(BQ) Part 2 book Essentials of modern business statistics has contents: has contents: Interval estimation, hypothesis tests, simple linear regression, multiple regression, comparisons involving proportions and a test of independence,...and other contents.
CHAPTER Interval Estimation CONTENTS Using Excel Practical Advice Using a Small Sample Summary of Interval Estimation Procedures STATISTICS IN PRACTICE: FOOD LION 8.1 8.2 POPULATION MEAN: σ KNOWN Margin of Error and the Interval Estimate Using Excel Practical Advice POPULATION MEAN: σ UNKNOWN Margin of Error and the Interval Estimate 8.3 DETERMINING THE SAMPLE SIZE 8.4 POPULATION PROPORTION Using Excel Determining the Sample Size Chapter STATISTICS 309 Interval Estimation in PRACTICE FOOD LION* SALISBURY, NORTH CAROLINA Founded in 1957 as Food Town, Food Lion is one of the largest supermarket chains in the United States with 1200 stores in 11 Southeastern and Mid-Atlantic states The company sells more than 24,000 different products and offers nationally and regionally advertised brand-name merchandise, as well as a growing number of high-quality private label products manufactured especially for Food Lion The company maintains its low price leadership and quality assurance through operating efficiencies such as standard store formats, innovative warehouse design, energy-efficient facilities, and data synchronization with suppliers Food Lion looks to a future of continued innovation, growth, price leadership, and service to its customers Being in an inventory-intense business, Food Lion made the decision to adopt the LIFO (last-in, first-out) method of inventory valuation This method matches current costs against current revenues, which minimizes the effect of radical price changes on profit and loss results In addition, the LIFO method reduces net income thereby reducing income taxes during periods of inflation Food Lion establishes a LIFO index for each of seven inventory pools: Grocery, Paper/Household, Pet Supplies, Health & Beauty Aids, Dairy, Cigarette/Tobacco, and Beer/Wine For example, a LIFO index of 1.008 for the Grocery pool would indicate that the company’s grocery inventory value at current costs reflects a 0.8% increase due to inflation over the most recent one-year period A LIFO index for each inventory pool requires that the yearend inventory count for each product be valued at the current year-end cost and at the preceding year-end cost To avoid excessive time and expense associated with counting the inventory *The authors are indebted to Keith Cunningham, Tax Director, and Bobby Harkey, Staff Tax Accountant, at Food Lion for providing this Statistics in Practice The Food Lion store in the Cambridge Shopping Center, Charlotte, North Carolina © Courtesy of Food Lion in all 1200 store locations, Food Lion selects a random sample of 50 stores Year-end physical inventories are taken in each of the sample stores The current-year and preceding-year costs for each item are then used to construct the required LIFO indexes for each inventory pool For a recent year, the sample estimate of the LIFO index for the Health & Beauty Aids inventory pool was 1.015 Using a 95% confidence level, Food Lion computed a margin of error of 006 for the sample estimate Thus, the interval from 1.009 to 1.021 provided a 95% confidence interval estimate of the population LIFO index This level of precision was judged to be very good In this chapter you will learn how to compute the margin of error associated with sample estimates You will also learn how to use this information to construct and interpret interval estimates of a population mean and a population proportion In Chapter 7, we stated that a point estimator is a sample statistic used to estimate a population parameter For instance, the sample mean x¯ is a point estimator of the population mean µ and the sample proportion p¯ is a point estimator of the population proportion p Because a point estimator cannot be expected to provide the exact value of the population parameter, an interval estimate is often computed by adding and subtracting a value, called the margin of error, to the point estimate The general form of an interval estimate is as follows: Point estimate Ϯ Margin of error 310 Chapter Interval Estimation The purpose of an interval estimate is to provide information about how close the point estimate, provided by the sample, is to the value of the population parameter In this chapter we show how to compute interval estimates of a population mean µ and a population proportion p The general form of an interval estimate of a population mean is x¯ Ϯ Margin of error Similarly, the general form of an interval estimate of a population proportion is p¯ Ϯ Margin of error The sampling distributions of x¯ and p¯ play key roles in computing these interval estimates 8.1 CD file Lloyd’s Population Mean: σ Known In order to develop an interval estimate of a population mean, either the population standard deviation σ or the sample standard deviation s must be used to compute the margin of error In most applications σ is not known, and s is used to compute the margin of error In some applications, however, large amounts of relevant historical data are available and can be used to estimate the population standard deviation prior to sampling Also, in quality control applications where a process is assumed to be operating correctly, or “in control,” it is appropriate to treat the population standard deviation as known We refer to such cases as the σ known case In this section we introduce an example in which it is reasonable to treat σ as known and show how to construct an interval estimate for this case Each week Lloyd’s Department Store selects a simple random sample of 100 customers in order to learn about the amount spent per shopping trip With x representing the amount spent per shopping trip, the sample mean x¯ provides a point estimate of µ, the mean amount spent per shopping trip for the population of all Lloyd’s customers Lloyd’s has been using the weekly survey for several years Based on the historical data, Lloyd’s now assumes a known value of σ ϭ $20 for the population standard deviation The historical data also indicate that the population follows a normal distribution During the most recent week, Lloyd’s surveyed 100 customers (n ϭ 100) and obtained a sample mean of x¯ ϭ $82 The sample mean amount spent provides a point estimate of the population mean amount spent per shopping trip, µ In the discussion that follows, we show how to compute the margin of error for this estimate and develop an interval estimate of the population mean Margin of Error and the Interval Estimate In Chapter we showed that the sampling distribution of x¯ can be used to compute the probability that x¯ will be within a given distance of µ In the Lloyd’s example, the historical data show that the population of amounts spent is normally distributed with a standard deviation of σ ϭ 20 So, using what we learned in Chapter 7, we can conclude that the sampling distribution of x¯ follows a normal distribution with an unknown mean µ, and a known standard error of σx¯ ϭ σ͙͞n ϭ 20͙͞100 ϭ This sampling distribution is shown in Figure 8.1.* *We use the fact that the population of amounts spent has a normal distribution to conclude that the sampling distribution of x¯ has a normal distribution If the population did not have a normal distribution, we could rely on the central limit theorem and the large sample size of n ϭ 100 to conclude that the sampling distribution of x¯ is approximately normal In either case, the sampling distribution of x¯ would appear as shown in Figure 8.1 8.1 FIGURE 8.1 Population Mean: σ Known 311 SAMPLING DISTRIBUTION OF THE SAMPLE MEAN AMOUNT SPENT FROM SIMPLE RANDOM SAMPLES OF 100 CUSTOMERS Sampling distribution of x σx = σ n = 20 = 100 x µ Because the sampling distribution shows how values of x¯ are distributed around the population mean µ, the sampling distribution of x¯ provides information about the possible differences between x¯ and µ Using the standard normal probability table, we find that 95% of the values of any normally distributed random variable are within Ϯ1.96 standard deviations of the mean Thus, when the sampling distribution of x¯ is normally distributed, 95% of the x¯ values must be within Ϯ1.96σx¯ of the mean µ In the Lloyd’s example we know that the sampling distribution of x¯ is normally distributed with a standard error of σx¯ ϭ Because Ϯ1.96σx¯ ϭ 1.96(2) ϭ 3.92, we can conclude that 95% of all x¯ values obtained using a sample size of n ϭ 100 will be within Ϯ3.92 of the population mean µ See Figure 8.2 FIGURE 8.2 SAMPLING DISTRIBUTION OF x¯ SHOWING THE LOCATION OF SAMPLE MEANS THAT ARE WITHIN 3.92 OF µ Sampling distribution of x σx = 95% of all x values x µ 3.92 1.96 σ x 3.92 1.96 σ x 312 Chapter Interval Estimation In the introduction to this chapter we said that the general form of an interval estimate of the population mean µ is x¯ Ϯ margin of error For the Lloyd’s example, suppose we set the margin of error equal to 3.92 and compute the interval estimate of µ using x¯ Ϯ 3.92 To provide an interpretation for this interval estimate, let us consider the values of x¯ that could be obtained if we took three different simple random samples, each consisting of 100 Lloyd’s customers The first sample mean might turn out to have the value shown as x¯1 in Figure 8.3 In this case, Figure 8.3 shows that the interval formed by subtracting 3.92 from x¯1 and adding 3.92 to x¯1 includes the population mean µ Now consider what happens if the second sample mean turns out to have the value shown as x¯ in Figure 8.3 Although this sample mean differs from the first sample mean, we see that the interval formed by subtracting 3.92 from x¯ and adding 3.92 to x¯ also includes the population mean µ However, consider what happens if the third sample mean turns out to have the value shown as x¯3 in Figure 8.3 In this case, the interval formed by subtracting 3.92 from x¯3 and adding 3.92 to x¯3 does not include the population mean µ Because x¯3 falls in the upper tail of the sampling distribution and is farther than 3.92 from µ, subtracting and adding 3.92 to x¯3 forms an interval that does not include µ Any sample mean x¯ that is within the darkly shaded region of Figure 8.3 will provide an interval that contains the population mean µ Because 95% of all possible sample means are in the darkly shaded region, 95% of all intervals formed by subtracting 3.92 from x¯ and adding 3.92 to x¯ will include the population mean µ Recall that during the most recent week, the quality assurance team at Lloyd’s surveyed 100 customers and obtained a sample mean amount spent of x¯ ϭ 82 Using x¯ Ϯ 3.92 to construct the interval estimate, we obtain 82 Ϯ 3.92 Thus, the specific interval estimate of FIGURE 8.3 INTERVALS FORMED FROM SELECTED SAMPLE MEANS AT LOCATIONS x¯1, x¯ , AND x¯3 Sampling distribution of x σx = 95% of all x values x µ 3.92 3.92 x1 Interval based on x1 ± 3.92 x2 Interval based on x2 ± 3.92 x3 The population mean µ Interval based on x3 ± 3.92 (note that this interval does not include µ) 8.1 This discussion provides insight as to why the interval is called a 95% confidence interval Population Mean: σ Known 313 µ based on the data from the most recent week is 82 Ϫ 3.92 ϭ 78.08 to 82 ϩ 3.92 ϭ 85.92 Because 95% of all the intervals constructed using x¯ Ϯ 3.92 will contain the population mean, we say that we are 95% confident that the interval 78.08 to 85.92 includes the population mean µ We say that this interval has been established at the 95% confidence level The value 95 is referred to as the confidence coefficient, and the interval 78.08 to 85.92 is called the 95% confidence interval Another term sometimes associated with an interval estimate is the level of significance The level of significance associated with an interval estimate is denoted by the Greek letter α The level of significance and the confidence coefficient are related as follows: α ϭ Level of Significance ϭ Ϫ Confidence Coefficient The level of significance is also referred to as the significance level The level of significance is the probability that the interval estimation procedure will generate an interval that does not contain µ For example, the level of significance corresponding to a 95 confidence coefficient is α ϭ Ϫ 95 ϭ 05 In Lloyd’s case, the level of significance (α ϭ 05) is the probability of drawing a sample, computing the sample mean, and finding that x¯ lies in one of the tails of the sampling distribution (see x¯3 in Figure 8.3) When the sample mean happens to fall in the tail of the sampling distribution (and it will 5% of the time), the confidence interval generated will not contain µ With the margin of error given by zα/2(σ͙͞n ), the general form of an interval estimate of a population mean for the σ known case follows INTERVAL ESTIMATE OF A POPULATION MEAN: σ KNOWN x¯ Ϯ zα/2 σ ͙n (8.1) where (1 Ϫ α) is the confidence coefficient and zα/2 is the z value providing an area of α/2 in the upper tail of the standard normal probability distribution Let us use expression (8.1) to construct a 95% confidence interval for the Lloyd’s example For a 95% confidence interval, the confidence coefficient is (1 Ϫ α) ϭ 95 and thus, α ϭ 05 Using the tables of areas for the standard normal distribution, an area of α/2 ϭ 05/2 ϭ 025 in the upper tail provides z.025 ϭ 1.96 With the Lloyd’s sample mean x¯ ϭ 82, σ ϭ 20, and a sample size n ϭ 100, we obtain 82 Ϯ 1.96 82 Ϯ 3.92 20 ͙100 Thus, using expression (8.1), the margin of error is 3.92 and the 95% confidence interval is 82 Ϫ 3.92 ϭ 78.08 to 82 ϩ 3.92 ϭ 85.92 Although a 95% confidence level is frequently used, other confidence levels such as 90% and 99% may be considered Values of zα/2 for the most commonly used confidence levels are shown in Table 8.1 Using these values and expression (8.1), the 90% confidence interval for the Lloyd’s example is 82 Ϯ 1.645 82 Ϯ 3.29 20 ͙100 314 Chapter TABLE 8.1 Interval Estimation VALUES OF zα/2 FOR THE MOST COMMONLY USED CONFIDENCE LEVELS Confidence Level α α/2 zα/2 90% 95% 99% 10 05 01 05 025 005 1.645 1.960 2.576 Thus, at 90% confidence, the margin of error is 3.29 and the confidence interval is 82 Ϫ 3.29 ϭ 78.71 to 82 ϩ 3.29 ϭ 85.29 Similarly, the 99% confidence interval is 82 Ϯ 2.576 82 Ϯ 5.15 20 ͙100 Thus, at 99% confidence, the margin of error is 5.15 and the confidence interval is 82 Ϫ 5.15 ϭ 76.85 to 82 ϩ 5.15 ϭ 87.15 Comparing the results for the 90%, 95%, and 99% confidence levels, we see that in order to have a higher degree of confidence, the margin of error and thus the width of the confidence interval must be larger Using Excel We will use the Lloyd’s Department Store data to illustrate how Excel can be used to construct an interval estimate of the population mean for the σ known case Refer to Figure 8.4 as we describe the tasks involved The formula worksheet is in the background; the value worksheet appears in the foreground Enter Data: A label and the sales data are entered into cells A1:A101 Enter Functions and Formulas: The sample size and sample mean are computed in cells D4:D5 using Excel’s COUNT and AVERAGE functions, respectively The value worksheet shows that the sample size is 100 and the sample mean is 82 The value of the known population standard deviation (20) is entered into cell D7 and the desired confidence coefficient (.95) is entered into cell D8 The level of significance is computed in cell D9 by entering the formula ϭ1ϪD8; the value worksheet shows that the level of significance associated with a confidence coefficient of 95 is 05 The margin of error is computed in cell D11 using Excel’s CONFIDENCE function The CONFIDENCE function has three inputs: the level of significance (cell D9); the population standard deviation (cell D7); and the sample size (cell D4) Thus, to compute the margin of error associated with a 95% confidence interval, the following formula is entered into cell D11: ϭCONFIDENCE(D9,D7,D4) The resulting value of 3.92 is the margin of error associated with the interval estimate of the population mean amount spent per week Cells D13:D15 provide the point estimate and the lower and upper limits for the confidence interval Because the point estimate is just the sample mean, the formula ϭD5 is entered into cell D13 To compute the lower limit of the 95% confidence interval, x¯ Ϫ (margin of error), we enter the formula ϭD13-D11 into cell D14 To compute the upper limit of the 95% confidence interval, x¯ ϩ (margin of error), we enter the formula ϭD13ϩD11 into cell D15 The value worksheet shows a lower limit of 78.08 and an upper limit of 85.92 In other words, the 95% confidence interval for the population mean is from 78.08 to 85.92 8.1 FIGURE 8.4 10 11 12 13 14 15 16 17 100 101 102 Population Mean: σ Known 315 EXCEL WORKSHEET: CONSTRUCTING A 95% CONFIDENCE INTERVAL FOR LLOYD’S DEPARTMENT STORE A Amount 72 91 74 115 71 120 37 96 91 105 104 89 70 125 43 61 71 84 Note: Rows 18–99 are hidden B C D Interval Estimate of a Population Mean: σ Known Case E Sample Size =COUNT(A1:A101) Sample Mean =AVERAGE(A1:A101) Population Standard Deviation 20 Confidence Coefficient 0.95 Level of Significance =1-D8 Margin of Error =CONFIDENCE(D9,D7,D4) Point Estimate =D5 Lower Limit =D13-D11 Upper Limit =D13+D11 A Amount 72 91 74 115 71 120 37 96 10 91 11 105 12 104 13 89 14 70 15 125 16 43 17 61 100 71 101 84 102 B C D Interval Estimate of a Population Mean: σ Known Case Sample Size Sample Mean 100 82 Population Standard Deviation Confidence Coefficient Level of Significance 20 0.95 0.05 Margin of Error 3.92 Point Estimate Lower Limit Upper Limit 82 78.08 85.92 E A Template for Other Problems To use this worksheet as a template for another problem of this type, we must first enter the new problem data in column A Then, the cell formulas in cells D4 and D5 must be updated with the new data range and the known population standard deviation must be entered into cell D7 After doing so, the point estimate and a 95% confidence interval will be displayed in cells D13:D15 If a confidence interval with a different confidence coefficient is desired, we simply change the value in cell D8 We can further simplify the use of Figure 8.4 as a template for other problems by eliminating the need to enter new data ranges in cells D4 and D5 To so we rewrite the cell formulas as follows: Cell D4: ϭCOUNT(A:A) Cell D5: ϭAVERAGE(A:A) With the A:A method of specifying data ranges, Excel’s COUNT function will count the number of numeric values in column A and Excel’s AVERAGE function will compute the 316 Chapter The Lloyd’s data set includes a worksheet titled Template that uses the A:A method for entering the data ranges average of the numeric values in column A Thus, to solve a new problem it is only necessary to enter the new data into column A and enter the value of the known population standard deviation in cell D7 This worksheet can also be used as a template for text exercises in which the sample size, sample mean, and the population standard deviation are given In this type of situation we simply replace the values in cells D4, D5, and D7 with the given values of the sample size, sample mean, and the population standard deviation Interval Estimation Practical Advice If the population follows a normal distribution, the confidence interval provided by expression (8.1) is exact In other words, if expression (8.1) were used repeatedly to generate 95% confidence intervals, exactly 95% of the intervals generated would contain the population mean If the population does not follow a normal distribution, the confidence interval provided by expression (8.1) will be approximate In this case, the quality of the approximation depends on both the distribution of the population and the sample size In most applications, a sample size of n Ն 30 is adequate when using expression (8.1) to develop an interval estimate of a population mean If the population is not normally distributed, but is roughly symmetric, sample sizes as small as 15 can be expected to provide good approximate confidence intervals With smaller sample sizes, expression (8.1) should only be used if the analyst believes, or is willing to assume, that the population distribution is at least approximately normal NOTES AND COMMENTS The interval estimation procedure discussed in this section is based on the assumption that the population standard deviation σ is known By σ known we mean that historical data or other information are available that permit us to obtain a good estimate of the population standard deviation prior to taking the sample that will be used to develop an estimate of the population mean So technically we don’t mean that σ is actually known with certainty We just mean that we obtained a good estimate of the standard deviation prior to sampling and thus we won’t be using the same sample to estimate both the population mean and the population standard deviation The sample size n appears in the denominator of the interval estimation expression (8.1) Thus, if a particular sample size provides too wide an interval to be of any practical use, we may want to consider increasing the sample size With n in the denominator, a larger sample size will provide a smaller margin of error, a narrower interval, and greater precision The procedure for determining the size of a simple random sample necessary to obtain a desired precision is discussed in Section 8.3 Exercises Methods A simple random sample of 40 items resulted in a sample mean of 25 The population standard deviation is σ ϭ a What is the standard error of the mean, σx¯ ? b At 95% confidence, what is the margin of error? SELF test A simple random sample of 50 items from a population with σ ϭ resulted in a sample mean of 32 a Provide a 90% confidence interval for the population mean b Provide a 95% confidence interval for the population mean c Provide a 99% confidence interval for the population mean 8.1 Population Mean: σ Known 317 A simple random sample of 60 items resulted in a sample mean of 80 The population standard deviation is σ ϭ 15 a Compute the 95% confidence interval for the population mean b Assume that the same sample mean was obtained from a sample of 120 items Provide a 95% confidence interval for the population mean c What is the effect of a larger sample size on the interval estimate? A 95% confidence interval for a population mean was reported to be 152 to 160 If σ ϭ 15, what sample size was used in this study? Applications CD file Restaurant CD file Nielsen In an effort to estimate the mean amount spent per customer for dinner at an Atlanta restaurant, data were collected for a sample of 49 customers The data collected are shown in the CD file named Restaurant Based upon past studies the population standard deviation is assumed known with σ ϭ $5 a At 95% confidence, what is the margin of error? b Develop a 95% confidence interval estimate of the mean amount spent for dinner Nielsen Media Research conducted a study of household television viewing times during the p.m to 11 p.m time period The data contained in the CD file named Nielsen are consistent with the findings reported (The World Almanac, 2003) Based upon past studies the population standard deviation is assumed known with σ ϭ 3.5 hours Develop a 95% confidence interval estimate of the mean television viewing time per week during the p.m to 11 p.m time period A survey of small businesses with Web sites found that the average amount spent on a site was $11,500 per year (Fortune, March 5, 2001) Given a sample of 60 small businesses and a population standard deviation of σ ϭ $4000, what is the margin of error? Use 95% confidence What would you recommend if the study required a margin of error of $500? The National Quality Research Center at the University of Michigan provides a quarterly measure of consumer opinions about products and services (The Wall Street Journal, February 18, 2003) A survey of 10 restaurants in the Fast Food/ Pizza group showed a sample mean customer satisfaction index of 71 Past data indicate that the population standard deviation of the index has been relatively stable with σ ϭ a What assumption should the researcher be willing to make if a margin of error is desired? b Using 95% confidence, what is the margin of error? c What is the margin of error if 99% confidence is desired? CD file GPA A study was conducted of students admitted to the top graduate business schools The data contained in the CD file named GPA show the undergraduate grade point average for students and is consistent with the findings reported (“Best Graduate Schools,” U.S News and World Report, 2001) Using past years’ data, the population standard deviation can be assumed known with σ ϭ 28 What is the 95% confidence interval estimate of the mean undergraduate grade point average for students admitted to the top graduate business schools? 10 Playbill magazine reported that the mean annual household income of its readers is $119,155 (Playbill, December 2003) Assume this estimate of the mean annual household income is based on a sample of 80 households and, based on past studies, the population standard deviation is known to be σ ϭ $30,000 a Develop a 90% confidence interval estimate of the population mean b Develop a 95% confidence interval estimate of the population mean c Develop a 99% confidence interval estimate of the population mean d Discuss what happens to the width of the confidence interval as the confidence level is increased Does this result seem reasonable? Explain 690 Appendix D Self-Test Solutions and Answers to Even-Numbered Exercises For Drive4: Because the p-value ϭ 0073, Drive4 is significant For EightCyl: Because the p-value ϭ 0104, EightCyl is significant Chapter 14 a 5.42 b UCL ϭ 6.09, LCL ϭ 4.75 R chart: UCL ϭ R¯ D4 ϭ 1.6(1.864) ϭ 2.98 LCL ϭ R¯ D3 ϭ 1.6(.136) ϭ 22 x¯ chart: UCL ϭ x¯ ϩ A2 R¯ ϭ 28.5 ϩ 373(1.6) ϭ 29.10 LCL ϭ x¯ Ϫ A2 R¯ ϭ 28.5 Ϫ 373(1.6) ϭ 27.90 When p ϭ 06, the probability of accepting the lot is 25! (.06)0(1 Ϫ 06)25 ϭ 2129 f(0) ϭ 0!(25 Ϫ 0)! 12 p0 ϭ 02; producer’s risk ϭ 0599 p0 ϭ 06; producer’s risk ϭ 3396 Producer’s risk decreases as the acceptance criterion c is increased 14 n ϭ 20, c ϭ 16 a 95.4 b UCL ϭ 96.07, LCL ϭ 94.73 c No 18 20.01, 082 a b c d UCL LCL R Chart 4.23 x¯ Chart 6.57 4.27 0470 UCL ϭ 0989, LCL ϭ Ϫ.0049 (use LCL ϭ 0) p¯ ϭ 08; in control UCL ϭ 14.826, LCL ϭ Ϫ0.726 (use LCL ϭ 0) Process is out of control if more than 14 defective e In control with 12 defective f np chart 20 n! p x(1 Ϫ p)nϪx x!(n Ϫ x)! When p ϭ 02, the probability of accepting the lot is 25! f(0) ϭ (.02)0(1 Ϫ 02)25 ϭ 6035 0!(25 Ϫ 0)! 22 a UCL ϭ 0817, LCL ϭ Ϫ.0017 (use LCL ϭ 0) Estimate of standard deviation ϭ 86 UCL LCL R Chart 1121 x¯ Chart 3.112 3.051 10 f(x) ϭ 24 a .03 b ϭ 0802 Appendix E: Using Excel Functions Excel provides a wealth of functions for data management and statistical analysis If we know what function is needed, and how to use it, we can simply enter the function into the appropriate worksheet cell However, if we are not sure what functions are available to accomplish a task or are not sure how to use a particular function, Excel can provide assistance Finding the Right Excel Function In earlier versions of Excel, the Paste Function dialog box serves the same purpose as the Insert Function dialog box in Excel 2003 To identify the functions available in Excel, select the Insert menu and then choose Function from the list of options Alternatively, select the fx button on the formula bar Either approach provides the Insert Function dialog box shown in Figure The Search for a function box at the top of the Insert Function dialog box enables us to type a brief description of what we want to After doing so and clicking Go, Excel will search for and display, in the Select a function box, the functions that may accomplish our task In many situations, however, we may want to browse through an entire category of functions to see what is available For this task, the Or select a category box is helpful It contains a drop-down list of several categories of functions provided by Excel Figure shows that we selected the Statistical category As a result, Excel’s statistical functions FIGURE INSERT FUNCTION DIALOG BOX 692 Appendix E In earlier versions of Excel, a similar dialog box will appear It serves the same purpose as the Function Arguments dialog box in Excel 2003 Using Excel Functions appear in alphabetic order in the Select a function box We see the AVEDEV function listed first, followed by the AVERAGE function, and so on The AVEDEV function is highlighted in Figure 1, indicating it is the function currently selected The proper syntax for the function and a brief description of the function appear below the Select a function box We can scroll through the list in the Select a function box to display the syntax and a brief description for each of the statistical functions available For instance, scrolling down farther, we select the COUNTIF function See Figure Note that COUNTIF is now highlighted, and that immediately below the Select a function box we see COUNTIF(range,criteria), which indicates that the COUNTIF function contains two arguments, range and criteria In addition, we see that the description of the COUNTIF function is “Counts the number of cells within a range that meet the given condition.” If the function selected (highlighted) is the one we want to use, we click OK; the Function Arguments dialog box then appears The Function Arguments dialog box for the COUNTIF function is shown in Figure This dialog box assists in creating the appropriate arguments for the function selected When finished entering the arguments, we click OK; Excel then inserts the function into a worksheet cell Inserting a Function into a Worksheet Cell We will now show how to use the Insert Function and Function Arguments dialog boxes to select a function, develop its arguments, and insert the function into a worksheet cell In Section 2.1, we used Excel’s COUNTIF function to construct a frequency distribution for soft drink purchases Figure displays an Excel worksheet containing the soft drink FIGURE DESCRIPTION OF THE COUNTIF FUNCTION IN THE INSERT FUNCTION DIALOG BOX Appendix E 693 Using Excel Functions FIGURE FUNCTION ARGUMENTS DIALOG BOX FOR THE COUNTIF FUNCTION FIGURE EXCEL WORKSHEET WITH SOFT DRINK DATA AND LABELS FOR THE FREQUENCY DISTRIBUTION WE WOULD LIKE TO CONSTRUCT CD file SoftDrink Note: Rows 11–44 are hidden 10 45 46 47 48 49 50 51 52 A Brand Purchased Coke Classic Diet Coke Pepsi-Cola Diet Coke Coke Classic Coke Classic Dr Pepper Diet Coke Pepsi-Cola Pepsi-Cola Pepsi-Cola Pepsi-Cola Coke Classic Dr Pepper Pepsi-Cola Sprite B C Soft Drink Coke Classic Diet Coke Dr Pepper Pepsi-Cola Sprite D Frequency E 694 Appendix E Using Excel Functions data and labels for the frequency distribution we would like to construct We see that the frequency of Coke Classic purchases will go into cell D2, the frequency of Diet Coke purchases will go into cell D3, and so on Suppose we want to use the COUNTIF function to compute the frequencies for these cells and would like some assistance from Excel Step Select cell D2 Step Click fx on the formula bar (or select Insert and then choose Function) Step When the Insert Function dialog box appears: Select Statistical in the Or select a category box Select COUNTIF in the Select a function box Click OK Step When the Function Arguments box appears (see Figure 5): Enter $A$2:$A$51 in the Range box Enter C2 in the Criteria box (At this point, the value of the function will appear on the next-to-last line of the dialog box Its value is 19.) Click OK Step Copy cell D2 to cells D3:D6 The worksheet then appears as in Figure The formula worksheet is in the background; the value worksheet appears in the foreground The formula worksheet shows that the COUNTIF function was inserted into cell D2 We copied the contents of cell D2 into cells D3:D6 The value worksheet shows the proper class frequencies as computed We illustrated the use of Excel’s capability to provide assistance in using the COUNTIF function The procedure is similar for all Excel functions This capability is especially helpful if you not know what function to use or forget the proper name and/or syntax for a function FIGURE COMPLETED FUNCTION ARGUMENTS DIALOG BOX FOR THE COUNTIF FUNCTION Appendix E FIGURE 6 10 45 46 47 48 49 50 51 52 695 Using Excel Functions EXCEL WORKSHEET SHOWING THE USE OF EXCEL’S COUNTIF FUNCTION TO CONSTRUCT A FREQUENCY DISTRIBUTION A Brand Purchased Coke Classic Diet Coke Pepsi-Cola Diet Coke Coke Classic Coke Classic Dr Pepper Diet Coke Pepsi-Cola Pepsi-Cola Pepsi-Cola Pepsi-Cola Coke Classic Dr Pepper Pepsi-Cola Sprite Note: Rows 11–44 are hidden B C Soft Drink Coke Classic Diet Coke Dr Pepper Pepsi-Cola Sprite 10 45 46 47 48 49 50 51 52 D Frequency =COUNTIF($A$2:$A$51,C2) =COUNTIF($A$2:$A$51,C3) =COUNTIF($A$2:$A$51,C4) =COUNTIF($A$2:$A$51,C5) =COUNTIF($A$2:$A$51,C6) A Brand Purchased Coke Classic Diet Coke Pepsi-Cola Diet Coke Coke Classic Coke Classic Dr Pepper Diet Coke Pepsi-Cola Pepsi-Cola Pepsi-Cola Pepsi-Cola Coke Classic Dr Pepper Pepsi-Cola Sprite B E C D Soft Drink Frequency Coke Classic 19 Diet Coke Dr Pepper Pepsi-Cola 13 Sprite E This page intentionally left blank Index A B Acceptance criterion, 633 Acceptance sampling, 614 attributes plans, 638–639 company example, 633 computing the probability of accepting a lot, 633–635 defined, 631 lot, 631 multiple plans, 637–638 risks in, 632–633 selecting a plan, 635–637 variables plans, 639 Acceptance sampling plan, 633, 635–637 Accounting, use of statistics in, Addition law, 172–175 Adjusted multiple coefficient of determination, 573–574 Alliance Data Systems (regression analysis), 485 Alternative hypothesis in decision-making situations, 350 defined, 348 developing, 349–350 research hypotheses and, 349 summary of forms of, 350–351 testing the validity of a claim with, 349–350 American Society for Quality (ASQ), 610 Analysis of variance assumptions for, 425 conceptual overview, 425–427 interpretation of, 523–524 introduction to, 424–425 testing for the equality of k population means, 428 ANOVA table, 433 between-treatments estimate of population variance, 429–430 comparing the variance estimates: the F test, 430–432 hypothesis test about population means, conducted with Excel, 433–435 within-treatments estimate of population variance, 430 ANOVA table, 433, 516–517 Area, as measure of probability, 243–244 Assignable causes (quality control), 614 Association between two variables See Numerical measures Attributes sampling plans, 638–639 Average, 13 Baldrige Index, 612 Bar graphs, 12, 13 constructing with Excel, 36–38 with SWStatϩ, 29 defined, 36 histogram compared to, 55 for qualitative data, 36, 37 Basic requirements for assigning probabilities, 162–164 Bayes, Thomas, 187 Bayes’ theorem, 164 computing posterior probabilities with Excel, 189–190 probability calculations using, 185–190 tabular approach for, 188–189 Between-treatments estimates of population variance, 426–427, 429–430 Bimodal data, 93 Binomial probability distribution, 215 for accepting a lot, 633–635 binomial experiment, 216–217 clothing store problem, 217–221 defined, 216 Excel for computation of, 221–223 expected value and variance for, 223–224 Binomial probability function, 217, 220 Box plot constructing generally, 121–122 with SWStatϩ, 151 defined, 121 Business, use of statistics in, 3–5 BusinessWeek (statistics use), C Categorical data, Census, defined, 14 Central limit theorem defined, 288 sampling distribution of p¯ and, 297–298 sampling distribution of x¯ and, 288–289 theoretical proof of, 294 Central location, 93, 99 Chebyshev’s theorem, 116, 117–118 Chi-square distribution, 463–465, 469 Citibank (discrete probability distributions), 201 Classes, in a frequency distribution, 38, 42–43, 55 Classical method, of assigning probabilities, 162–163, 169 Clemance, Philip, 485 Cluster sampling, 301–302 Coefficient of determination See Simple linear regression Coefficient of variation, 108 Coefficients, interpretation of, 567–568 Colgate-Palmolive Company (statistics use), 31 Collectively exhausted events, 188n Combinations, in probability, 161 Common causes (quality control), 614–615 Comparisons involving means, 393 See also Analysis of variance case problems, 446–449 inferences about the difference between two population means: matched samples about, 417–419 hypothesis test conducted with Excel, 419–421 inferences about the difference between two population means: σ1 and σ2 known confidence interval constructed with Excel, 397–399 hypothesis test conducted with Excel, 401–402 hypothesis tests about µ1 Ϫ µ2, 399–401 interval estimation of µ1 Ϫ µ2, 395–397 practical advice, 403 inferences about the difference between two population means: σ1 and σ2 unknown, 405 confidence interval constructed with Excel, 407–409 hypothesis test conducted with Excel, 411–413 hypothesis tests about µ1 Ϫ µ2, 409–411 interval estimation of µ1 Ϫ µ2, 406–407 practical advice, 413 pooled sample variance, 413 Statistics in Practice: Fisons Corporation, 394 Comparisons involving proportions See also Test of independence hypothesis tests for proportions of a multinominal population about, 461–466 goodness of fit test conducted with Excel, 466–467 inferences about the difference between two population proportions confidence interval constructed with Excel, 454–455 698 Index Comparisons involving proportions (cont.) hypothesis test conducted with Excel, 457–458 hypothesis tests about p1 Ϫ p2, 456–457 interval estimation of p1 Ϫ p2, 452–454 Complement of an event, in probability, 171–172 Conditional probability, 177–182 Confidence coefficient, 313 Confidence interval, 313 for (simple linear regression), 514–515 estimates, skewed population and, 325–326 hypothesis testing and, 366–367 multiple regression analysis, 584–585, 608 regression analysis with SWStatϩ, 557 in simple linear regression, 527–529 Confidence level, 313 Consumer’s risk, in acceptance sampling, 632–633 Continuous probability distributions, 240 case problem, 269–270 exponential probability distribution computing probabilities for, 262–263 defined, 261 density function, 261 Excel for computation of, 263–264 Poisson distribution related to, 263 normal probability distribution computing probabilities for, 253–254 Excel for computation of, 256–258 normal curve, 246–248, 249 standard normal probability distribution, 248–253 tire company problem, 254–256 Statistics in Practice: Procter & Gamble, 241 uniform probability distribution area, as measure of probability, 243–245 defined, 242 density function, 242 Continuous quantitative data, 8, 55 Continuous random variables, 203 Control charts, 615–629 Convenience sampling, 303 Correlation coefficient Excel for computation of, 132–133 interpretation of, 130–131 population data, 130 sample data, 129 in simple linear regression, 505–506 Counting rules, in probability, 158–162 Covariance Excel for computation of, 132–133 interpretation of, 127–129 population, 127 sample, 125–126 Critical value approach See One-tailed tests; Two-tailed tests Crosby, Philip B., 611 Cross-sectional data, 7–8 Crosstabulation about, 63–65 constructing, with Excel, 66–69 defined, 63 Simpson’s paradox, 69–70 Cumulative frequency distributions, 48–51, 55 Cumulative percent frequency distribution, 50 Cumulative relative frequency distribution, 50 D Data bimodal, 93 multimodal, 93 Data acquisition errors, 12 Data and statistics, applications in business and economics, 3–5 data cross-sectional, 7–8 defined, elements, 6, observations, 6, qualitative, quantitative, 7, scales of measurement, 6–7 time series, 7–8 variables, data sources data acquisition errors, 12 existing, 8–9, 10 statistical studies, 9, 11–12 descriptive statistics, 12–14 kinds of statements of, 2–3 statistical analysis using Microsoft Excel, 15–19 statistical inference, 14–15, 16 Statistics in Practice: BusinessWeek, SWStatϩ and, 27–29 Data collection, scales of measurement and, 6–7 Data set, in Excel, 16–18 Data sources See Data and statistics Decision-making, hypothesis testing in, 350 Defect, defined, 612 de Fermat, Pierre, 156 Degree of belief, 163 Degrees of freedom, 318–321 Deming, W Edwards, 611 de Moivre, Abraham, 246 Dependent variables, 486 Descriptive statistics, 12 See also Numerical measures; Tabular and graphical presentations Deviation about the mean, 105 Discrete probability distributions, 200 auto dealer example, 204–207 binomial probability distribution, 215 binomial experiment, 216–217 clothing store problem, 217–221 Excel for computation of, 221–223 expected value and variance for, 223–224 expected value, 210 Excel for computation of, 211–212 hypergeometric probability distribution, 231–233 Excel for computation of, 233 Poisson probability distribution Excel for computation of, 228–230 function of, 226 length/distance intervals example, 227–228 properties of, 226 time intervals example, 226–227 random variables, 201 continuous, 203 defined, 202 discrete, 202 standard deviation, Excel for computation of, 211–212 Statistics in Practice: Citibank, 201 variance about, 210–211 Excel for computation of, 211–212 Discrete quantitative data, 8, 55 Discrete random variables, 202 Discrete uniform probability distribution, 206 Distance intervals, Poisson probability distribution and, 227–228 Distribution shape, measures of, 113–114 Dow Chemical (statistical quality control), 610 Dummy variables, 587 E Economics, use of statistics in, 4–5 Elements, 6, 8, 14 Empirical rule, 116–117 Errors See Data acquisition errors; Hypothetical testing Estimated multiple regression equation, 561–562 Estimated regression equation See Multiple regression; Simple linear regression Estimated regression line, 488 Estimation See Interval estimation; Multiple regression; Simple linear regression Events, in probability about, 167–169 addition law and, 172–175 complement of, 171–172 defined, 167 independent, 181–182 intersection of two, 173 multiplication law and, 181–182 mutually exclusive, 175 union of two, 172–173 Excel See also SWStatϩ ANOVA and testing for population means, 433–435 bar graphs, 36–38 binomial probability distribution, 221–223 coefficient of determination, 505, 506 699 Index confidence interval estimated regression equation, 531–533 population means, 397–399, 407–409 population proportions, 454–455 covariance and correlation coefficient, 132–133 crosstabulations, 66–69 data sets, 16–18 Descriptive Statistics tool, 108, 109, 110 deviations and squared deviations about the mean, 107 estimated multiple regression equation, 564–567 estimated regression equation, 493–495 expected value, 211–212 exponential probabilities, 263–264 frequency distribution, 33–34, 43–45, 51–54 F test and multiple regression, 578–579 goodness of fit test, 466–467 histograms, 46–48, 49, 51–54 hypergeometric probabilities, 233 hypothesis tests about the difference between population means, 401–402, 411–413, 419–421, 433–435 about the difference between population proportions, 457–458 population mean: σ known, 363–364 population mean: σ unknown, 374–376 population proportion, 382–383 interval estimate of the difference between population proportions, 454–455 population mean: σ known, 314–316 population mean: σ unknown, 322–323 population proportion, 332–334 mean, median, and mode, 94–95 multiple regression analysis, 578–579, 587–589 normal probabilities, 256–258 percent frequency distribution, 35 percentiles, 97–99 pie charts, 36–38 Poisson probability distribution, 227, 228–230 posterior probabilities, 189–190 prediction interval estimate, 531–533 quality control, 623–626 quartiles, 97–99 random numbers selection with, 274–278 R chart, 623–626 Regression tool ANOVA output, 523–524 estimated regression output, 523 F test, 524 regression statistics output, 524 residual plots, 535–540 restaurant problem example, 521–524 t test, 523 relative frequency distribution, 35 residual plots, 539–540 sample standard deviation, 108, 109 sample variance scatter diagram, 72–74, 493–495 simple random sampling, 274–278 skewness, 114 standard deviation, 211–212 statistical analysis using, 15–19 test of independence, 473–474 trendline, 72–74 2003, 27, 122 variance, 211–212 worksheets, 16–18 x¯ chart, 623–626 z-score, 115 Expected value for binomial probability distribution, 223–224 of a discrete random variable, 210 Excel and, 211–212 of p¯, in sampling distribution, 296–297 of x¯ , in sampling distribution, 286–287 Experimental outcomes, 157 Experimental statistical study, 11 Experiments, in probability, 157–158, 165 Exploratory data analysis advantage of, 122 box plot, 121–122 five-number summary, 120–121 stem-and-leaf display, 58–61 Exponential probability distribution computing probabilities for, 262–263 defined, 261 density function, 261 Excel for computation of, 263–264 Poisson distribution related to, 263 F Factorial, 161 Feigenbaum, A V., 611 Finance, use of statistics in, Finite population, sampling from, 274–278 Finite population correction factor, 287 Fisons Corporation (testing with statistical procedures), 394 Five-number summary, 120–121 Food Lion (interval estimation), 309 Formula worksheet, 18–19 Frequency distribution constructing, with Excel, 33–34, 43–45, 51–54 defined, 32 developing, 34 qualitative data and, 32–34 quantitative data and, 41–45 F test Excel’s Regression tool and, 524 in multiple regression, 577–580 for significance in regression, 515–517 variance estimates comparisons and, 430–432 G Galton, Francis, 486 Gauss, Carl Friedrich, 491 Goodness of fit test Excel for conducting, 466–467 multinomial population and, 462–466 Gosset, William Sealy, 318 Graphical presentations See Tabular and graphical presentations Grouped data, 136–139 H Histogram, 13, 14 bar graph compared to, 55 constructing, with Excel, 46–48, 49, 50, 51–54 for quantitative data, 45–48, 49, 50 Hypergeometric probability distribution defined, 231 Excel for computation of, 233 function, 231–233 Hypothesis testing, 347 See also Comparisons involving means; Comparisons involving proportions case problem, 390–392 interval estimation related to, 366–367 null and alternative hypotheses in decision-making situations, 350 defined, 348 developing, 349–350 forms of, 350–351 research hypotheses and, 349 testing the validity of a claim, 349–350 population mean: σ known and unknown See One-tailed tests; Two-tailed tests population proportion Excel for conducting, 382–383 how to conduct, 380–382 summary of tests about, 384 test statistic for, 381 procedures and steps of, 364–365 for proportions of a multinomial population about, 461–466 Excel and, 466–467 goodness of fit test, 462–467 Statistics in Practice: John Morrell & Company, 348 Type I and Type II errors, 351–353 I Independence See Test of independence Independent events, in probability, 181 Independent samples, 417 Independent variables, 486 Indicator (variables), 587 Individual significance, 577 Inferences about population means See Comparisons involving means 700 Index Infinite population, sampling from, 278–279 International Organization for Standardization (ISO), 612 International Paper (multiple regression analysis), 560 Internet, as source of data and statistical information, Interquartile range (IQR), 104 Intersection of two events, 173 Interval estimation, 308 See also Comparisons involving means; Comparisons involving proportions case problems, 343–346 general form of, 309 hypothesis testing related to, 366–367 population mean: σ known Excel for constructing, 314–316 margin of error and, 310–314 practical advice, 316 population mean: σ unknown about, 318–319, 320 Excel for constructing, 322–323 margin of error and, 319, 321 practical advice, 323 small sample for, 323–324 population proportion about, 331–332 Excel for constructing, 332–334 sample size determination, 334–335 procedures summary, 325 purpose of, 310 sample size determination, 328–329 in simple linear regression, 527 Statistics in Practice: Food Lion, 309 Interval scale of measurement, Ishikawa, Karou, 611 ISO 9000, 612 ith residual, 501 J John Morrell & Company (hypothesis testing), 348 Joint probabilities, 178–179 Joint probability table, 178–179 Judgment sampling, 303 Juran, Joseph, 611 L Leaf unit (stem-and-leaf display), 61 Least squares method See Multiple regression; Simple linear regression Length intervals, Poisson probability distribution and, 227–228 Level of significance, 313 in hypothesis testing, 352–353 observed, 357 Location, measures of See Numerical measures Lot acceptance sampling, 223 Lot, in acceptance sampling, 631, 633–635 Lower tail test See One-tailed tests M Malcolm Baldrige National Quality Award, 611–612 Marginal probabilities, 179 Margin of error, 309 See also Interval estimation Marketing, use of statistics in, Matched sample design, 421 Matched samples, 417–419 McClain, John O., 524, 533 MeadWestvaco Corporation (sampling), 272 Mean, 13 See also Comparisons involving means defined, 91 Excel for computation of, 94–95 sample and population, 91–92 trimmed, 99 Mean square, 577 Mean square due to error (MSE), 430 Mean square due to regression (MSR), 515–516 Mean square due to treatments (MSTR), 429–430 Mean square error (MSE), 512 Measurement area, as measure of probability, 243–244 scales of, 6–7 Median defined, 92–93 Excel for computation of, 94–95 Microsoft Excel See Excel Mode defined, 93–94 Excel for computation of, 94–95 Model assumptions in multiple regression, 575–577 in regression analysis, 510–511 Morton International (probability testing), 156 Multicollinearity, 581 Multimodal data, 93 Multinomial population Excel and, 466–467 goodness of fit test, 462–467 proportions of, hypothesis testing for, 461–467 Multiple coefficient of determination, 572–574 Multiple regression, 559 case problems, 603–607 complex qualitative independent variables, 591–592 estimated regression equation for estimation and prediction, 584–585 least squares method, 562 coefficients, interpretation of, 567–568 Excel’s Regression tool and, 564–567 trucking company example, 563–565 model estimated multiple regression equation, 561–562 regression equation, 561 regression model, 561 model assumptions, 575–577 multiple coefficient of determination, 572–574 qualitative independent variables, 586 parameter interpretation, 589–590 water filtration systems example, 587–589 significance tests for F test, 577–580 multicollinearity, 581 t test, 580 Statistics in Practice: International Paper, 560 SWStatϩ use with, 607–608 Multiple sampling plans, 637–638 Multiple-step experiments, in probability, 158–161 Multiplication law, in probability, 181–182 Mutually exclusive events, 175 Myerson, Roger, 246n N Nominal scale of measurement, Nonexperimental statistical study, 11–12 Nonprobability sampling technique, 303 Normal curve, in normal probability distribution, 246–248, 249 Normal probability distribution computing probabilities for, 253–254 defined, 246 density function, 247 Excel for computation of, 256–258 normal curve, 246–248, 249 standard normal probability distribution, 248–253 tire company problem, 254–256 np chart (quality control method), 223, 616, 628 Null hypothesis in decision-making situations, 350 defined, 348 developing, 349–350 research hypotheses and, 349 summary of forms for, 350–351 testing the validity of a claim, 349–350 Numerical measures (descriptive statistics), 89 association between two variables correlation coefficient, 129–131 covariance, 125–129 Excel and computing the covariance and correlation coefficient, 132–133 case problems, 149–151, 152 distribution shape, 113–114 exploratory data analysis box plot, 121–122 701 Index box plot construction using SWStatϩ, 151, 153–154 five-number summary, 120–121 grouped data, 136–139 location Excel and computing mean, median, and mode, 94–95 Excel and computing percentiles and quartiles, 97–99 mean, 91–92 median, 92–93 mode, 93–94 percentiles, 95–96 quartiles, 96–97 relative location Chebyshev’s theorem, 116, 117–118 detecting outliers, 117 empirical rule, 116–117 z-scores, 115 Statistics in Practice: Small Fry Design, 90 variability, 103 coefficient of variation, 108 Excel and computing sample variance and sample standard deviation, 108, 109 Excel’s Descriptive Statistics tool, 108, 109, 110 interquartile range, 104 range, 104 standard deviation, 106–107 variance, 105–106, 107 weighted mean, 135–136 O Observational statistical study, 11–12 Observations, defined, 6, Observed level of significance, 357 Ogive, 50–51 One-tailed tests (hypothesis) population mean: σ known critical value, 356 critical value approach, 358–359 Excel and, 363–364 lower tail test, 354, 359 p-value approach, 356–358, 363–364 summary of, 365 test statistic, 355–356 upper tail test, 354, 359 population mean: σ unknown about, 371–372 Excel and, 374–376 summary and practical advice, 377 Open-end class, 55 Operating characteristic (OC) curve, 635 Ordinal scale of measurement, Outliers box plots for identifying, 121–122 detecting, 117 Out of control (quality control), 615 Overall sample mean, 426, 435 Overall significance, 577 P Parameters defined, 272–273 in multiple regression, 589–590 Partitioning, 433 Pascal, Blaise, 156 p chart (quality control method), 616, 626–628 Pearson, Karl, 486 Pearson product moment correlation coefficient population data, 130 sample data, 129 Percent frequency distribution constructing, with Excel, 35 defined, 34 for qualitative data, 34–35 for quantitative data, 45 Percentiles calculating, 95–96 defined, 95 Excel for computation of, 97–99 Permutations, in probability, 161–162 Personal interview surveys, 11–12 Pie charts constructing, with Excel, 36–38 defined, 36 for qualitative data, 36, 37 Planning value, 329 Point estimate, 281–282, 309 Point estimation in sampling, 280–282 in simple linear regression, 527 Point estimator, 91, 281–282 Poisson, Siméon, 226 Poisson probability distribution defined, 226 Excel for computation of, 228–230 exponential distributions related to, 263 length/distance intervals example, 227–228 properties of experiment of, 226 time intervals example, 226–227 Pooled estimates, 427 Pooled estimator of p, 456 Pooled sample variance, 413 Population defined, 14, 272 finite, sampling from, 274–278 infinite, sampling from, 278–279 Population covariance, 127 Population mean, 92 Population parameters, defined, 91 Population proportion See also Comparisons involving proportions constructing, with Excel, 332–334 hypothesis tests about how to conduct, 380–382 summary of, 384 using Excel, 382–383 interval estimation and, 331–335 sample size determination, 334–335 Population standard deviation, 106 Population variance, 105 Posterior probabilities, 185 Excel for computation of, 189–190 Prediction See Multiple regression; Simple linear regression Prediction interval In multiple regression, 584–585, 608 in simple linear regression, 527, 529–531, 557 PredInt.xls macro, 524, 533 Prior probability, 185 Probability, 155 See also Discrete probability distributions addition law, 172–175 assigning, 162–164, 169 Bayes’ theorem and, 185–190 case problem, 198–199 combinations, 161 complement of an event, 171–172 conditional, 177–182 counting rules, 158–162 defined, 156 events and, 167–169 example, 164–165 experiments in, 157–158 joint, 178–179 marginal, 179 permutations, 161–162 posterior, 185, 189–190 prior, 185 Statistics in Practice: Morton International, 156 Probability distribution, for a random variable, 204–207 Probability function, 204–207 Probability sampling techniques, 303 Probability tree, 186–187 Procter & Gamble (continuous probability distribution), 241 Producer’s risk, in acceptance sampling, 632–633 Production, use of statistics in, Proportions See Comparisons involving proportions p-value approach See One-tailed tests; Twotailed tests Q Qualitative data, See also Tabular and graphical presentations Qualitative independent variables, 586–592 Qualitative variable, Quality, defined, 610, 611 Quality assurance, 613 Quality control, 614 See also Statistical methods for quality control Quality engineering, 613–614 Quantitative data, See also Tabular and graphical presentations discrete or continuous, 8, 55 702 Index Quantitative variable, Quartiles defined, 96–97 Excel for computation of, 97–99 Questionnaires, 11–12 R Random experiments, 165 Random variables, 201 continuous, 203 defined, 202 discrete, 202 Range, 104 Ratio scale of measurement, R chart (quality control method), 616 Excel for constructing, 623–626 statistical process control, 621–623 Regression analysis/equation/model, 486, 487, 489 See also Multiple regression; Simple linear regression Relative frequency distribution constructing, with Excel, 35 defined, 34 for qualitative data, 34–35 for quantitative data, 45 Relative frequency method, of assigning probabilities, 163 Relative location See Numerical measures Research hypothesis testing, 349 Residual analysis See Simple linear regression Residual plots See Simple linear regression Risk, in acceptance sampling, 632–633 S Sample correlation coefficient, 129 Sample covariance, 125–126 Sample, defined, 14, 272 Sample mean, 91–92, 273, 282 Sample point, 157 Sample size, determining, in interval estimation, 328–329, 334–335 Sample space, 157 Sample standard deviation, 106 Excel for computation of, 108, 109 Sample statistic, in point estimation, 280 Sample statistics, defined, 91 Sample survey, defined, 14 Sample variance, 105 Excel for computation of, 108, 109 pooled, 413 Sampling, 271 See also Acceptance sampling company problem example, 273–274 parameters, 272–273 point estimation and, 280–282 population, defined, 272 sample, defined, 272 simple random sampling from a finite population, 274–278 from an infinite population, 278–279 Statistics in Practice: MeadWestvaco Corporation, 272 Sampling distributions defined, 284 introduction to, 283–286 of p¯ expected value, 296–297 form of, 297–298 formula for, 296 practical value of, 298–299 standard deviation, 297 of x¯ company problem example, 290 defined, 286 expected value, 286–287 form of, 288–290 practical value of, 290–292 sample size related to, 292–293 standard deviation, 287–288 Sampling methods cluster, 301–302 convenience, 303 judgment, 303 stratified random, 301, 302 systematic, 302–303 Scales of measurement, 6–7 Scatter diagram constructing, with Excel, 72–74, 493–495 defined, 70–71 for regression analysis, 489–490 types of relationships depicted by, 72 Shadow stocks, 83 Significance See Level of significance Significance, testing for See Multiple regression; Simple linear regression Significance tests (hypothesis testing), 353 Simple linear regression, 484 case problems, 551–555, 556 coefficient of determination about, 501–505 correlation coefficient, 505–506 Excel for computation of, 505, 506 defined, 486 equation, 487 estimated regression equation, 487–488 confidence interval estimate of the mean value of y, 527–529 for estimation and prediction, 527–533 Excel for developing confidence and prediction interval estimates, 531–533 interval estimation, 527 point estimation, 527 prediction interval estimate of an individual value of y, 529–531 Excel’s Regression tool ANOVA output, interpretation of, 523–524 estimated regression equation output, interpretation of, 523 F test, 524 regression statistics output, interpretation of, 524 restaurant problem example, 521–524 t test, 523 least squares method about, 489–493 defined, 489 Excel for scatter diagrams and computing estimated regression analysis, 493–495 model, 486–488 model assumptions, 510–511 regression analysis with SWStatϩ, 555–558 regression model and regression equation, 486–487 residual analysis, 535 Excel’s Regression tool for constructing residual plot, 539–540 residual plot against x, 536–539 residual plot against ˆy, 539 standardized residual plot with SWStatϩ, 558 Statistics in Practice: Alliance Data Systems, 485 testing for significance, 511 confidence interval for 1, 514–515 estimate of σ2, 512 F test, 515–517 interpretation of, cautions about, 517–518 t test, 512–514 Simple random sampling from a finite population, 274–278 from an infinite population, 278–279 selecting a sample with Excel, 274–278 Simpson’s paradox, 69–70 Single-sample plan, 637 Six Sigma, 612–614 Skewness confidence interval estimates and, 325–326 in a histogram, 46, 47 in shape of a distribution, 113–114 Small Fry Design (descriptive statistics), 90 Squared deviations about the mean, 105 Standard deviation Excel for computation of, 211–212 as measure of variability, 106–108 of p¯, in sampling distribution, 297 as risk measure, 110 in variance, 211 of x¯ , in sampling distribution, 287–288 Standard error, 297 Standard error of the estimate, 512 Standard error of the mean, 288 Standard error of the proportion, 297 Standardized value (z-score), 115, 117 Standard normal probability distribution, 248–253 Stationarity assumption, 217 Statistical analysis using Microsoft Excel, 15–19 Statistical inference, defined, 14–15, 16 Statistical methods for quality control, 609 acceptance sampling, 631–632 company example, 633 computing the probability of accepting a lot, 633–635 703 Index multiple plans, 637–638 selecting a plan, 635–637 philosophies and frameworks ISO 9000, 612 Malcolm Baldrige National Quality Award, 611–612 Six Sigma, 612–614 statistical process control, 614 control charts, 615–616 Excel for constructing an R chart and an x¯ chart, 623–626 interpretation of control charts, 629 np chart, 628 p chart, 626–628 R chart, 621–623 x¯ chart: process mean and standard deviation known and unknown, 616–621 Statistics in Practice: Dow Chemical, 610 total quality (TQ), 610–611 Statistical process control See Statistical methods for quality control Statistical studies experimental, nonexperimental/observational, 11–12 Statistics, See also Data and statistics Stem-and-leaf display, 58–61 Strata, 301 Stratified random sampling, 301, 302 Stretched stem-and-leaf display, 60 Subjective method, of assigning probabilities, 163–164 Sum of squares due to error (SSE), 430, 501–502 Sum of squares due to regression (SSR), 504 Sum of squares due to treatments (SSTR), 429–430 Surveys, 11–12 SWStatϩ (Excel add-in), 27 bar graph construction using, 29 box plot construction using, 151, 153–154 installing and running, 28 multiple regression analysis with, 607–608 regression analysis with, 555–558 standardized residual plot with, 558 using, 28–29 Symmetric histogram, 46, 47 Systematic sampling, 302–303 T Tabular and graphical presentations (descriptive statistics), 12–14, 30 case problem, 87–88 crosstabulations, 63–65 constructing, with Excel, 66–69 Simpson’s paradox, 69–70 exploratory data analysis: stem-and-leaf display, 58–61 qualitative data bar graphs, 36 bar graphs constructed with Excel, 36–38 frequency distribution, 32–33 frequency distribution constructed with Excel, 33–34 percent frequency distribution, 34 percent frequency distribution constructed with Excel, 35 pie charts, 36, 37 pie charts constructed with Excel, 36–38 relative frequency distribution, 34 relative frequency distribution constructed with Excel, 35 quantitative data cumulative distributions, 48–51 frequency distribution, 41–43 frequency distribution constructed with Excel, 43–45, 51–54 histogram constructed with Excel, 48, 49, 50, 51–54 histogram presentation of, 45–46, 47 percent frequency distribution, 45 relative frequency distribution, 45 scatter diagram about, 70–71 constructed with Excel, 72–74 Statistics in Practice: Colgate-Palmolive Company, 31 summary of methods, 79 trendline about, 70–71 constructed with Excel, 72–74 Tabular approach for Bayes’ theorem calculations, 188–189 Taguchi, Genichi, 611 t distribution, 318–321 Testing for significance See Simple linear regression Test of independence See also Comparisons involving proportions about, 469–473 case problem, 483 Excel for conducting, 473–474 Statistics in Practice: United Way, 451 Time intervals, Poisson probability distribution and, 226–227 Time series data, 7–8 Total quality (TQ), 610–611 Total sum of squares (SST), 502–503 Tree diagram, 159–160 Trendline constructing, with Excel, 72–74 defined, 70–71 Trimmed mean, 99 t test, 512–514, 523, 580 Two-tailed tests (hypothesis) population mean: σ known critical value approach, 362 Excel and, 363–364 general form for, 360 p-value approach, 361–364 summary of, 365 population mean: σ unknown, 372–374, 377 Type I and Type II errors See Hypothesis testing U Uniform probability distribution area, as measure of probability, 243–244 defined, 242 density function, 242 Union of two events, 172–173 United Way (test of independence), 451 Upper tail test See One-tailed tests (hypothesis) V Validity of a claim (hypothesis testing), 349–350 Value worksheet, 18 Variability, measures of See Numerical measures Variables See also Numerical measures; Random variables defined, dependent, in regression analysis, 486 dummy, 587 independent, in regression analysis, 486 qualitative, qualitative independent, 586–592 quantitative, Variables sampling plans, 639 Variance for binomial probability distribution, 223–224 defined, 105 of a discrete random variable, 210–211 Excel for computation of, 211–212 as measure of variability, 105–106, 107 Venn diagram, 171–172 W Weighted mean, 135–136 Whiskers (in box plots), 121 Within-treatments estimates of population variance, 427, 430 Worksheets See Excel X x¯ chart (quality control method) defined, 615–616 Excel for constructing, 623–626 process mean and standard deviation known and unknown, 616–621 Z z-scores, 115 for detecting outliers, 117 Essentials of Modern Business Statistics 3e Data Disk Chapter BWS&P Hotel Minisystems Music Norris Shadow02 Table 1.1 Table 1.6 Table 1.7 Exercise 14 Table 1.5 Table 1.8 Chapter ApTest Audit AutoData Baseball Broker BWBooks CEOs Client Computer Comstock Concerts Crosstab Dow Fortune Frequency Golf HighLow IBD Income Marathon Names NFL OccupSat PelicanStores Restaurant RevEmps Scatter Shadow SoftDrink Spending Stereo StockPrices TVMedia Wageweb Table 2.9 Table 2.5 Exercise 38 Exercise Exercise 26 Exercise Exercise Exercise 10 Exercise 21 Exercise 42 Exercise 20 Exercise 29 Exercise 41 Exercise 51 Exercise 11 Exercise 40 Exercise 46 Exercise 34 Exercise 44 Exercise 28 Exercise Exercise 37 Exercise 48 Case Problem Table 2.10 Exercise 49 Exercise 30 Exercise 43 Table 2.1 Exercise 17 Table 2.13 Exercise 27 Exercise Exercise 18 Chapter Asian Beer Broker Cameras Cities DowS&P Health Hotels Income Case Problem Exercise 65 Exercise & 22 Exercise 12 Exercise 64 Exercise 50 Case Problem Exercise Exercise 62 MPG Mutual NCAA Notebook Orders Payroll PCs PelicanStores Property Retainer Salary Speakers Stereo Temperature Visa WageWeb Websites Exercise 11 Exercise 44 Exercise 34 Exercise 23 Exercise 20 Exercise 42 Exercise 49 Case Problem Exercise 40 Exercise 59 Table 3.1 Exercise 35 Table 3.6 Exercise 51 Exercise 58 Exercise 33 Exercise Chapter Judge Case Problem Chapter American League Dining EAI MutualFund National League Exercise Exercise 38 Section 7.1 Exercise 10 Figure 7.1 Chapter ActTemps Auto Balance Bock FastFood Flights GPA GulfProp JobSatisfaction Lloyd's Miami Nielsen NYSEStocks OpenEndFunds Restaurant Scheer TeeTimes TVtime Exercise 49 Case Problem Table 8.3 Case Problem Exercise 18 Exercise 48 Exercise Case Problem Exercise 37 Section 8.1 Exercise 17 Exercise Exercise 47 Exercise 22 Exercise Table 8.4 Section 8.4 Exercise 20 BLS Coffee Diamonds Drowsy Fowle GolfTest Orders Quality RentalRates SuperBowl UsedCars Viewers WageRate WomenGolf Case Problem Section 9.3 Exercise 29 Exercise 44 Exercise 21 Section 9.3 Section 9.4 Case Problem Exercise 16 Exercise 40 Exercise 32 Exercise 30 Exercise 19 Section 9.5 Chapter 10 Cargo CheckAcct Digital Earnings ExamScores Funds Florida Golf HomeStyle IDSalary Matched Medical1 Medical2 Mortgage Mutual NCP NFL Resorts SAT SATVerbal Ships SoftwareTest Stress Technology Traffic Trucks TVRadio Exercise 13 Section 10.2 Exercise 39 Exercise 26 Section 10.1 Exercise 43 Exercise 42 Case Problem Section 10.1 Case Problem Table 10.2 Case Problem Case Problem Exercise Exercise 40 Table 10.3 Exercise 46 Exercise 45 Exercise 18 Exercise 16 Exercise 36 Table 10.1 Exercise 35 Exercise 34 Exercise 33 Exercise 44 Exercise 25 Chapter 11 Alber's NYReform Research TaxPrep Section 11.3 Case Problem Section 11.2 Section 11.1 Airport Alumni Armand's Boats Boots Cars Cities EmpRev HoursPts Hydration1 IPO IRSAudit Jensen JobSat MktBeta MLB MtnBikes NAEP OffRates Options PCs Printers Safety Salaries Sales VPSalary Chapter 13 Alumni Auto2 Backpack Brokers Butler Consumer Enquirer Exer2 Football ForFunds FuelEcon HomeValue Johnson MLB NBA Repair Schools Showtime SportsCar Stroke Trucks Chapter 12 Chapter AirRating Applicant Section 9.4 Exercise 47 Absent ADRs AgeCost Exercise 58 Exercise 49 Exercise 59 Exercise 11 Case Problem Table 12.1 Exercise Exercise 27 Exercises & 19 Exercise 20 Exercise 12 Exercise 60 Exercise 35 Exercise 52 Exercise 61 Exercise 56 Exercise 55 Exercise 54 Case Problem Exercise Case Problem Exercise 36 Exercise 53 Exercises 10, 28, & 41 Exercise 22 & 30 Case Problem Exercise 14 Exercise Exercise Case Problem Exercise 45 Exercise Exercise 25 Tables 13.1 & 13.2 Case Problem Case Problem Exercise Exercise 37 Exercise Exercise 46 Exercise 44 Table 13.6 Exercises 6, 16, & 24 Exercises 10, 18, & 26 Exercise 35 & 36 Exercises 9, 17, & 30 Exercises 5, 15, 23, & 29 Exercise 31 Exercise 38 Exercise 47 Chapter 14 Coffee Jensen Tires Exercise 20 Table 14.2 Exercise ... 2. 060 2. 056 2. 0 52 2.048 2. 045 2. 0 42 2. 021 2. 009 2. 000 1.990 1.984 1.960 2. 764 2. 718 2. 681 2. 650 2. 624 2. 6 02 2.583 2. 567 2. 5 52 2.539 2. 528 2. 518 2. 508 2. 500 2. 4 92 2.485 2. 479 2. 473 2. 467 2. 4 62. .. 2. 4 62 2.457 2. 423 2. 403 2. 390 2. 374 2. 364 2. 326 3.169 3.106 3.055 3.0 12 2.977 2. 947 2. 921 2. 898 2. 878 2. 861 2. 845 2. 831 2. 819 2. 807 2. 797 2. 787 2. 779 2. 771 2. 763 2. 756 2. 750 2. 704 2. 678 2. 660 2. 639... 2. 365 2. 306 2. 2 62 3.365 3.143 2. 998 2. 896 2. 821 4.0 32 3.707 3.499 3.355 3 .25 0 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 80 100 ϱ 879 876 873 870 868 866 865 863 862