Solution Manual For Applied Statistics For Engineers And Physical Scientists, Coursesmart Etextbook, 3rd Edition

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1SOLUTIONS MANUALTO ACCOMPANYLEDOLTER AND HOGG:APPLIED STATISTICS FOR ENGINEERS AND PHYSICAL SCIENTISTSJOHANNES LEDOLTERUniversity of IowaCONTENTS1. INTRODUCTIONpage22. LISTING OF DATA FILES STORED ON WEBSITESpage33. A BRIEF INTRODUCTION TO MINITABpage64. A BRIEF INTRODUCTION TO Rpage175. SOLUTIONS TO PROBLEMSpage29Chapter 1page29Chapter 2page69Chapter 3page88Chapter 4page 118Chapter 5page 170Chapter 6page 193Chapter 7page 216Chapter 8page 254Acknowledgments:I would like to thank Professor Erdogan Gunel of the Department ofStatistics at West Virginia University and my son Thomas Ledolter, an engineeringstudent at Northwestern University, for the careful checking of the solutions. My

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1. Introduction21. INTRODUCTIONIn preparing this instructor’s manual, we have selected about eighty-five percent of theexercises and projects. Along with the answers and the "show that" type problems in thetext, this manual provides the reader with answers to most of the exercises and projects.The data sets used in this book are availableat www.pearsonhighered.com/datasetsas wellas on Ledolter’s website, www.biz.uiowa.edu/faculty/jledolter/AppliedStatistics. Asummary and a brief description of the stored files are given in part 2 of this manual. Thenames of the files refer to the section and the location where the data are first used. Forexample, Section1.5Table1.5-1Cars refers to the data in Table 1.5-1 of Section 1.5, andthe data set refers to cars. We have stored two versions of each file: A Minitab version(with file extension .MTW) and a text version (with file extension .TXT).MINITAB, a popular statistical computer software, is used for most of the data analysisin this book. A short description and introduction to MINITAB are given in part 3 of thismanual. Other software programs such as SAS, SPSS, and JMP could be used as well.Since these programs work on spreadsheets, using one or the other package should notcause any difficulties.Part 4 of this manual gives an introduction to the R Project for Statistical Computing. TheR approach to statistical computing is somewhat different from the spreadsheet approachof Minitab, as R analyses are executed through simple R-language instructions. Thereasons for including R are two-fold: (1) The program is free. It can be downloaded,without charge, from the R website http://www.r-project.org. (2) The R statisticalcomputing environment is very flexible and more comprehensive than that ofcommercially available software programs; R software is developed and maintained bymathematicians and statisticians who are experts in their areas of specialization. Mostengineering students are also familiar with programming in Matlab, and hence the switchto the R language should cause few difficulties.Part 5 of this manual includes the detailed solutions to most of the exercises and projects.

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2. Listing of Data Files32. LISTING OF DATA FILES STORED ON WEBSITESThere are two versions of each listed file: A Minitab worksheet (with extension .MTW)and a textfile (with extension .TXT).Chapter 1Section1.2Exercise1.2-1ThermostatSection1.2TemperaturesSection1.3Exercise1.3-2MeltingPointSection1.3Exercise1.3-3Lead1976Section1.3Exercise1.3-8ThicknessSection1.3Exercise1.3-13HurricaneSection1.3Exercise1.3-14BatchYieldSection1.3Exercise1.3-18SurveySection1.3Table1.3-1StrengthSection1.3TestScoresN=58Section1.3TestScoresN=44Section1.4Exercise1.4-3JaffeSection1.4Exercise1.4-6TwoProcessesSection1.4Table1.4-1MisfeedingLeadsSection1.4LakeNeusiedlSection1.4Lead1976&1977Section1.5Exercise1.5-FisherSection1.5Exercise1.5-5TukeySection1.5Exercise1.5-6AirPollutionSection1.5Exercise1.5-7TimeSection1.5Exercise1.5-8ACTSection1.5Exercise1.5-9SalarySection1.5Table1.5-1CarsChapter1Project1MetalCuttingChapter1Project2IowaFatalitiesChapter1Project2IowaVMTChapter1Project3TrucksChapter1Project9NHTempChapter 3Section3.1Exercise3.1-2HoursSection3.1Exercise3.1-3WindSection3.6Exercise3.6-1ObservationsSection3.6Exercise3.6-11SurgeryChapter 4Chapter4Project7ZieglerStudy1

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2. Listing of Data Files4Chapter4Project10BootstrapChapter4Project11PermutationChapter4Project14TrucksChapter 5Section5.1Exercise5.1-1GrantSection5.1Exercise5.1-2AstroSection5.1Exercise5.1-10CartonsSection5.1Table5.1-1ChartSection5.2Table5.2-CapabilityChapter5Project1BreedingChapter 6Section6.1Exercise6.1-2CuckooSection6.1Exercise6.1-3StrengthSection6.1Exercise6.1-6StrengthSection6.1Exercise6.1-10GPASection6.1Exercise6.1-11SalarySection6.1Table6.1-2DeflectionSection6.2Exercise6.2-4BakerySection6.2Exercise6.2-9Youden1Section6.2Exercise6.2-10Youden2Section6.3Exercise6.3-1WearTesterSection6.3Exercise6.3-2ResistorsSection6.3Exercise6.3-3ProductivitySection6.3Exercise6.3-4ReactionSection6.3Exercise6.3-6SodiumSection6.3Table6.3-1StrengthSection6.4Exercise6.4-2MarigoldSection6.4Exercise6.4-3TireSection6.4Exercise6.4-4CheeseSection6.4Table6.4-5YieldChapter 7Section7.1Exercise7.1-3SalesSection7.1Exercise7.1-4BrightnessSection7.1Exercise7.1-5BreakSection7.1Exercise7.1-9StressTestSection7.1Table7.1-5PopcornSection7.2Exercise7.2-1PigmentSection7.2Exercise7.2-4SteelBeamSection7.2Table7.2-1RodSection7.3Exercise7.3-4Conversion

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2. Listing of Data Files5Section7.3Exercise7.3-5SmoothnessSection7.3Exercise7.3-6LossSection7.3Exercise7.3-7ImpuritySection7.3Exercise7.3-9YieldSection7.3Exercise7.3-10ConversionSection7.3Exercise7.3-11MeredithSection7.3Table7.3-5FabricSection7.4Exercise7.4-8ColorSection7.4Exercise7.4-9ViscositySection7.4Table7.4-2FractFact1Section7.4Table7.4-3FractFact2Chapter7Project6MotherJonesChapter 8Section8.1Exercise8.1-1BetsSection8.1Exercise8.1-2YieldSection8.1Exercise8.1-3SnedecorSection8.1Table8.1-1CarsSection8.3Exercise8.3-2AnscombeSection8.3Exercise8.3-5AerosolSection8.3Exercise8.3-6CarsSection8.3Exercise8.3-8EnrollmentSection8.3Exercise8.3-9EggsSection8.3Exercise8.3-11WeldStrengthSection8.3Table8.3-1SteamSection8.4Exercise8.4-4CarsNewSection8.4Exercise8.4-7TrainStoppingDistancesSection8.5Exercise8.5-1GrowthRateSection8.5Exercise8.5-2CloudPointSection8.5Exercise8.5-5SoilSection8.5Exercise8.5-6GPASection8.5Exercise8.5-7TractionSection8.5Exercise8.5-8OxygenSection8.5Exercise8.5-10YieldSection8.5Exercise8.5-11ToolLifeSection8.5Exercise8.5-15TreesSection8.5Table8.5-6DurabilitySection8.6Exercise8.6-3YieldSection8.6Exercise8.6-4ReactionSection8.6Exercise8.6-5PercYieldSection8.6Exercise8.6-7SynthesisChapter8Project2WineChapter8Project4NeusiedlChapter8Project5BeansChapter8Project8Assay

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3. A Brief Introduction to Minitab63. A BRIEF INTRODUCTION TO MINITABMinitab Inc. (http://www.minitab.com/) is a leading global provider of software andservices for quality improvement and statistics education. Their mission is to provide thetools and resources professionals need to analyze complex problems, improve theirprocesses, and train their students.Minitab is best known for its flagship product, the Minitab® Statistical Software. Thepackage was originally created in 1972 to help professors teach statistics, but has sinceevolved into the premier software organizations use when analyzing business data toimprove the quality of their goods and services. It has driven virtually every major SixSigma improvement initiative around the world, and is the package students use to learnstatistics in more than 4,000 colleges and universities.Basic principles and simple tools for data analysisThe Minitab software is very easy to use. Minitab is a spreadsheet program. Data getentered into columns and rows of a spreadsheet, and calculation and graphing operationsare executed through simple pull-down menus. As of July 2008, the latest version ofMinitab is Version 15. Minitab divides its worksheet (spreadsheet) into columns (labeledC1, C2, C3, .... ) and rows (labeled 1, 2, 3, …). Take, for example, the data in Table 1.5-1which lists the weights (in 1000 pounds), the fuel efficiencies (in gallons per 100travelled miles), and the names of n = 10 cars. Weights are entered into column 1, fuelefficiencies are entered into column 2, and labels are entered into column 3. Each row ofthe spreadsheet represents a different car. The data can be entered through the keyboard,or it can be entered by clicking (and opening up) the Minitab file that has been preparedfor this particular data set (see section 2 of this manual). Click on the fileSection1.5Table1.5-1Cars. The Minitab program will open and one of its windows willshow a worksheet with three columns of data. Informative labels are attached to thecolumns; here column C1 is labeled as X=Weight, column C2 as Y=GPM, and columnC3 as Car.The Worksheet containing the data is one of the windows that you see when calling upMinitab. Another window that you see is the Session window. The Session windowcollects the output that is generated during a Minitab analysis. Some operations willgenerate Graph windows. All windows can be saved to files.The command line on the top of the Minitab window (with its tabs: File, Data, Edit, Calc,Stat, Graph, Editor, Tools, Windows, and Help) contains pull-down menus for carryingout operations. For example, click on the tab File. The commands within this folder allowyou to save the worksheet, enter previous worksheets (such as the worksheet files wehave prepared for the data sets in this text), print the worksheet, and save the currentproject (which consists of the worksheet and the output that has been generated by thecurrent Minitab session).

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3. A Brief Introduction to Minitab7The commands under the tab Data allow you to sort, rank, and copy the information inspecified columns to other columns of the worksheet. The commands under the tab Calcallow you to create new variables (using calculator), generate random variables, and carryout probability calculations. The commands under the tab Stat carry out the variousstatistical analyses, and commands under the tab Graph provide many of the displays thatwe discuss in our book. The instruction Enable under the Editor tab enables a record ofall instructions that are carried out during a Minitab session. The Help tab is important forgetting information on how Minitab works; it gives detailed descriptions of theprocedures and explains how to carry them out. We suggest that you start with thesimplest versions of the commands, before learning how to tweak each procedure to getthe maximum benefit. It should take you almost no time to get familiar with the basicfeatures of the software, and you will become very proficient in a matter of days.Click on the prepared file Section1.5Table1.5-1Cars. You will see the worksheet with thedata, and a session window. Go to the command line on the top of the Minitab windowand click on Editor and then on Enable commands. You will see the greater symbol( “>”) on the command line of the session window. Minitab uses “>” as its prompt. Enterthe line “print C1 C2 C3” (without the quotes; you can also use either lower or upper caseletters) and hit return. This will print out the three columns. There are two ways ofexecuting tasks. You can execute commands from the session window by entering certaintext instructions, or from the command line by clicking on tasks that are included underthe various tabs. Enabling commands (under the Editor tab) translates the instructionsfrom the pull down menus into text instructions in the session window.Click on the Stat tab, and then click on Basic Statistics and on Display DescriptiveStatistics. We indicate this path by writing “Stat > Basic Statistics > Display DescriptiveStatistics”. A dialog box will open. Enter C1 and C2 for the variables (you can do this byclicking on the columns in the area on the left). Running this command (by clicking“OK”) provides the summary statistics on these two columns. You can change the desiredoutput by clicking on the tab Statistics; for example, you can calculate the trimmed meanif you wish. Also, you can stratify the analysis by adding a categorical variable into the“by variable” box. (In this data set, no such variable is available). Note that the textvariable (label) does not show up as a variable in the dialog box; this makes sense as youwould not want to calculate numerical statistics for text data.Go to the Graph tab next. You can get dot plots of the data in the two columns (use“Graph > Dotplot”). A scatter plot of the fuel efficiency against the weight of the car canbe obtained from “Graph > Scatterplot.” Select the simplest version to get started. All youneed is to enter the variables into the dialog box. If you wish, you can add labels andtitles. With time and practice you will find that there are many other useful options. Forexample you may want to add to this graph the least squares line. For that you need to goto the window “With Regressions”. You may want to overlay two scatter plots on thesame graph. For that you have to go to “With Groups.” Selecting the “Multiple Graphs”window in the following dialog box will give you many graphing options.

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3. A Brief Introduction to Minitab8The correlation coefficient is calculated from the “Stat > Basic Statistics > Correlation”dialog window. You can achieve the same from the session window by typing in “corr c1c2” and hitting the return key.Time sequence plots can be obtained through “Graph > Time Series Plot”. For exampleclick on the file Section1.2Exercise1.2-1Thermostat. It contains the sales from 52consecutive weeks. You can add informative labels. You can also change the labels in thegraph by double clicking on the labels and changing them. You can change the scales ofthe axes by pointing your mouse to the desired axis (either x or y), clicking the rightbutton of your mouse, and going to the ”Edit X Scale” tab for further instructions.Consider the file Section1.3Exercise1.3-8Thickness. It contains the thicknessmeasurements of n = 150 ears of paint cans. Calculate summary statistics (by “Stat >Basic Statistics > Display Descriptive Statistics”), construct a dotplot (“Graph >Dotplot”), a histogram (“Graph > Histogram”), a steam-and-leaf display (“Graph >“Stem-and-Leaf”), and a box plot (“Graph > Boxplot”). Look at the available options forconstructing histograms. Once you have created the histogram, you can point your mouseto the x-axis, right click, and edit scale. The options in the “binning” tab will allow you tochange the number of bins as well as the cut- and mid-points of the histogram.Consider the data on the lead concentrations in the file Section1.4Lead1976&1977. Thefirst column contains the lead concentration for 1976, while the second column containsthe data for 1977. The numbers of observations happen to be the same in the two groups.Stratification is important in this analysis as we want to compare the two distributionsHence dot diagrams, box plots and histograms should be graphed for each of the twoyears on the same sheet and on the same scale. You can construct comparative plots byexecuting “Graph > Dotplot,” entering C1 and C2 into the dialog box, and clicking“Multiple Y’s”.Sometimes observations are missing. Consider the file Chapter8Project2Wine. The pricefor the 1954 and 1956 vintage is missing. Minitab uses the symbol “*”. Commands willskip over the rows that contain the missing value. For example the summary statistics forcolumn C5 (price) calculates the statistics from the 27 available rows in that column.Summary statistics on the other columns (such as rain in C3) use all available rows forthat column (that is, not just those that have information on all columns). Command suchas scatter plot of price (in C5) on rain (in C3) use the 27 available pairs.Determining probabilities and percentiles of various distributionsThe “Calc > Probability Distributions” tab can calculate cumulative probabilities andpercentiles for all distributions discussed in this text. For example, selecting Normal willopen up a dialog box. For cumulative probabilities, we click “Cumulative probability,”enter the mean and standard deviation, and a constant (which specifies the argument ofthe c.d.f.). For example with mean 3, standard deviation 2, and constant 1, we obtain thecumulative probability0.158655)1(=≤XP. For percentiles, we need to click “Inverse

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3. A Brief Introduction to Minitab9cumulative probability,” enter the mean and standard deviation, and a constant (which isnow the specified proportion). For example with mean 3, standard deviation 2, andconstant 0.80, we obtain the 80thpercentile as 4.68324. Clicking on the “probabilitydensity” will give us the value of the density at a specified constant. For example,0.120985)0.1(==xffor the normal distribution with mean 2 and standard deviation 3.The same type of operations (“cumulative probability” for probabilities, “inversecumulative probability” for percentiles, and “probability density” for the value of thedensity function) apply to all other continuous distributions. The only changes are in theparameters of the distribution. For example, for the Gamma distribution with parameters2=α(shape) and5=β(scale), we obtain0.593994)10(=≤XP; note the mean of thisdistribution is (2)(5) = 10. The 90thpercentile of this distribution is 19.4486.The same instructions are carried out for discrete distributions such as the binomialdistribution with parameters n = 20 and p = 0.1. “Cumulative probability” provides thecumulative probabilities up to the selected constant c,)(cXP≤. For example,0.391747)1()4.1(=≤=≤XPXP. “Probability density” provides the probabilities,)(cXP=. For example,270170.0)1(==XPand0)4.1(==XP, since this particularbinomial is a discrete distribution with support on the integers from 0 to 20. “Inversecumulative probability” provides the percentiles; the 80thpercentile equals 3 [Minitablists the cumulative probabilities86704.0)3(=≤XPand676927.0)2(=≤XP]It’s easy to draw the p.d.f. of the binomial distribution with n = 20 and p = 0.1. First enterthe integers 0, 1, 2, …, 20 into the 21 rows of column C1 [you can do this manually, oryou can use the command “Calc > Make Patterned Data > Simple Set of Numbers”.]Then use the “Calc > Probability Distributions” tab and go to the binomial distribution.Click “probability” and add C1 into the input column field. Enter C2 (optional storage) tostore the probabilities20,...,1,0),(==xxXP. Then go to “Graph > Barchart,” click“values from a table,” and enter columns C2 and C1.Generating random variables“Calc > Random Data > Normal” can be used to generate a fixed number of realizationsfrom a normal distribution with specified mean and standard deviation. The data can bestored in any column(s). Use this command to generate 1,000 realizations from a normaldistribution with mean 10 and standard deviation 3. Calculate the summary statistics, andplot the histogram to convince yourself that this function is doing the right thing.You can use the “Calc > Random Data” command to generate realizations from manyother distributions. For illustration, generate realizations (say n = 100) from suchdistributions as the geometric with probability 0.2 (a discrete distribution), the continuousuniform between 0 and 1, the exponential with mean 6, the gamma with2=α(shape)and5=β(scale), and the chi-square with 10 degrees of freedom. If you want to learnmore about these distributions, click the help feature in the dialog box.

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3. A Brief Introduction to Minitab10There is no carry-over among consecutive random numbers. You can check this bycomputing the autocorrelations of the generated data sequence at lags 1, 2, … Use “Stat >Time Series > Autocorrelations” to obtain the autocorrelations. They should be within the2 sigma limits that are indicated on the graph. You can also lag the series C1 by oneperiod (using “Stat > Time Series > Lag”), store the lagged series in another column, sayC2, construct a scatter plot of C1 against C2, and calculate the correlation coefficient(corr C1 C2). You should see no patterns in the scatter plot, and the correlation (which isthe lag 1 autocorrelation) should be small.Constructing probability plotsProbability plots are easy to obtain. Take the generated normal random variables (incolumn C1), for example, and check whether or not these observations are from a normaldistribution. Use “Graph > Probability Plot > Single,” enter the column that contains thedata, and check Normal under the Distribution tab. You can check many otherdistributions such as the Weibull, gamma, etc. The added line in the Minitab probabilityplot helps you judge whether the data can be modeled with the selected distribution.Violations from the linearity indicate that the selected distribution is not a good fit.Confidence intervals and testing hypothesesThe “Stat > Basic Statistics” folder includes procedures for the calculation of confidenceintervals and the testing of hypotheses. “1-Sample Z” is used for the inference about apopulation mean from a single random sample assuming that the standard deviation isknown; it uses the normal distribution. “1-Sample t” is used for the inference about apopulation mean from a single random sample with estimated standard deviation; it usesthe t-distribution. “2-Sample t” is used for the comparison of two means when thesamples are independent. “Paired t” is used in the paired or blocked situation. “1Proportion” is used for the inference about a single proportion, while “2 Proportions” isused for the comparison of two proportions. “1 Variance” is used for the inference abouta single variance, while “2 Variances” covers the comparison of two variances.For example, take the 1976 lead concentrations (with n = 64) in fileSection1.4Lead1976&1977 and obtain a 95 percent confidence interval for the mean leadconcentration in 1976. Furthermore, test the research hypothesis that the mean leadconcentration is different than 7 ppm. The following result is obtained with the command“Stat > Basic Statistics > 1-Sample t”. You need to enter the column that contains thedata, the hypothesized mean, and you need to specify a 2-sided test alternative (under theOptions tab).Test of mu = 7 vs not = 7VariableNMeanStDevSE Mean95% CITPLead1976647.2912.0250.253(6.785, 7.797)1.150.255

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3. A Brief Introduction to Minitab11Next, assume that you want to test whether the mean concentration for 1976 is smallerthan the mean concentration for 1977. The result is obtained with the command “Stat >Basic Statistics > 2-Sample t”. You need to specify the columns that contain the 1976 and1977 data (two different columns) and, under Options, you need to specify the confidencecoefficient (say 95 percent) and the fact that the alternative hypothesis specifies that themean in the first column (1976) is smaller than the mean in the second column (1977).The Welch approximation (see book Section 4.6-1) is used if you can not assume that thevariances are equal. The pooled standard deviation is used if the variances are assumedequal. You can also create graphs that support your test conclusions.Two-sample T for Lead1976 vs Lead1977NMeanStDevSE MeanLead1976647.292.030.25Lead1977649.422.080.26Difference = mu (Lead1976) - mu (Lead1977)Estimate for difference:-2.13195% upper bound for difference:-1.530T-Test of difference=0 (vs <):T-Value=-5.87 P-Value=0.000 DF=125Finally, let us illustrate how you can use “Stat > Basic Statistics > 2 Variances” to testwhether the variances of the observations in 1976 and 1977 are the same. The F-testexplained in Section 4.6-4 is calculated by Minitab. (In addition, Minitab calculates theLevine test which is another useful procedure). The large p value (0.832 being muchlarger than 0.05) indicates that we cannot reject the null hypothesis that the two variancesare equal.Lead1977Lead19762.62.42.22.01.81.695% Bonferroni Confidence Intervals for StDevsLead1977Lead197617.515.012.510.07.55.02.50.0DataTest Statistic0.95P-Value0.832Test Statistic0.19P-Value0.665F-TestLevene's TestTest for Equal Variances for Lead1976, Lead1977

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3. A Brief Introduction to Minitab12Power calculationsIn the simplest case, use the command “Stat > Power and Sample Size > 1-Sample Z.”For example, assume that the standard deviation of a measurement is 2. You want to testfor an increase in the mean (one-sided alternative) and want to detect a one-unit increasewith power 0.8; that is, the probability of not detecting such a change is 0.20 if the meanis indeed increased by one unit (type II error). The type I error probability is selected as0.05. Then the required sample size is 25. The output is shown below.1-Sample Z TestTesting mean = null (versus > null)Calculating power for mean = null + differenceAlpha = 0.05Assumed standard deviation = 2SampleTargetDifferenceSizePowerActual Power1250.80.803765Chi-square test for independenceUse the command “Stat > Table > Chi-Square Test (Two way Table in Worksheet).” Forexample, with the data from Table 4.7-2 stored in columns C1, C2, and C3 of theworksheet, the output is as follows:Expected counts are printed below observed countsChi-Square contributions are printed below expected countsC1C2C3Total115187408.8917.7813.334.2010.0033.0082522235011.1122.2216.673.3610.0022.407Total20403090Chi-Sq = 12.982, DF = 2, P-Value = 0.002Use the command “Stat > Table > Cross Tabulation and Chi-Square” if you work withraw (not summarized) data. Minitab will create the table of counts and perform the chi-square test.Control charts“Stat > Control Charts > Variables Charts for Subgroups” can be used for the x-bar andR-charts, while “Stat > Control Charts > Attributes Charts” can be used for the p- and c-charts discussed in Section 5.1.

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3. A Brief Introduction to Minitab13Consider the information in the file Section5.1Table5.1-1Chart. Go to “Stat > ControlCharts > Variables Charts for Subgroups” and then to “Xbar-R”. Indicate that theobservations for a subgroup are in one row of columns. Go to “Xbar-R chart Options;”under the “Estimate” tab, specify that you want to omit the subgroups when estimatingparameters; here 11:12) and click “Rbar” so thatRis used for the calculation of thecontrol limits. This will result in Figures 5.1-1 and 5.1-2 of the text.For the p-chart in Example 5.1-1 of the text, you enter the numbers of defectives ”1 1 3…. 0 2” into a column of the worksheet, say C1. Then go to “Stat > Control Charts >Attributes Charts,” specify “P”(chart), and enter the column with the number ofdefectives (here C1) and the subgroup size (here n = 50). This will give you the chart inFigure 5.1-3.Minitab includes numerous other useful control charts that have not been discussed inthis text.Analyzing data from designed experiments: ANOVA methods“Stat > ANOVA” with its commands “One Way”, “One Way (unstacked)”, “Two Way”,“Balanced ANOVA”, “General Linear Model”, and “Fully Nested ANOVA” arerelevant.Consider the data in file Section6.1Table6.1-2Deflection. Use the “Stat > ANOVA > OneWay” (the stacked) command. The column of deflections (C1) is entered as the response,while the column of beam indicators (1, 2, 3 in C2) is entered as the factor. TheComparison tab allows us to obtain “Tukey” multiple comparisons, and under the Graphtab we can ask for useful graphs (such as an individual value plot or box plots of thedata). Try it, and check that this will give you the output that we discussed in Sections 6.1and 6.2.Consider the data in file Section6.3Table6.3-1Strength. Use the “Stat > ANOVA > TwoWay” command. Strength in column C1 is entered as the response, while the column oftreatments (1, 2, 3 in C2) and batch (1, 2, …,5 in C3) are entered as the factors. TheGraph tab allows for useful graphs (such as an individual value plot or box plots of thedata). Try it, and check that this will give you the output that we discussed in Section 6.3.Consider the data in file Section7.1Table7.1-5Popcorn. For our analysis we use the “Stat> ANOVA > Two Way” command. Yield in column C1 is entered as the response, whilethe popper (1 and 2 in C2) and brand (1, 2, 3 in C3) are entered as the factors. The outputcoincides with the one shown in Section 7.1 of the book. The commands “Stat > ANOVA> Main Effects Plot” and “Stat > ANOVA > Interactions Plot” display main effects andinteractions graphically. The output of the interaction plot is shown in Figure 7.1-2 of thetext.

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3. A Brief Introduction to Minitab14The function “Stat > ANOVA > Balanced ANOVA” allows the analysis of balancedmulti-factor experiments like the ones in Section 7.1 (2-factor factorial designs) andSection 7.3 (general factorial designs with more than two factors). We can declare factorsas either fixed or random. If this approach is used, one must specify the model in terms ofmain effects (say C2 and C3 for popper and brand) and the interaction (C2*C3).The commands “Stat > ANOVA > Two Way” and “Stat > ANOVA > BalancedANOVA” do not work for unbalanced situations. Convince yourself of this fact bydeleting the last row in the file Section7.1Table7.1-5Popcorn. For unbalanced designs wemust use “Stat > ANOVA > General Linear Models”. The interpretation of the sums ofsquares and their associated tests is more complicated than the one in the balancedsituation. Now sums of squares need to be interpreted as partial (that is, partial to all otherfactors in the model). We explain this concept in Section 8.5 (chapter on regression).Consider the data in file Section7.2Table7.2-1Rods. For our analysis we use the “Stat >ANOVA > Fully Nested ANOVA” command. Column C3 containing the measurementsis entered as the response, while rods (1, 2, …, 5 in C1) and forging (1, 2, 3 in C2) areentered as the factors. The output of the analysis (ANOVA table and estimated variancecomponents) is shown in Table 7.2-3 of the text.Factorial and fractional factorial designsMinitab constructs the designs (that is, tells us about the factor levels of the design runsand the order in which the runs should be carried out), and helps us analyze the resultingdata once the experiment has been performed. Minitab’s features are included under the“Stat > DOE > Factorial” tab. Minitab makes it easy for the user to construct 2-levelfactorial (Section 7.3) and fractional factorial designs (Section 7.4). The user enters thenumber of factors and is then presented a list of full and fractional designs with theirrespective run sizes. After deciding on the number of runs, the user can construct thedesign either through default generators that optimize the resolution of the design, or bywriting out specific generators. In either situation, Minitab indicates the design columns,displays – if desired – a randomized arrangement of the runs, and lists the confoundingpatterns of the particular fraction that is being selected. If default generators are used,Minitab displays the adopted generators. The user can add center points, replicate thedesign, block the experiment by specifying blocking generators, and modify the designby considering fold-overs. These can be complete (full) fold-overs where the signs of allfactors are changed, or fold-overs of individual factors.Once the design has been carried out, Minitab facilitates an efficient analysis of the data.The analysis of 2-level factorial and fractional factorial designs includes estimates of theeffects. Two different estimates are listed: estimates labeled “Coef” which are theestimates discussed in this book, and estimates called “Effects” (which are twice the sizeof the coefficients expressing the differences in the response averages at high and lowlevels). Standard errors of the estimated coefficients are calculated if replications areavailable. Normal probability plots and Pareto plots for assessing the importance of the

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Solution Manual For Applied Statistics For Engineers And Physical Scientists, Coursesmart Etextbook, 3rd Edition - Page 16 preview image

3. A Brief Introduction to Minitab15effects, and main effects and interaction plots for assessing the nature of the relationshipsare also available.Consider the data file Section7.3Table7.3-5Fabric. This is a 24full factorial in 16 runs.The runs are listed in standard order. We have already run the experiment and havecollected the observations. Hence we first need to define the experiment by carrying outthe command “Stat > DOE > Factorial > Define Custom Factorial Design”. Wecommunicate to the program the columns that contain the factors. As far as thesubsequent calculations are concerned, we can ignore the information that gets added intocolumns C6 through C9. After this first “define” step we execute the “Stat > DOE >Factorial > Analyze Factorial Design” command, and indicate to Minitab the column thatcontains the response (here it is in C5). The output lists the effects and estimatedcoefficients (see the above discussion). Note that there is no error since the model is fullysaturated, with 16 coefficients (average, main effects and interactions) estimated from 16observations. Under the “Graph” tab one can request a normal probability plot of theestimated coefficients. This resulting graph is shown in Figure 7.4-2 of the text. Onecould specify a model with just main effects and 2-factor interactions. This is achieved bygoing to the tab “Terms” and selecting main effects and 2-factor interactions for themodel. An estimate of the standard deviation of the error is obtained from pooling thehigher-order interactions; in this case the ANOVA table lists an error mean square andtests for main effects and 2-factor interactions.Next, let Minitab select a “good” fractional factorial design for, say, 8 factors in 16 runs.This amounts to a 28-4fractional factorial design. We run the “Stat > DOE > Factorial >Create Factorial Design” command. We enter the number of factors (there are 8) andlearn, under available factorial designs, that a resolution IV design is possible.We click on “Design”, and select from the offered choices the 1/16 resolution IV fraction.Under “Options”, we click the tab for getting the runs in standard (that is, notrandomized) order. The output lists the factor levels of the 16 runs. It also reveals thegenerating relationship (design generators: E = BCD, F = ACD, G = ABC, H = ABD)and the alias structure that is implied by this particular fraction. Once we have obtainedthe responses at these experimental conditions, we add them to the worksheet. [Forillustration, generate 16 realizations of the standard normal distribution]. Then execute“Stat > DOE > Factorial > Analyze Factorial Design”, and indicate to the program thecolumn that contains the responses. Minitab then calculates the effects and estimatedcoefficients, and lists the alias relationships for the estimates.Consider the data file Section7.4Table7.4-3FractFact2. It shows a 27-4fractional factorialdesign in 8 runs. Since we already have run the experiment and have collected theobservations we need to define the experiment by carrying out the “Stat > DOE >Factorial > Define Custom Factorial Design” command. We basically tell Minitab wherethe factors are. Minitab recognizes the arrangement as the fully saturated factorial design.After this step, we execute the “Stat > DOE > Factorial > Analyze Factorial Design”command and indicate to Minitab the response column. The resulting output lists theeffects and estimated coefficients (see above) and the alias patterns, and provides, ifasked for, the normal probability graph of the estimated effects. The coefficients and the

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