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Solution Manual for Business Statistics: A First Course, 8th Edition

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    S OLUTIONS M ANUAL G AIL I LLICH P AUL I LLICH McLennan Community College Southeast Community College B USINESS S TATISTICS : A F IRST C OURSE E IGHTH E DITION David M. Levine Baruch College, City University of New York Kathryn A. Szabat La Salle University David F. Stephan Two Bridges Instructional Technology

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    Table of Contents Teaching Tips...................................................................................................................................1 Chapter 1 Defining and Collecting Data ...................................................................................29 Chapter 2 Organizing and Visualizing Variables ......................................................................37 Chapter 3 Numerical Descriptive Measures ............................................................................145 Chapter 4 Basic Probability .....................................................................................................187 Chapter 5 Discrete Probability Distributions...........................................................................197 Chapter 6 The Normal Distribution .........................................................................................229 Chapter 7 Sampling Distributions ...........................................................................................263 Chapter 8 Confidence Interval Estimation ..............................................................................289 Chapter 9 Fundamentals of Hypothesis Testing: One-Sample Tests ......................................327 Chapter 10 Two-Sample Tests and One-Way ANOVA ...........................................................373 Chapter 11 Chi-Square Tests .....................................................................................................447 Chapter 12 Simple Linear Regression .......................................................................................469 Chapter 13 Multiple Regression ................................................................................................519 Chapter 14 Business Analytics ..................................................................................................557 Chapter 15 Statistical Applications in Quality Management (Online) ......................................595

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    6 Teaching Tips Chapter 1 You need to continue the discussion of the Define task by establishing the types of variables. Mention the importance of having an operational definition for each variable. Be sure to discuss the different types carefully since the ability to distinguish between categorical and numerical variables will be crucial later in the course. Go over examples of each type of variable and have students provide examples of each type. Then, if you wish, you can cover the different measurement scales. Then move on to the C of the DCOVA approach, collecting data. Mention the different sources of data and make sure to cover the fact that data often needs to be cleaned of errors. Then, you could spend some time discussing sampling, even if it is just using the table of random numbers to select a random sample. You may want to take a bit more time and discuss the types of survey sampling methods and issues involved with survey sampling results. The Think About This essay discusses the important issue of the use of Web-based surveys. There is also a section on Data Cleaning that discusses the issues that occur in data collection. This is followed by a section on data formatting that includes the important concepts of stacking and unstacking variables and recoding variables. The last section discusses the types of errors that occur in surveys. The chapter also introduces three continuing cases -- the Managing Ashland MultiComm Services , CardioGood Fitness , and Clear Mountain State Student Surveys that appear at the end of many chapters. The Digital cases are introduced in this chapter also. In these cases, students visit Web sites related to companies and issues raised in the Using Statistics scenarios that start each chapter. The goal of the Digital cases is for students to develop skills needed to identify misuses of statistical information. As would be the situation with many real world cases, in Digital cases, students often need to sift through claims and assorted information in order to discover the data most relevant to a case task. They will then have to examine whether the conclusions and claims are supported by the data. (Instructional tips for using the Managing Ashland MultiComm Services, and Digital cases and solutions to the Managing Ashland MultiComm Service, CardioGood Fitness, Clear Mountain State Student Surveys, and Digital cases are included in this Instructor’s Solutions Manual .). Make sure that students read the Excel Guide and/or JMP Guide or Minitab Guide at the end of each chapter. The In-Depth Excel instructions provide step-by-step instructions and live worksheets that automatically update when data changes. The PHStat add-in instructions provide instructions for using the PHStat add-in. Analysis ToolPak instructions provide instructions for using the Analysis ToolPak, the Excel add-in package that is included with many versions of Excel.

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    Teaching Tips 7 Chapter 2 This chapter moves on to the organizing and visualizing steps of the DCOVA framework. If you are going to collect sample data to use in Chapters 2 and 3, you can illustrate sampling by conducting a survey of students in your class. Ask each student to collect his or her own personal data concerning the time it takes to get ready to go to class in the morning or the time it takes to get to school or home from school. First, ask the students to write down a definition of how they plan to measure this time. Then, collect the various answers and read them to the class. Then, a single definition could be provided (such as the time to get ready is the time measured from when you get out of bed to when you leave your home, recorded to the nearest minute). In the next class, select a random sample of students and use the data collected (depending on the sample size) in class when Chapters 2 and 3 are discussed. Then, move on to the Organize step that involves setting up your data in an Excel, JMP, Minitab, or Tableau. Show the summary worksheet and develop tables to help you prepare charts and analyze your data. Begin your discussion for categorical data with the example on p. 45 concerning the percentage of the time millennials use different devices for watching television and then if you wish, explain that you can sometimes organize the data into a two-way table that has one variable in the row and another in the column. Continue with organizing data (but now for numerical data) by referring to the cost of a restaurant meal on p. 49. Show the simple ordered array and how a frequency distribution, percentage distribution, or cumulative distribution can summarize the raw data in a way that is more useful. Now you are ready to tackle the Visualize step. A good way of starting this part of the chapter is to display the following quote. "A picture is worth a thousand words." Students will almost certainly be familiar with Microsoft ® Word and may have already used Excel to construct charts that they have pasted into Word documents. Now you will be using Excel or JMP or Minitab or Tableau to construct many different types of charts. Return to the data previously discussed on what devices millennials use to watch television and illustrate how a bar chart and pie chart can be constructed. Mention their advantages and disadvantages. A good example is to show the data on incomplete ATM transactions on p. 60 and how the Pareto chart enables you to focus on the vital few categories. If time permits, you can discuss the side-by-side bar chart for a contingency table. To examine charts for numerical variables you can either use the restaurant data previously mentioned or data that you have collected from your class. You may want to begin with a simple stem- and-leaf display that both organizes the data and shows a bar type chart. Then move on to the histogram and the various polygons, pointing out the advantages and disadvantages of each.

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    8 Teaching Tips If time permits, you can discuss the scatter plot and the time-series plot for two numerical variables. Otherwise, you can wait until you get to regression analysis. Also, you may want to discuss how multidimensional tables allow you to see several variables simultaneously. If the opportunity is available, we believe that it is worth the time to cover Section 2.9 on Pitfalls in Organizing and Visualizing Data. This is a topic that students very much enjoy since it allows for a great deal of classroom interaction. After discussing the fundamental principles of good graphs, try to illustrate the improper display shown in Figure 2.31. Ask students what is “bad” about this figure. Follow up with a homework assignment involving Problems 2.69 – 2.73 ( USA Today is a great source). You will find that the chapter review problems provide large data sets with numerous variables. Report writing exercises provide the opportunity for students to integrate written and/or oral presentation with the statistics they have learned. The Managing Ashland MultiComm Services case enables students to examine the use of statistics in an actual business environment. The Digital case refers to the EndRun Financial Services and claims that have been made. The CardioGood Fitness case focuses on developing a customer profile for a market research team. The Choice Is Yours Follow-up expands on the chapter discussion of the mutual funds data. The Clear Mountain State Student Survey provides data collected from a sample of undergraduate students and a separate sample of graduate students. The Excel Guide and the JMP and Minitab and Tableau Guides for this and the remaining chapters are organized according to the sections of the chapter. They are quite extensive since they cover both organizing and visualizing many different graphs. The Excel Guide includes instructions for In- Depth Excel, PHStat, and the Analysis ToolPak. Pick and allows you to choose the approaches that you prefer.

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    Teaching Tips 9 Chapter 3 This chapter on descriptive numerical statistical measures represents the initial presentation of statistical symbols in the text. Students who need to review arithmetic and algebraic concepts may wish to refer to Appendix A for a quick review or to appropriate texts (see www.pearson.com ) or videos ( www.videoaidedinstruction.com ). Once again, as with the tables and charts constructed for numerical data, it is useful to provide an interesting set of data for classroom discussion. If a sample of students was selected earlier in the semester and data concerning student time to get ready or commuting time were collected (see Chapters 1 and 2), use these data in developing the numerous descriptive summary measures in this chapter. (If they have not been developed, use other data for classroom illustration.) Discussion of the chapter begins with the property of central tendency. We have found that almost all students are familiar with the arithmetic mean (which they know as the average) and most students are familiar with the median. A good way to begin is to compute the mean for your classroom example. Emphasize the effect of extreme values on the arithmetic mean and point out that the mean is like the center of a seesaw -- a balance point. Note that you will return to this concept later when you discuss the variance and the standard deviation. You might want to introduce summation notation at this point and express the arithmetic mean in formula notation as in Equation (3.1). (Alternatively, you could wait until you cover the variance and standard deviation.) A classroom example in which summation notation is reviewed is usually worthwhile. Remind the students again that Appendix A includes a review of arithmetic and algebra and summation notation [or refer them to other text sources such as those found at www.pearson.com or videos (see www. videoaidedinstruction.com )]. The next statistic to compute is the median. Be sure to remind the students that the median as a measure of position must have all the values ranked in order from lowest to highest. Be sure to have the students compare the arithmetic mean to the median and explain that this tells us something about another property of data (skewness). Following the median, the mode can be briefly discussed. Once again, have the students compare this result to those of the arithmetic mean and median for your data set. If time permits, you can also discuss the geometric mean which is heavily used in finance. The completion of the discussion of central tendency leads to the second characteristic of data, variability. Mention that all measures of variation have several things in common: (1) they can never be negative, (2) they will be equal to 0 when all values are the same, (3) they will be small when there isn't much variation, and (4) they will be large when there is a great deal of variation. The first measure of variability to consider is the simplest one, the range. Be sure to point out that the range only provides information about the extremes, not about the distribution between the extremes. Point out that the range lacks one important ingredient, the ability to take into account each data value. Bring up the idea of computing the differences around the mean, but then return to the fact that as

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    10 Teaching Tips the balance point of the seesaw, these differences add up to zero. At that point, ask the students what they can do mathematically to remove the negative sign for some of the values. Most likely, they will answer by telling you to square them (although someone may realize that the absolute value could be taken). Next, you may want to define the squared differences as a sum of squares. Now you need to have the students realize that the number of values being considered affects the magnitude of the sum of squared differences. Therefore, it makes sense to divide by the number of values and compute a measure called the variance. If a population is involved, you divide by N , the population size, but if you are using a sample, you divide by n – 1, to make the sample result a better estimate of the population variance. You can finish the development of variation by noting that since the variance is in squared units, you need to take the square root to compute the standard deviation. Another measure of variation that can be discussed is the coefficient of variation. Be sure to illustrate the usefulness of this as a measure of relative variation by using an example in which two data sets have vastly different standard deviations, but also vastly different means. A good example is one that involves the volatility of stock prices. Point out that the variation of the price should be considered in the context of the magnitude of the arithmetic mean. By changing values in the data provided, students can observe how the mean, median, and standard deviation are affected. The final measure of variation is the Z score. Point out that this provides a measure of variation in standard deviation units. You can also say that you will return to Z scores in Chapter 6 when the normal distribution will be discussed. You are now ready to move on to the third characteristic of data, shape. Be sure to clearly define and illustrate both symmetric and skewed distributions by comparing the mean and median. You may also want to briefly mention the property of kurtosis which is the relative concentration of values in the center of the distribution as compared to the tails. This statistic is provided by Excel through an Excel function or the Analysis Toolpak and by JMP or Minitab. Once these three characteristics have been discussed, you are ready to show how they can be computed using Excel or JMP or Minitab. Now that these measures are understood, you can further explore data by computing the quartiles, the interquartile range, the five number summary, and constructing a boxplot. You begin by determining the quartiles. Reference here can be made to the standardized exams that most students have taken, and the quantile scores that they have received (97th percentile, 48th percentile, 12th percentile, …, etc.). Explain that the 1 st and 3 rd quartiles are merely two special quantiles -- the 25th and 75th, that unlike the median (the 2 nd quartile), are not at the center of the distribution. Once the quartiles have been computed, the interquartile range can be determined. Mention that the interquartile range computes the variation in the center of the distribution as compared to the difference in the extremes computed by the range.

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    Teaching Tips 11 You can then discuss the five-number summary of minimum value, first quartile, median, third quartile, and maximum value. Then, you construct the boxplot. Present this plot from the perspective of serving as a tool for determining the location, variability, and symmetry of a distribution by visual inspection, and as a graphical tool for comparing the distribution of several groups. It is useful to display Figure 3.9 on page 142 that indicates the shape of the boxplot for four different distributions. Then, use PHStat or JMP or Minitab or Tableau to construct a boxplot. Note that you can construct the boxplot for a single group or for multiple groups. If you desire, you can discuss descriptive measures for a population and introduce the empirical rule and the Chebyshev rule. If time permits, and you have covered scatter plots in Chapter 2, you can briefly discuss the covariance and the coefficient of correlation as a measure of the strength of the association between two numerical variables. Point out that the coefficient of correlation has the advantage as compared to the covariance of being on a scale that goes from –1 to +1. Figure 3.11 on p. 157 is useful in depicting scatter plots for different coefficients of correlation. Once again, you will find that the chapter review problems provide large data sets with numerous variables. The Managing Ashland MultiComm Services case enables students to examine the use of descriptive statistics in an actual business environment. The Digital case continues the evaluation of the EndRun Financial Services discussed in the Digital case in Chapter 2. The CardioGood Fitness case focuses on developing a customer profile for a market research team. More Descriptive Choices Follow- up expands on the discussion of the mutual funds data. The Clear Mountain State Student Survey provides data collected from a sample of undergraduate students and a separate sample of graduate students. The Excel Guide for the chapter includes instructions on using different Excel functions to compute various statistics. Alternatively, you can use PHStat or the Analysis ToolPak to compute a list of statistics. PHStat can be used to construct a boxplot. Or you can use JMP or Minitab. The Tableau guide covers the five number summary and the boxplot

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    12 Teaching Tips Chapter 4 The chapter on probability represents a bridge between the descriptive statistics already covered and the topics of statistical inference and regression to be covered in subsequent chapters. In many traditional statistics courses, often a great deal of time is spent on probability topics that are of little direct applicability in basic statistics. The approach in this text is to cover only those topics that are of direct applicability in the remainder of the text. You need to begin with a relatively concise discussion of some probability rules. Essentially, students really just need to know that (1) no probability can be negative, (2) no probability can be more than 1, and (3) the sum of the probabilities of a set of mutually exclusive events adds to 1.0. Students often understand the subject best if it is taught intuitively with a minimum of formulas, with an example that relates to a business application shown as a two-way contingency table (see the Using Statistics example). If desired, you can use In-Depth Excel or PHStat to compute probabilities from the contingency table. Once these basic elements of probability have been discussed, if there is time and you desire, conditional probability and Bayes’ theorem can be covered. The Think About This concerning email SPAM is a wonderful way of helping students realize the application of probability to everyday life. In addition, you may wish to spend a bit of time going over counting rules, especially if the binomial distribution will be covered in Chapter 5. Be aware that in a one-semester course where time is particularly limited, these topics may be of marginal importance. The Digital case in this chapter extends the evaluation of the EndRun Financial Services to consider claims made about various probabilities. The CardioGood Fitness, More Descriptive Choices Follow-up, and Clear Mountain State Student Survey each involve developing contingency tables to be able to compute and interpret conditional and marginal probabilities.

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    Teaching Tips 13 Chapter 5 Now that the basic principles of probability have been discussed, the probability distribution is developed and the expected value and variance (and standard deviation) are computed and interpreted. Given that a probability distribution has been defined, you can now discuss some specific distributions. Although every introductory course undoubtedly covers the normal distribution to be discussed in Chapter 6, the decision about whether to cover the binomial or Poisson distributions is a matter of personal choice and depends on whether the course is part of a two-course sequence. If the binomial distribution is covered, an interesting way of developing the binomial formula is to follow the Using Statistics example that involves an accounting information system. Note, in this example, the value for p is 0.10. (It is best not to use an example with p = 0.50 since this represents a special case). The discussion proceeds by asking how you could get three tagged order forms in a sample of 4. Usually a response will be elicited that provides three items of interest out of four selections in a particular order such as Tagged Tagged Not Tagged Tagged. Ask the class, what would be the probability of getting Tagged on the first selection? When someone responds 0.1, ask them how they found that answer and what would be the probability of getting Tagged on the second selection. When they answer 0.1 again, you will be able to make the point that in saying 0.1 again, they are assuming that the probability of Tagged stays constant from trial to trial. When you get to the third selection and the students respond 0.9, point out that this is a second assumption of the binomial distribution -- that only two outcomes are possible -- in this case Tagged and Not Tagged, and the sum of the probabilities of Tagged and Not Tagged must add to 1.0. Now you can compute the probability of three out of four in this order by multiplying (0.1)(0.1)(0.9)(0.1) to get 0.0036. Ask the class if this is the answer to the original question. Point out that this is just one way of getting three Tagged out of four selections in a specific order, and, that there are four ways to get three Tagged out of four selections This leads to the development of the binomial formula Equation (5.4). You might want to do another example at this point that calls for adding several probabilities such as three or more Tagged, less than three Tagged, etc. Complete the discussion of the binomial distribution with the computation of the mean and standard deviation of the distribution. Be sure to point out that for samples greater than five, computations can become unwieldy and the student should use PHStat, an Excel function, the binomial tables (See the Online Binomial.pdf tables), JMP, or Minitab. Once the binomial distribution has been covered, if time permits, other discrete probability distributions can be presented. If you cover the Poisson distribution, point out the distinction between the binomial and Poisson distributions. Note that the Poisson is based on an area of opportunity in which you are counting occurrences within an area such as time or space. Contrast this with the binomial distribution in which each value is classified as of interest or not of interest. Point out the equations for the mean and

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    14 Teaching Tips standard deviation of the Poisson distribution and indicate that the mean is equal to the variance. Since the computation of probabilities from these discrete probability distributions can become tedious for other than small sample sizes, it is important to discuss PHStat, an Excel function the Poisson tables (See the Online Poisson.pdf tables) or JMP or Minitab. The Managing Ashland MultiComm Services case for this chapter relates to the binomial distribution. The Digital case involves the expected value and standard deviation of a probability distribution and applications of the covariance in finance.

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    Teaching Tips 15 Chapter 6 Now that probability and probability distributions have been discussed in Chapters 4 and 5, you are ready to introduce the normal distribution. We recommend that you begin by mentioning some reasons that the normal distribution is so important and discuss several of its properties. We would also recommend that you do not show Equation (6.1) in class as it will just intimidate some students. You might begin by focusing on the fact that any normal distribution is defined by its mean and standard deviation and display Figure 6.3 on p. 234. Then, an example can be introduced and you can explain that if you subtracted the mean from a particular value, and divided by the standard deviation, the difference between the value and the mean would be expressed as a standardized normal or Z score that was discussed in Chapter 3. Next, use Table E.2, the cumulative normal distribution, to find probabilities under the normal curve. In the text, the cumulative normal distribution is used since this table is consistent with results provided by Excel, JMP, and Minitab. Make sure that all the students can find the appropriate area under the normal curve in their cumulative normal distribution tables. If anyone cannot, show them how to find the correct value. Be sure to remind the class that since the total area under the curve adds to 1.0, the word area is synonymous with the word probability. Once this has been accomplished, a good approach is to work through a series of examples with the class, having a different student explain how to find each answer. The example that will undoubtedly cause the most difficulty will be finding the values corresponding to known probabilities. Slowly go over the fact that in this type of example, the probability is known and the Z value needs to be determined, which is the opposite of what the student has done in previous examples. Also point out that in cases in which the unknown X value is below the mean, the negative sign must be assigned to the Z value. Once the normal distribution has been covered, you can use PHStat, or various Excel functions or JMP or Minitab to compute normal probabilities. You can also use the Visual Explorations in Statistics Normal distribution procedure on p. 239. This will be useful if you intend to use examples that explore the effect on the probabilities obtained by changing the X value, the population mean,  , or the standard deviation, .  The Think About This essay provides a historical perspective of the application of the normal distribution. If you have sufficient time in the course, the normal probability plot can be discussed. Be sure to note that all the data values need to be ranked in order from lowest to highest and that each value needs to be converted to a normal score. Again, you can either use PHStat to generate a normal probability plot, use Excel functions with Excel charts, or use JMP or Minitab. The Managi n g Ashland MultiComm Services case for this chapter relates to the normal distribution. The Digital case involves the normal distribution and the normal probability plot. The CardioGood Fitness, More Descriptive Choices Follow-up, and Clear Mountain State Student Survey each involve developing normal probability plots.

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    16 Teaching Tips Chapter 7 The coverage of the normal distribution in Chapter 6 flows into a discussion of sampling distributions. Point out the fact that the concept of the sampling distribution of a statistic is important for statistical inference. Make sure that students realize that problems in this section will find probabilities concerning the mean, not concerning individual values. It is helpful to display Figure 7.4 on p. 265 to show how the Central Limit Theorem applies to different shaped populations. A useful classroom or homework exercise involves using PHStat, Excel, or JMP or Minitab to form sampling distributions. This reinforces the concept of the Central Limit Theorem. The Managing Ashland MultiComm Services case for this chapter relates to the sampling distribution of the mean. The Digital case also involves the sampling distribution of the mean. You might want to have students experiment with using the Visual Explorations add-in workbook to explore sampling distributions. You can also use either Excel functions, the PHStat add-in, the Analysis ToolPak, or JMP or Minitab to develop sampling distribution simulations.

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    Teaching Tips 17 Chapter 8 You should begin this chapter by reviewing the concept of the sampling distribution covered in Chapter 7. It is important that the students realize that (1) an interval estimate provides a range of values for the estimate of the population parameter, (2) you can never be sure that the interval developed does include the population parameter, and (3) the proportion of intervals that include the population parameter within the interval is equal to the confidence level. Note that the Using Statistics example for this chapter, which refers to the Ricknel Home Centers is actually a case study that relates to every part of the chapter. This scenario is a good candidate for use as the classroom example demonstrating an application of statistics in accounting. It also enables you to use the DCOVA approach of Define, Collect, Organize, Visualize, and Analyze in the context of statistical inference. When introducing the t distribution for the confidence interval estimate of the population mean, be sure to point out the differences between the t and normal distributions, the assumption of normality, and the robustness of the procedure. It is useful to display Table E.3 in class to illustrate how to find the critical t value. When developing the confidence interval for the proportion, remind the students that the normal distribution may be used here as an approximation to the binomial distribution as long as the assumption of normality is valid [when n  and (1 ) n   are at least 5]. Having covered confidence intervals, you can move on to sample size determination by turning the initial question of estimation around, and focusing on the sample size needed for a desired confidence level and width of the interval. In discussing sample size determination for the mean, be sure to focus on the need for an estimate of the standard deviation. When discussing sample size determination for the proportion, be sure to focus on the need for an estimate of the population proportion and the fact that a value of 0.5   can be used in the absence of any other estimate. Since the formulas for the confidence interval estimates and sample sizes discussed in this chapter are straightforward, using PHStat or In-Depth Excel can remove much of the tedious nature of these computations or you can use JMP or Minitab. The Managing Ashland MultiComm Services case for this chapter involves developing various confidence intervals and interpreting the results in a marketing context. The Digital case also relates to confidence interval estimation. This chapter marks the first appearance of the Sure Value Convenience Store case which places the student in the role of someone working in the corporate office of a nationwide c0nvenience store franchise. This case will appear in the next two chapters, Chapters 9 – 10. The CardioGood Fitness, More Descriptive Choices Follow-up, and Clear Mountain State Student Survey each involve developing confidence interval estimates.

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