercise 6-7. Does it seem reasonable to assume that piston ring diameter is normally distributed?Data from 6-7Eight
An article in the Journal of Pharmaceuticals Sciences (1991, Vol. 80, pp. 971€“977) presented data on the observed mole fraction solubility of a solute at a constant temperature and the
An article in Technometrics by S. C. Narula and J. F. Wellington [€œPrediction, Linear Regression, and a Minimum Sum of Relative Errors€ (1977, Vol. 19)] presents data on the
Table 6E.6 presents data on the ratings of quarterbacks for the 2008 National Football League season (source: The Sports Network). It is suspected that the rating (y) is related to the average number
Table 6E.5 contains the global mean surface air temperature anomaly and the global CO2concentration for the years 1880€“2004. The temperature is measured at a number of locations around the
Table 6E.3 shows the number of earthquakes per year of magnitude 7.0 and higher since 1900 (source: Earthquake Data Base System of the U.S. Geological Survey, National Earthquake Information Center,
Table 6E.4 shows U.S. petroleum imports as a percentage of the totals, and Persian Gulf imports as a percentage of all imports by year since 1973 (source: U.S. Department of Energy Web site,
The 100 annual Wolfer sunspot numbers from 1770 to 1869 follow. (For an interesting analysis and interpretation of these numbers, see the book by Box, Jenkins, and Reinsel referenced in Exercise
In their book Time Series Analysis, Forecasting, and Control (Prentice Hall, 1994), G. E. P. Box, G. M. Jenkins, and G. C. Reinsel present chemical process concentration readings made every two
The pull-off force for a connector is measured in a laboratory test. Data for 40 test specimens follow (read down, then left to right). Construct and interpret either a digidot plot or a separate
The following data are the viscosity measurements for a chemical product observed hourly (read down, then left to right). Construct and interpret either a digidot plot or a separate stem-and-leaf and
In Exercise 6-69, data were presented on the cold start ignition time of a particular gasoline used in a test vehicle. A second formulation of the gasoline was tested in the same vehicle, with the
Reconsider the semiconductor speed data in Exercise 6-42. Construct a box plot of the data and write an interpretation of the plot. How does the box plot compare in interpretive value to the original
Reconsider the weld strength data in Exercise 6-39. Construct a box plot of the data and write an interpretation of the plot. How does the box plot compare in interpretive value to the original
Reconsider the energy consumption data in Exercise 6-37. Construct a box plot of the data and write an interpretation of the plot. How does the box plot compare in interpretive value to the original
An article in Nature Genetics [€œTreatment-specifi c Changes in Gene Expression Discriminate in Vivo Drug Response in Human Leukemia Cells€ (2003, Vol. 34(1), pp.
Exercise 6-19 presented the joint temperatures of the O-rings (°F) for each test firing or actual launch of the space shuttle rocket motor. In that exercise, you were asked to find the sample mean
The nine measurements that follow are furnace temperatures recorded on successive batches in a semiconductor manufacturing process (units are °F): 953, 950, 948, 955, 951, 949, 957, 954,
An article in Transactions of the Institution of Chemical Engineers (1956, Vol. 34, pp. 280–293) reported data from an experiment investigating the effect of several process variables on the vapor
Construct a histogram for the pinot noir wine rating data in Exercise 6-43. Comment on the shape of the histogram. Does it convey the same information as the stem-andleaf display?Data from 6-43

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