Đang tải... (xem toàn văn)
(BQ) Part 2 book “Statistics for business and economics” has contents: Tests of goodness of fit and independence, simple linear regression, multiple regression, index numbers, nonparametric methods, statistical methods for quality control, statistical methods for quality control,… and other contents.
Find more at www.downloadslide.com CHAPTER 12 Tests of Goodness of Fit and Independence CONTENTS STATISTICS IN PRACTICE: UNITED WAY 12.1 GOODNESS OF FIT TEST: A MULTINOMIAL POPULATION 12.2 TEST OF INDEPENDENCE 12.3 GOODNESS OF FIT TEST: POISSON AND NORMAL DISTRIBUTIONS Poisson Distribution Normal Distribution Find more at www.downloadslide.com Statistics in Practice STATISTICS 473 in PRACTICE UNITED WAY* ROCHESTER, NEW YORK United Way of Greater Rochester is a nonprofit organization dedicated to improving the quality of life for all people in the seven counties it serves by meeting the community’s most important human care needs The annual United Way/Red Cross fund-raising campaign, conducted each spring, funds hundreds of programs offered by more than 200 service providers These providers meet a wide variety of human needs— physical, mental, and social—and serve people of all ages, backgrounds, and economic means Because of enormous volunteer involvement, United Way of Greater Rochester is able to hold its operating costs at just eight cents of every dollar raised The United Way of Greater Rochester decided to conduct a survey to learn more about community perceptions of charities Focus-group interviews were held with professional, service, and general worker groups to get preliminary information on perceptions The information obtained was then used to help develop the questionnaire for the survey The questionnaire was pretested, modified, and distributed to 440 individuals; 323 completed questionnaires were obtained A variety of descriptive statistics, including frequency distributions and crosstabulations, were provided from the data collected An important part of the analysis involved the use of contingency tables and chisquare tests of independence One use of such statistical tests was to determine whether perceptions of administrative expenses were independent of occupation The hypotheses for the test of independence were: H0: Perception of United Way administrative expenses is independent of the occupation of the respondent *The authors are indebted to Dr Philip R Tyler, marketing consultant to the United Way, for providing this Statistics in Practice United Way programs meet the needs of children as well as adults © Ed Bock/CORBIS Ha: Perception of United Way administrative expenses is not independent of the occupation of the respondent Two questions in the survey provided the data for the statistical test One question obtained data on perceptions of the percentage of funds going to administrative expenses (up to 10%, 11–20%, and 21% or more) The other question asked for the occupation of the respondent The chi-square test at a 05 level of significance led to rejection of the null hypothesis of independence and to the conclusion that perceptions of United Way’s administrative expenses did vary by occupation Actual administrative expenses were less than 9%, but 35% of the respondents perceived that administrative expenses were 21% or more Hence, many had inaccurate perceptions of administrative costs In this group, productionline, clerical, sales, and professional-technical employees had more inaccurate perceptions than other groups The community perceptions study helped United Way of Rochester to develop adjustments to its programs and fund-raising activities In this chapter, you will learn how a statistical test of independence, such as that described here, is conducted In Chapter 11 we showed how the chi-square distribution could be used in estimation and in hypothesis tests about a population variance In Chapter 12, we introduce two additional hypothesis testing procedures, both based on the use of the chi-square distribution Like other hypothesis testing procedures, these tests compare sample results with those that are expected when the null hypothesis is true The conclusion of the hypothesis test is based on how “close” the sample results are to the expected results Find more at www.downloadslide.com 474 Chapter 12 Tests of Goodness of Fit and Independence In the following section we introduce a goodness of fit test for a multinomial population Later we discuss the test for independence using contingency tables and then show goodness of fit tests for the Poisson and normal distributions 12.1 The assumptions for the multinomial experiment parallel those for the binomial experiment with the exception that the multinomial has three or more outcomes per trial Goodness of Fit Test: A Multinomial Population In this section we consider the case in which each element of a population is assigned to one and only one of several classes or categories Such a population is a multinomial population The multinomial distribution can be thought of as an extension of the binomial distribution to the case of three or more categories of outcomes On each trial of a multinomial experiment, one and only one of the outcomes occurs Each trial of the experiment is assumed to be independent, and the probabilities of the outcomes remain the same for each trial As an example, consider the market share study being conducted by Scott Marketing Research Over the past year market shares stabilized at 30% for company A, 50% for company B, and 20% for company C Recently company C developed a “new and improved” product to replace its current entry in the market Company C retained Scott Marketing Research to determine whether the new product will alter market shares In this case, the population of interest is a multinomial population; each customer is classified as buying from company A, company B, or company C Thus, we have a multinomial population with three outcomes Let us use the following notation for the proportions pA ϭ market share for company A pB ϭ market share for company B pC ϭ market share for company C Scott Marketing Research will conduct a sample survey and compute the proportion preferring each company’s product A hypothesis test will then be conducted to see whether the new product caused a change in market shares Assuming that company C’s new product will not alter the market shares, the null and alternative hypotheses are stated as follows H0: pA ϭ 30, pB ϭ 50, and pC ϭ 20 Ha: The population proportions are not pA ϭ 30, pB ϭ 50, and pC ϭ 20 If the sample results lead to the rejection of H0 , Scott Marketing Research will have evidence that the introduction of the new product affects market shares Let us assume that the market research firm has used a consumer panel of 200 customers for the study Each individual was asked to specify a purchase preference among the three alternatives: company A’s product, company B’s product, and company C’s new product The 200 responses are summarized here The consumer panel of 200 customers in which each individual is asked to select one of three alternatives is equivalent to a multinomial experiment consisting of 200 trials Company A’s Product Observed Frequency Company B’s Product Company C’s New Product 48 98 54 We now can perform a goodness of fit test that will determine whether the sample of 200 customer purchase preferences is consistent with the null hypothesis The goodness Find more at www.downloadslide.com 12.1 475 Goodness of Fit Test: A Multinomial Population of fit test is based on a comparison of the sample of observed results with the expected results under the assumption that the null hypothesis is true Hence, the next step is to compute expected purchase preferences for the 200 customers under the assumption that pA ϭ 30, pB ϭ 50, and pC ϭ 20 Doing so provides the expected results Company A’s Product Expected Frequency Company B’s Product Company C’s New Product 200(.30) ϭ 60 200(.50) ϭ 100 200(.20) ϭ 40 Thus, we see that the expected frequency for each category is found by multiplying the sample size of 200 by the hypothesized proportion for the category The goodness of fit test now focuses on the differences between the observed frequencies and the expected frequencies Large differences between observed and expected frequencies cast doubt on the assumption that the hypothesized proportions or market shares are correct Whether the differences between the observed and expected frequencies are “large” or “small” is a question answered with the aid of the following test statistic TEST STATISTIC FOR GOODNESS OF FIT ( fi Ϫ ei )2 ei iϭ1 k ϭ ͚ (12.1) where fi ϭ observed frequency for category i ei ϭ expected frequency for category i k ϭ the number of categories Note: The test statistic has a chi-square distribution with k Ϫ degrees of freedom provided that the expected frequencies are or more for all categories The test for goodness of fit is always a one-tailed test with the rejection occurring in the upper tail of the chi-square distribution An introduction to the chi-square distribution and the use of the chi-square table were presented in Section 11.1 Let us continue with the Scott Market Research example and use the sample data to test the hypothesis that the multinomial population retains the proportions pA ϭ 30, pB ϭ 50, and pC ϭ 20 We will use an α ϭ 05 level of significance We proceed by using the observed and expected frequencies to compute the value of the test statistic With the expected frequencies all or more, the computation of the chi-square test statistic is shown in Table 12.1 Thus, we have ϭ 7.34 We will reject the null hypothesis if the differences between the observed and expected frequencies are large Large differences between the observed and expected frequencies will result in a large value for the test statistic Thus the test of goodness of fit will always be an upper tail test We can use the upper tail area for the test statistic and the p-value approach to determine whether the null hypothesis can be rejected With k Ϫ ϭ Ϫ ϭ degrees of freedom, the chi-square table (Table of Appendix B) provides the following: Area in Upper Tail Value (2 df) 10 05 025 01 005 4.605 5.991 7.378 9.210 10.597 ϭ 7.34 Find more at www.downloadslide.com 476 TABLE 12.1 Chapter 12 Tests of Goodness of Fit and Independence COMPUTATION OF THE CHI-SQUARE TEST STATISTIC FOR THE SCOTT MARKETING RESEARCH MARKET SHARE STUDY Category Hypothesized Proportion Observed Frequency ( fi ) Company A Company B Company C 30 50 20 48 98 54 Total Expected Frequency (ei ) 60 100 40 Difference ( fi ؊ ei ) Squared Difference ( fi ؊ ei )2 Squared Difference Divided by Expected Frequency ( fi ؊ ei )2/ei Ϫ12 Ϫ2 14 144 196 2.40 0.04 4.90 200 ϭ 7.34 The test statistic ϭ 7.34 is between 5.991 and 7.378 Thus, the corresponding upper tail area or p-value must be between 05 and 025 With p-value Յ α ϭ 05, we reject H0 and conclude that the introduction of the new product by company C will alter the current market share structure Minitab or Excel procedures provided in Appendix F at the back of the book can be used to show ϭ 7.34 provides a p-value ϭ 0255 Instead of using the p-value, we could use the critical value approach to draw the same conclusion With α ϭ 05 and degrees of freedom, the critical value for the test statistic is 2.05 ϭ 5.991 The upper tail rejection rule becomes Reject H0 if Ն 5.991 With 7.34 Ͼ 5.991, we reject H0 The p-value approach and critical value approach provide the same hypothesis testing conclusion Although no further conclusions can be made as a result of the test, we can compare the observed and expected frequencies informally to obtain an idea of how the market share structure may change Considering company C, we find that the observed frequency of 54 is larger than the expected frequency of 40 Because the expected frequency was based on current market shares, the larger observed frequency suggests that the new product will have a positive effect on company C’s market share Comparisons of the observed and expected frequencies for the other two companies indicate that company C’s gain in market share will hurt company A more than company B Let us summarize the general steps that can be used to conduct a goodness of fit test for a hypothesized multinomial population distribution MULTINOMIAL DISTRIBUTION GOODNESS OF FIT TEST: A SUMMARY State the null and alternative hypotheses H0: The population follows a multinomial distribution with specified probabilities for each of the k categories Ha: The population does not follow a multinomial distribution with the specified probabilities for each of the k categories Select a random sample and record the observed frequencies fi for each category Assume the null hypothesis is true and determine the expected frequency ei in each category by multiplying the category probability by the sample size Find more at www.downloadslide.com 12.1 477 Goodness of Fit Test: A Multinomial Population Compute the value of the test statistic ( fi Ϫ ei )2 ei iϭ1 k ϭ ͚ Rejection rule: Reject H0 if p-value Յ α p-value approach: Critical value approach: Reject H0 if Ն 2α where α is the level of significance for the test and there are k Ϫ degrees of freedom Exercises Methods SELF test Test the following hypotheses by using the goodness of fit test H 0: pA ϭ 40, pB ϭ 40, and pC ϭ 20 H a: The population proportions are not pA ϭ 40, pB ϭ 40, and pC ϭ 20 A sample of size 200 yielded 60 in category A, 120 in category B, and 20 in category C Use α ϭ 01 and test to see whether the proportions are as stated in H0 a Use the p-value approach b Repeat the test using the critical value approach Suppose we have a multinomial population with four categories: A, B, C, and D The null hypothesis is that the proportion of items is the same in every category The null hypothesis is H0: pA ϭ pB ϭ pC ϭ pD ϭ 25 A sample of size 300 yielded the following results A: 85 B: 95 C: 50 D: 70 Use α ϭ 05 to determine whether H0 should be rejected What is the p-value? Applications SELF test During the first 13 weeks of the television season, the Saturday evening 8:00 p.m to 9:00 p.m audience proportions were recorded as ABC 29%, CBS 28%, NBC 25%, and independents 18% A sample of 300 homes two weeks after a Saturday night schedule revision yielded the following viewing audience data: ABC 95 homes, CBS 70 homes, NBC 89 homes, and independents 46 homes Test with α ϭ 05 to determine whether the viewing audience proportions changed M&M/MARS, makers of M&M® chocolate candies, conducted a national poll in which more than 10 million people indicated their preference for a new color The tally of this poll resulted in the replacement of tan-colored M&Ms with a new blue color In the Find more at www.downloadslide.com 478 Chapter 12 Tests of Goodness of Fit and Independence brochure “Colors,” made available by M&M/MARS Consumer Affairs, the distribution of colors for the plain candies is as follows: Brown Yellow Red Orange Green Blue 30% 20% 20% 10% 10% 10% In a follow-up study, samples of 1-pound bags were used to determine whether the reported percentages were indeed valid The following results were obtained for one sample of 506 plain candies Brown Yellow Red Orange Green Blue 177 135 79 41 36 38 Use α ϭ 05 to determine whether these data support the percentages reported by the company Where women most often buy casual clothing? Data from the U.S Shopper Database provided the following percentages for women shopping at each of the various outlets (The Wall Street Journal, January 28, 2004) Outlet Percentage Wal-Mart Traditional department stores JC Penney 24 11 Outlet Kohl’s Mail order Other Percentage 12 37 The other category included outlets such as Target, Kmart, and Sears as well as numerous smaller specialty outlets No individual outlet in this group accounted for more than 5% of the women shoppers A recent survey using a sample of 140 women shoppers in Atlanta, Georgia, found 42 Wal-Mart, 20 traditional department store, JC Penney, 10 Kohl’s, 21 mail order, and 39 other outlet shoppers Does this sample suggest that women shoppers in Atlanta differ from the shopping preferences expressed in the U.S Shopper Database? What is the p-value? Use α ϭ 05 What is your conclusion? The American Bankers Association collects data on the use of credit cards, debit cards, personal checks, and cash when consumers pay for in-store purchases (The Wall Street Journal, December 16, 2003) In 1999, the following usages were reported In-Store Purchase Credit card Debit card Personal check Cash Percentage 22 21 18 39 A sample taken in 2003 found that for 220 in-stores purchases, 46 used a credit card, 67 used a debit card, 33 used a personal check, and 74 used cash a At α ϭ 01, can we conclude that a change occurred in how customers paid for in-store purchases over the four-year period from 1999 to 2003? What is the p-value? b Compute the percentage of use for each method of payment using the 2003 sample data What appears to have been the major change or changes over the four-year period? c In 2003, what percentage of payments was made using plastic (credit card or debit card)? Find more at www.downloadslide.com 12.2 479 Test of Independence The Wall Street Journal’s Shareholder Scoreboard tracks the performance of 1000 major U.S companies (The Wall Street Journal, March 10, 2003) The performance of each company is rated based on the annual total return, including stock price changes and the reinvestment of dividends Ratings are assigned by dividing all 1000 companies into five groups from A (top 20%), B (next 20%), to E (bottom 20%) Shown here are the one-year ratings for a sample of 60 of the largest companies Do the largest companies differ in performance from the performance of the 1000 companies in the Shareholder Scoreboard? Use α ϭ 05 A B C D E 15 20 12 How well airline companies serve their customers? A study showed the following customer ratings: 3% excellent, 28% good, 45% fair, and 24% poor (BusinessWeek, September 11, 2000) In a follow-up study of service by telephone companies, assume that a sample of 400 adults found the following customer ratings: 24 excellent, 124 good, 172 fair, and 80 poor Is the distribution of the customer ratings for telephone companies different from the distribution of customer ratings for airline companies? Test with α ϭ 01 What is your conclusion? 12.2 Test of Independence Another important application of the chi-square distribution involves using sample data to test for the independence of two variables Let us illustrate the test of independence by considering the study conducted by the Alber’s Brewery of Tucson, Arizona Alber’s manufactures and distributes three types of beer: light, regular, and dark In an analysis of the market segments for the three beers, the firm’s market research group raised the question of whether preferences for the three beers differ among male and female beer drinkers If beer preference is independent of the gender of the beer drinker, one advertising campaign will be initiated for all of Alber’s beers However, if beer preference depends on the gender of the beer drinker, the firm will tailor its promotions to different target markets A test of independence addresses the question of whether the beer preference (light, regular, or dark) is independent of the gender of the beer drinker (male, female) The hypotheses for this test of independence are: H0: Beer preference is independent of the gender of the beer drinker Ha: Beer preference is not independent of the gender of the beer drinker Table 12.2 can be used to describe the situation being studied After identification of the population as all male and female beer drinkers, a sample can be selected and each individual TABLE 12.2 CONTINGENCY TABLE FOR BEER PREFERENCE AND GENDER OF BEER DRINKER Gender Male Female Light Beer Preference Regular Dark cell(1,1) cell(2,1) cell(1,2) cell(2,2) cell(1,3) cell(2,3) Find more at www.downloadslide.com 480 Chapter 12 TABLE 12.3 Tests of Goodness of Fit and Independence SAMPLE RESULTS FOR BEER PREFERENCES OF MALE AND FEMALE BEER DRINKERS (OBSERVED FREQUENCIES) Light Gender To test whether two variables are independent, one sample is selected and crosstabulation is used to summarize the data for the two variables simultaneously Male Female Total Beer Preference Regular Dark Total 20 30 40 30 20 10 80 70 50 70 30 150 asked to state his or her preference for the three Alber’s beers Every individual in the sample will be classified in one of the six cells in the table For example, an individual may be a male preferring regular beer (cell (1,2)), a female preferring light beer (cell (2,1)), a female preferring dark beer (cell (2,3)), and so on Because we have listed all possible combinations of beer preference and gender or, in other words, listed all possible contingencies, Table 12.2 is called a contingency table The test of independence uses the contingency table format and for that reason is sometimes referred to as a contingency table test Suppose a simple random sample of 150 beer drinkers is selected After tasting each beer, the individuals in the sample are asked to state their preference or first choice The crosstabulation in Table 12.3 summarizes the responses for the study As we see, the data for the test of independence are collected in terms of counts or frequencies for each cell or category Of the 150 individuals in the sample, 20 were men who favored light beer, 40 were men who favored regular beer, 20 were men who favored dark beer, and so on The data in Table 12.3 are the observed frequencies for the six classes or categories If we can determine the expected frequencies under the assumption of independence between beer preference and gender of the beer drinker, we can use the chi-square distribution to determine whether there is a significant difference between observed and expected frequencies Expected frequencies for the cells of the contingency table are based on the following rationale First we assume that the null hypothesis of independence between beer preference and gender of the beer drinker is true Then we note that in the entire sample of 150 beer drinkers, a total of 50 prefer light beer, 70 prefer regular beer, and 30 prefer dark beer In terms of fractions we conclude that ⁵⁰⁄₁₅₀ ϭ ¹⁄₃ of the beer drinkers prefer light beer, ⁷⁰⁄₁₅₀ ϭ ⁷⁄₁₅ prefer regular beer, and ³⁰⁄₁₅₀ ϭ ¹⁄₅ prefer dark beer If the independence assumption is valid, we argue that these fractions must be applicable to both male and female beer drinkers Thus, under the assumption of independence, we would expect the sample of 80 male beer drinkers to show that (¹⁄₃)80 ϭ 26.67 prefer light beer, ( ⁷⁄₁₅)80 ϭ 37.33 prefer regular beer, and (¹⁄₅)80 ϭ 16 prefer dark beer Application of the same fractions to the 70 female beer drinkers provides the expected frequencies shown in Table 12.4 Let eij denote the expected frequency for the contingency table category in row i and column j With this notation, let us reconsider the expected frequency calculation for males TABLE 12.4 EXPECTED FREQUENCIES IF BEER PREFERENCE IS INDEPENDENT OF THE GENDER OF THE BEER DRINKER Light Gender Male Female Total Beer Preference Regular Dark Total 26.67 23.33 37.33 32.67 16.00 14.00 80 70 50.00 70.00 30.00 150 Find more at www.downloadslide.com 12.2 481 Test of Independence (row i ϭ 1) who prefer regular beer (column j ϭ 2), that is, expected frequency e12 Following the preceding argument for the computation of expected frequencies, we can show that e12 ϭ ( ⁷ ₁₅)80 ϭ 37.33 This expression can be written slightly differently as e12 ϭ ( ⁷ ₁₅)80 ϭ ( ⁷⁰ ₁₅₀)80 ϭ (80)(70) ϭ 37.33 150 Note that 80 in the expression is the total number of males (row total), 70 is the total number of individuals preferring regular beer (column total), and 150 is the total sample size Hence, we see that e12 ϭ (Row Total)(Column Total) Sample Size Generalization of the expression shows that the following formula provides the expected frequencies for a contingency table in the test of independence EXPECTED FREQUENCIES FOR CONTINGENCY TABLES UNDER THE ASSUMPTION OF INDEPENDENCE eij ϭ (Row i Total)(Column j Total) Sample Size (12.2) Using the formula for male beer drinkers who prefer dark beer, we find an expected frequency of e13 ϭ (80)(30)/150 ϭ 16.00, as shown in Table 12.4 Use equation (12.2) to verify the other expected frequencies shown in Table 12.4 The test procedure for comparing the observed frequencies of Table 12.3 with the expected frequencies of Table 12.4 is similar to the goodness of fit calculations made in Section 12.1 Specifically, the value based on the observed and expected frequencies is computed as follows TEST STATISTIC FOR INDEPENDENCE ϭ ͚͚ i j ( fij Ϫ eij)2 eij (12.3) where fij ϭ observed frequency for contingency table category in row i and column j eij ϭ expected frequency for contingency table category in row i and column j based on the assumption of independence Note: With n rows and m columns in the contingency table, the test statistic has a chisquare distribution with (n Ϫ 1)(m Ϫ 1) degrees of freedom provided that the expected frequencies are five or more for all categories Find more at www.downloadslide.com 1068 Appendix F Computing p-Values Using Minitab and Excel Step Choose t Step When the t Distribution dialog box appears: Select Cumulative probability Enter 59 in the Degrees of freedom box Select Input Constant Enter 1.84 in the Input Constant box Click OK Minitab provides a cumulative probability of 9646, and hence the lower tail p-value ϭ 9646 The Heathrow Airport example is an upper tail test; the upper tail p-value is Ϫ 9646 ϭ 0354 In the case of a two-tailed test, we would use the minimum of 9646 and 0354 to compute p-value ϭ 2(.0354) ϭ 0708 test statistic We use the St Louis Metro Bus example from Section 11.1 as an illustration; the value of the test statistic is ϭ 28.18 with 23 degrees of freedom The Minitab steps used to compute the cumulative probability corresponding to ϭ 28.18 follow The Step Step Step Step Select the Calc menu Choose Probability Distributions Choose Chi-Square When the Chi-Square Distribution dialog box appears: Select Cumulative probability Enter 23 in the Degrees of freedom box Select Input Constant Enter 28.18 in the Input Constant box Click OK Minitab provides a cumulative probability of 7909, which is the lower tail p-value The upper tail p-value ϭ Ϫ the cumulative probability, or Ϫ 7909 ϭ 2091 The two-tailed pvalue is times the minimum of the lower and upper tail p-values Thus, the two-tailed p-value is 2(.2091) ϭ 4182 The St Louis Metro Bus example involved an upper tail test, so we use p-value ϭ 2091 The F test statistic We use the Dullus County Schools example from Section 11.2 as an illustration; the test statistic is F ϭ 2.40 with 25 numerator degrees of freedom and 15 denominator degrees of freedom The Minitab steps to compute the cumulative probability corresponding to F ϭ 2.40 follow Step Step Step Step Select the Calc menu Choose Probability Distributions Choose F When the F Distribution dialog box appears: Select Cumulative probability Enter 25 in the Numerator degrees of freedom box Enter 15 in the Denominator degrees of freedom box Select Input Constant Enter 2.40 in the Input Constant box Click OK Minitab provides the cumulative probability and hence a lower tail p-value ϭ 9594 The upper tail p-value is Ϫ 9594 ϭ 0406 Because the Dullus County Schools example is a two-tailed test, the minimum of 9594 and 0406 is used to compute p-value ϭ 2(.0406) ϭ 0812 Find more at www.downloadslide.com Appendix F 1069 Computing p-Values Using Minitab and Excel Using Excel WEB file p-Value Excel functions and formulas can be used to compute p-values associated with the z, t, 2, and F test statistics We provide a template in the data file entitled p-Value for use in computing these p-values Using the template, it is only necessary to enter the value of the test statistic and, if necessary, the appropriate degrees of freedom Refer to Figure F.1 as we describe how the template is used For users interested in the Excel functions and formulas being used, just click on the appropriate cell in the template The z test statistic We use the Hilltop Coffee lower tail hypothesis test in Section 9.3 as an illustration; the value of the test statistic is z ϭ Ϫ2.67 To use the p-value template for this hypothesis test, simply enter Ϫ2.67 into cell B6 (see Figure F.1) After doing so, p-values for all three types of hypothesis tests will appear For Hilltop Coffee, we would use the lower tail p-value ϭ 0038 in cell B9 For an upper tail test, we would use the p-value in cell B10, and for a two-tailed test we would use the p-value in cell B11 The t test statistic We use the Heathrow Airport example from Section 9.4 as an illustra- tion; the value of the test statistic is t ϭ 1.84 with 59 degrees of freedom To use the p-value template for this hypothesis test, enter 1.84 into cell E6 and enter 59 into cell E7 (see Figure F.1) After doing so, p-values for all three types of hypothesis tests will appear FIGURE F.1 EXCEL WORKSHEET FOR COMPUTING p-VALUES A B Computing p-Values Using the Test Statistic z Enter z > Ϫ2.67 p-value (Lower Tail) 0.0038 10 p-value (Upper Tail) 0.9962 11 p-value (Two Tail) 0.0076 12 13 14 15 16 Using the Test Statistic Chi Square 17 18 Enter Chi Square > 28.18 19 df > 23 20 21 22 p-value (Lower Tail) 0.7909 23 p-value (Upper Tail) 0.2091 24 p-value (Two Tail) 0.4181 C D E Using the Test Statistic t Enter t > df > p-value (Lower Tail) p-value (Upper Tail) p-value (Two Tail) 1.84 59 0.9646 0.0354 0.0708 Using the Test Statistic F Enter F > Numerator df > Denominator df > 2.40 25 15 p-value (Lower Tail) p-value (Upper Tail) p-value (Two Tail) 0.9594 0.0406 0.0812 Find more at www.downloadslide.com 1070 Appendix F Computing p-Values Using Minitab and Excel The Heathrow Airport example involves an upper tail test, so we would use the upper tail p-value ϭ 0354 provided in cell E10 for the hypothesis test test statistic We use the St Louis Metro Bus example from Section 11.1 as an illustration; the value of the test statistic is ϭ 28.18 with 23 degrees of freedom To use the p-value template for this hypothesis test, enter 28.18 into cell B18 and enter 23 into cell B19 (see Figure F.1) After doing so, p-values for all three types of hypothesis tests will appear The St Louis Metro Bus example involves an upper tail test, so we would use the upper tail p-value ϭ 2091 provided in cell B23 for the hypothesis test The The F test statistic We use the Dullus County Schools example from Section 11.2 as an illustration; the test statistic is F ϭ 2.40 with 25 numerator degrees of freedom and 15 denominator degrees of freedom To use the p-value template for this hypothesis test, enter 2.40 into cell E18, enter 25 into cell E19, and enter 15 into cell E20 (see Figure F.1) After doing so, p-values for all three types of hypothesis tests will appear The Dullus County Schools example involves a two-tailed test, so we would use the two-tailed p-value ϭ 0812 provided in cell E24 for the hypothesis test Find more at www.downloadslide.com Index Note: Chapter 22 can be found with the Online Content for this book Index entries found in this chapter are denoted by chapter number 22, hyphen, and page number Page numbers followed by a n indicate a footnote A Acceptable quality level (AQL), 930n2 Acceptance criterion, 924 Acceptance sampling, 922–931 binomial probability function for, 925 computing the probability of accepting a lot, 924–927 KALI, Inc example, 924 selecting plans for, 928–929 Accounting, Addition law, 165–168 Additive decomposition models, 829–830 Adjusted multiple coefficient of determination, 655, 655n1 Aggregate price indexes, 765–767 computing from price relatives, 769–770 Air traffic controller stress test, 531–532 Alliance Data Systems, 561 Alpha to enter, 739–740, 743n1 Alpha to remove, 743n1 Alternative hypothesis, 349 as research hypothesis, 350–351 American Military Standard Table (MIL-STD-105D), 929 American Society for Quality (ASQ), 904 American Statistical Association “Ethical Guidelines for Statistical Practice,” 18–19 Analysis of variance (ANOVA), 508–537, 513n3, 513n4 assumptions for, 510 completely randomized designs and, 513–524 computer results for, 519–520 experimental design and, 508–513 for factorial experiments, 539 for randomized block design, 532–533 ANOVA See Analysis of variance (ANOVA) ANOVA tables, 518–519, 589–590 Approximate class width, formula for, 65 Area as a measure of probability, 235–236 Assignable causes, 909 Association between two variables, measures of, 115–124 Attributes sampling plans, 930n3 Autocorrelation, 750 Average outgoing quality limit (AOQL), 930n2 Averages, 14–15 B Backward elimination, 741 Baldridge, Malcolm, 906 Baldridge Index, 906 Baldridge National Quality Program (BNQP), 906 Bar charts, 14f1.5, 34–36, 45n1 Barnett, Bob, 906 Bayes’ theorem, 157, 178–182, 183n1, 183n2 computing branch probabilities using, 960–965 tabular approach, 182 two-event case, 181 Bell curve See also Normal curve, 238–240 Bell Labs, 218 Bell Telephone Company, 905 Bernoulli, Jakob, 208 Bernoulli process, 208 Best-subsets regression, 741–742 Between-treatments estimate of population variance, 514–515 Between-treatments estimate of σ2, 511–512, 521n2 Bimodal data, 89 Binomial distribution for acceptance sampling, 930n1 expected value and variance for, 214–215 Binomial experiments, 208–209 Binomial probabilities normal approximation of, 250–252 tables, 213–214, 215n1, 215n2 Binomial probability distributions, 208 Binomial probability functions, 209, 212 Binomial sampling distribution, 861n2 Blocking, 530, 531 Bonferroni adjustment, 527–528 Bound on the sampling error, 22–7 Box plots, 110–111, 112n2 Burke Marketing Services, Inc., 507 Business Week, Butler Trucking Company, 646–648 C Case problems Air Force Training Program, 469 Alumni Giving, 705 alumni giving, 633 Bipartisan Agenda for Change, 501–502 business schools of Asia-Pacific, 139 compensation for sales professionals, 553–554 Consumer Research, Inc., 704–705 ethical behavior of business students, 397–398 forecasting food and beverage sales, 846–847 forecasting lost sales, 847–848 fuel economy for cars, 759–760 Gulf Real Estate Properties, 339–341 Hamilton County judges, 190–192 Find more at www.downloadslide.com 1072 Index Heavenly Chocolates website transactions, 139–141 lawsuit defense strategy, 969 measuring stock market risk, 631–632 Metropolitan Research, Inc., 341 motion picture industry, 72–73, 138–139 Par, Inc., 441–442 Pelican Stores, 71–72, 137–138 PGA tour statistics, 633–635, 705–707, 758–759 prediction winning percentage for the NFL, 708–709 Quality Associates, Inc., 396–397 Specialty Toys, Inc., 261–262 U.S Department of Transportation, 632–633 Wentworth Medical Center, 552–553 Young Professional magazine, 338–339 Categorical data, 7, 33–39 Categorical independent variables, 668–673 Categorical variables, Census, 15 Central limit theorem, 281–283, 286n2 Central location, measures of, 297n1 Chance events, 939 Chance nodes, 940 Chebyshev’s theorem, 104–105, 106–107n1 Chi-square distribution, 450–454 Chi-square test, 483n1 Cincinnatti Enquirer, 190 Citibank, 194 Classes, 39, 40 Class limits, 45n2 Class midpoints, 41, 127n1 Clusters, 298 Cluster sampling, 22–21–22–29, 298, 300n1 determining sample size, 22–26 population mean, 22–23–22–24 population total, 22–24–22–25 Coefficient of determination, 576–583, 579, 580n1, 692n2 Coefficient of variation, 99 Coefficients, interpretation of, 648–649 Colgate-Palmolive Company, 32 Combinations, 154 Common causes, 909 Company records, internal, 10 Comparisonwise Type I error rate, 527 Complements, 164, 165 Complete block design, 534 Completely randomized designs, 508, 513–524 Computers, 17 Conditional probabilities, 171–175, 960 Confidence coefficients, 313 Confidence intervals, 313, 594 for 1, 587–588 estimates, 323n2 for mean value of y, 595–596 Confidence levels, 313 Consequences, 939 Consistency, 297 Consumer Price Index (CPI), 764, 771 Consumer’s risk, 923 Contingency tables, 480 Continuity correction factor, 251 Continuous improvement, 909 Continuous random variables, 196 Control charts, 909–910 interpretation of, 920 np charts, 919–920 p charts, 917–919 R charts, 915–917 x chart, 910–915 Convenience sampling, 22–4, 299, 300n1 Cook’s distance measure, 679–681, 681n2 Correlation coefficient, 119–121, 579–580 Counting rules for combinations, 154 for multiple-step experiments, 151 for permutations, 154–155 Covariance, 115–119 Cravens, David W., 735 Critical value, 360 Critical value approach, 360–361 Crosby, Philip B., 905 Cross-sectional data, Cross-sectional regression, 786 Crosstabulations, 53–55 Cumulative frequency distributions, 43–44, 45n4 Cumulative percent frequency distributions, 44 Cumulative relative frequency distributions, 44 Customer’s Afternoon Letter, 772 Cyclical patterns, 789–791 D Data applications of, 580n1 bimodal and multimodal, 89 sources of, 10–13 types of, 5–8 Data acquistion errors, 13 Data errors, 681n1 Data mining, 17–18 Data sets, Data validity, 107n2 Data warehousing, 17 Decision analysis decision making with probabilities, 941–949 decision strategies, 951–954 decision trees, 940–941 payoff tables, 940 problem formulation, 939–941 with sample information, 949–960 Decision making, 381–382, 941–949 Decision nodes, 940 Decision strategies, 951–954 Decision trees, 940–941, 941n1, 941n2, 950–951 Deflating the series, 773–775 Degrees of belief, 156 Degrees of freedom, 316, 317, 319, 416, 535n1 DelGuzzi, Kristen, 190 Deming, W Edwards, 905 Dependent events, 175n1 Dependent variables, 562, 720–724 Descriptive statistics, 13–15, 127n1 Deseasonalized time series, 834–835, 837n2 Deviation about the mean, 97 Find more at www.downloadslide.com 1073 Index Discrete probability distributions, 197–200 Discrete probability functions, 198 Discrete random variables, 195 Discrete uniform probability distribution, 199 Discrete uniform probability function, 199 Distance intervals, 220 Distribution-free methods, 857 Distribution shape, 102–103 Doctrine of Chances, The (Moivre), 238–240 Dot plots, 41 Double-blind experiments, 513n2 Double-sample plans, 930 Dow, Charles Henry, 772 Dow Chemical Company, 904 Dow Jones averages, 772 Dow Jones Industrial Average (DJIA), 772 Duke Energy, Ch22–2 Dummy variables, 669 Duncan’s multipe range test, 528 Dunnhumby, 643 Durbin-Watson test, 751 E EAI problem, 283 Economics, Elements, 4–8, 5–6, 22–2 Empirical rule, 105–106 Error degrees of freedom, 535n1 Estimated logistic regression equation, 685–687 Estimated logit, 691 Estimated multiple regression equations, 644–645, 665–666 Estimated regression equations, 563–565, 567, 594, 612n2 Estimated standard deviation of b1, 586 Ethical behavior, 18–19 "Ethical Guidelines for Statistical Practice" (ASA), 18–19 Events, 160–162, 162n1, 164, 174 Excel analysis of variance with, 555–557 bar charts, 76–77 completely randomized design, 555 continuous probability distributions with, 262–263 crosstabulation, 79–81 descriptive statistics tool, 145–146 descriptive statistics using, 143–146 difference between two population means: σ1 and σ2 known, 444–445 difference between two population means: σ1 and σ2 unknown, 444–445 difference between two population means with matched samples, 445–446 discrete probability distributions with, 230–231 exponential smoothing, 851–852 factorial experiments, 556–557 forecasting with, 851–852 frequency distribution, 75–76, 77–79 goodness of fit test, 503, 504 histograms, 77–79 hypothesis testing with, 400–404 inferences about two populations, 444–446 interpretation of ANOVA output, 640 interpretation of estimated regression equation output, 639–640 interpretation of regression statistics output, 640 interval estimation using, 343–346 moving averages, 851 multiple regression with, 709–710 nonparametric methods with, 899–900 PivotChart, 77–79 PivotTable, 77–79 population mean: σ known, 343, 400–401 population mean: σ unknown, 344, 402–403 population proportion, 345–346, 403–404 population variances with, 470–471 Precision Tree add-in, 969–974 randomized block design, 555 random sampling with, 306–307 regression analysis, 638–640 scatter diagrams, 81–84 sign test, 899–900 Spearman rank correlation, 900 test of independence, 503, 505 trend projection, 852 using functions of, 143–145 Excel StatTools See StatTools, 17 Expected value, 202–203 binomial distribution and, 214–215 of p´, 289–290 of sample information, 954–956 of sample information (EVSI), 954–956 of x´, 279–280, 304 Expected value approach, 941–943 Expected value (EV), 942, 943–945 Experimental designs, 508–559 analysis of variance (ANOVA), 508–513 data collection, 509–510 multiple regression approach to, 745–749 Experimental studies, 11–12, 507 Experimental units, 508 Experiments, 150, 158n1 Experimentwise Type I error rate, 527 Exploratory data analysis, 48–51, 109–114, 112n1 Exponential distribution, 256n1, 258 Exponential probability density function, 258 Exponential probability distribution, 253–256 Exponential smoothing forecasting method, 800–804, 804n2 Exponential trend equation, 816 F Factorial experiments, 537–544 ANOVA procedure for, 539 F statistics for, 539–542 Factors, 508 Factors of interest, 531 F distribution, 460, 464n1, 516 Feigenbaum, A V., 905 Finance, Find more at www.downloadslide.com 1074 Index Finite population correction factor, 280 Fisher, Ronald Alymer, Sir, 508 Fisher’s least significant difference (LSD), 524–527 Fitness for use, 905 Five-number summary, 109–110 Food Lion, 309 Forecast accuracy, 792–797, 799, 800, 802 mean absolute error (MAE), 793 mean absolute percentage error (MAPE), 793 mean squared error, 794 mean squared error (MSE), 793 Forecast errors, 792 Forecasting methods exponential smoothing, 800–804 moving averages, 797–800 seasonality and trend, 820–829 trend projection, 807–820 weighted moving averages, 800 Forecasts, 785 Forward selection, 740–741 Frames, 22–3, 267 Frequencies, 13t1.4 Frequency distributions, 33–34 classes, 39–41 number of classes in, 36n1 sum of, 36n2 F statistic, 732n1 F tests, 516, 588–590 for multiple regression models, 658–661 F test statistics, 461 F(x), 234 G Galton, Francis, Sir, 562 Gauss, Carl Friedrich, 567 General linear model, 714–729 curvilinear relationship modeling, 714–717 interaction, 718–720 nonlinear models that are intrinsically linear, 724–725 second-order model with one predictor variable, 715 simple first-order model with one predictor variable, 714 transformations involving the dependent variable, 720–724 Goodness of fit test, 474–477, 692n2 multinomial distribution, 476–477 normal distribution, 491–495 poisson distribution, 487–491 test statistic for, 475 Gossett, William Sealy, 316 Government agencies, 10–11 Grear Tire Company problem, 246–248 Grouped data, 125–127 population mean for, 127 population variance for, 127 sample mean for, 126 sample variance for, 126 G statistic, 692n1 H High leverage points, 617 Histograms, 14f1.6, 41–43, 45n1 Holt’s linear exponential smooting, 812–814, 817n1 Horizontal patterns, 786–788 Hypergeometric probability distribution, 221–223, 223n1 Hypergeometric probability function, 221–222 Hypothesis testing, 861n1 about a population median, 857–861 about μ1 Ϫ μ2, 410–412, 417–419 about p1 Ϫ p2, 431–433 confidence interval approach, 366 decision making and, 381–382 interval estimation and, 366–367 with matched samples, 862–863 null and alternative hypotheses, 349–353 one-tailed tests, 356–361, 371–372 population mean σ known, 356–370 population mean σ unknown, 370–376 population proportion, 376–381 for population variance, 454–457 steps of, 365 two-tailed tests, 362–365 Type I and Type II errors, 353–355 I Incomplete block design, 534 Independent events, 174, 175, 175n1 Independent sample design, 426n2 Independent simple random samples, 407 Independent variables, 508, 562, 662–663, 668–673, 743n2 Index numbers aggregate price indexes, 765–767 computing aggregate price index from price relatives, 769–779 deflating series by price indexes, 773–775 price indexes, 771–773 price relatives, 765 quantity indexes, 778–779 Index of Industrial Production, 779 Indicator variables, 669 Indifference quality level (IQL), 930n2 Influential observations, 616–618, 679, 681n1 using Cook’s distance measure to identify, 679–681 Interactions, 538–539, 718–720 International Organization for Standardization, 906 Interquartile range (IQR), 96–97 Intersection of two events, 166 Interval estimates, 309, 310–314, 594 of difference between two population means, 430 of population variance, 450–454 procedures for, 322–323 Interval estimation, 314n1, 409 difference between two population means: σ1 and σ2 known, 410 difference between two population means: σ1 and σ2 unknown, 416 Find more at www.downloadslide.com Index hypothesis testing and, 366–367 of μ1 Ϫ μ2, 407–412, 415 of population mean: σ known, 313 of population proportion, 329, 330 Interval scales, Ishikawa, Karou, 905 ISO 9000, 906 Ith residual, 576 J John Morrell & Company, 349 Joint probabilities, 172, 962 Judgment sampling, 22–4, 299, 300n1 Juran, Joseph, 905 K K population means, 513n3 Kruskal-Wallis test, 882–884, 884n1 L Laspeyres index, 767 Least squares criterion, 567, 569n1, 645 Least squares estimated regression equation, 580n1 Least squares formulas, 635–636 Least squares method, 565–576, 569n1, 645–649 Levels of significance, 354 Leverage of observation i, 617, 676 LIFO (last-in, first-out) method of inventory valuation, 309 Linear exponential smoothing, 812–814 Linear trend regression forecast method, 807–812, 817n1 Logistic regression, 683–692, 692n2 Logit, 691 Logit transformation, 691 Lots, 922, 924 Lot tolerance percent defective (LPTD), 930n2 Lower control limits (LCL), 910 Lower tail tests, 356, 361 M Malcolm Baldridge National Quality Award, 906 Mann-Whitney-Wilcoxon (MWW) test, 871–882, 878n1 Marginal probabilities, 172 Margin of error, 309, 310–314, 323n1, 331n1 Marketing, Martin Clothing Store problem, 209–213 Matched samples, 423, 426n1, 426n2 Wilcoxon signed-rank test, 865–871 MeadWestvaco Corporation, 266 Mean, 14–15, 87–88, 124–125, 219 Mean absolute error (MAE), 793 Mean absolute percentage error (MAPE), 793 Mean square due to error (MSE), 521n3, 585, 793, 794 Mean square due to regression (MSR), 588 1075 Mean square due to treatments (MSTR), 514–515 Mean square regression (MSR), 588 Means square due to error (MSE), 515 Measures of association between two variables, 115–124 Measures of location, 87–92 Measures of variability, 95–102 Median, 88–89 Minitab, 17 Alpha to enter, 739–740 analysis of variance, 554–555 backward elimination procedure, 761 best-subsets procedure, 761 box plots, 143 completely randomized design, 554 continuous probability distributions with, 262–263 control charts, 935 covariance and correlation, 143 crosstabulation, 74–75 descriptive statistics using, 142–143 difference between two population means: σ1 and σ2 unknown, 442–443 difference between two population means with matched samples, 443 difference between two population proportions, 443–444 discrete probability distributions with, 230 dot plots, 73 exponential smoothing, 849 factorial experiments, 554–555 forecasting with, 848–851 forward selection procedure, 761 goodness of fit test, 502–503 histograms, 73–74 Holt’s linear exponential smoothing, 850 hypothesis testing with, 398–400 inferences about two populations, 442–444 interval estimation with, 341–343 Kruskal-Wallis test, 898–899 logistic regression with, 710 Mann-Whitney-Wilcoxon test, 898 moving averages, 848–849 multiple regression with, 708–709 nonparametric methods with, 896–899 population mean: σ known, 341–342, 398–399 population mean: σ unknown, 342, 399 population proportion, 342–343, 399–400 population variances with, 470 randomized block design, 554 random sampling with, 306 regression analysis with, 637–638 scatter diagrams, 74 sign test for a hypothesis test about a population mean, 896–897 sign test for a hypothesis test with matched samples, 897 Spearman rank correlation, 899 stem-and-leaf displays, 73–74 stepwise procedure, 760 test of independence, 503 time series decomposition, 850–851 trend projection, 849–850 Find more at www.downloadslide.com 1076 Index using for tabular and graphical presentations, 73–75 variable selection procedures, 760–761 Wilcoxon signed-rank test with matched samples, 897–899 Mode, 89 Model assumptions about the error term ε in the regression model, 583–584 confidence interval for 1, 587–588 F test, 588–590 for regression model, 584–585 t tests, 586 Model reliability, 18 Moivre, Abraham de, 238–240 Monsanto Company, 713 Motorola, Inc., 906 Moving averages forecasting method, 797–800, 804n2 MSE See Mean square due to error (MSE) MSE See Mean square due to error (MSE); Mean square error (MSE) MSR See Mean square due to regression (MSR); Mean square regression (MSR), 588 MSTR See Mean square due to treatments (MSTR) Multicollinearity, 662–663, 663n1 Multimodal data, 89 Multinomial distribution goodness of fit test, 476–477 Multinomial populations, 474 Multiple coefficient of determination, 654–655 Multiple comparison procedures Fisher’s least significant difference (LSD), 524–527 Type error rates, 527–528 Multiple regression analysis, 644, 692n2 Multiple regression equation, 644 Multiple regression model, 644, 657 Multiple sampling plans, 930 Multiplication law, 174–175 Multiplicative decomposition models, 830 Mutually exclusive events, 168, 175n1 N Nevada Occupational Health Clinic, 785 Nodes, 940 Nominal scales, Nonlinear trend regression, 814–816 Nonparametric methods, 857 Kruskal-Wallis test, 882–884 Mann-Whitney-Wilcoxon (MWW) test, 871–882 sign test, 857–865, 861n1 Spearman rank-correlation coefficient, 887–889 Wilcoxon signed-rank test, 865–871 Nonprobabilistic sampling, 22–4, 299, 300n1 Nonsampling errors, 22–5 Nonstationary time series, 804n2 Normal curve See also Bell curve, 238–240 Normal distribution goodness of fit test, 491–495 Normal probability density function, 239, 258 Normal probability distribution, 238–248 Normal probability plots, 610–612, 612n1 Normal scores, 610–612 Norris Electronics, 15–16, 19 Np chart, 910, 919–920, 920n2 Null hypothesis, 349–353 O Observational studies, 12–13, 507 testing for the equality of k population mean, 520–521 Observations, 6, 8n1 Oceanwide Seafood, 149 Odds in favor of an event occurring, 688 Odds ratio, 688–691, 692n1 Ogives, 44–45 Ohio Edison Company, 938 One-tailed tests, 371–372, 475 population mean σ known, 355–361 population mean σ unknown, 371–372 Open-end classes, 45n3 Operating characteristic (OC) curves, 925 Ordinal scales, Outliers, 106, 107n2, 614, 678, 681n1 Overall sample mean, 511 P Paasche index, 767 Parameters, 268 Parametric methods, 856 Partitioning, 518 Payoff, 940 Payoff tables, 940 P chart, 910 Pearson, Karl, 562 Pearson product moment correlation coefficient, 119–120, 889n1 Percent frequencies, 13t1.4 Percent frequency distributions, 34, 41 Percentiles, 90–91 Permutations, 154–155 Pie charts, 34–36 Planning values, 326 Point estimates, 274, 594 Point estimation, 273–275 Point estimators, 87, 274 consistency, 297 of difference between two population means, 409 of difference between two population proportions, 430 efficiency, 296–297 unbiased, 295–296 Poisson, Siméon, 218 Poisson probability distribution, 218–221 exponential distribution and, 255–256 goodness of fit test, 487–491 Poisson probability function, 218, 488 Pooled estimate of σ2, 512 Pooled estimator of p, 432 Pooled sample variance, 419n1 Population mean, 22–6–22–7, 22–12–22–14 Find more at www.downloadslide.com 1077 Index approximate 95% confidence interval estimate of, 22–13, 22–25 for grouped data, 127 inference about difference between: matched samples, 423–426 inference about difference between: σ1 and σ2 known, 407–412 inference about difference between: σ1 and σ2 unknown, 415 point estimator of, 22–12, 22–23–22–24 sample size when estimating, 22–17 σ known, 310–314 σ unknown, 316–323 Population mean σ known interval estimates, 310–314 margin of error, 310–314 one-tailed tests, 355–361 Population mean σ unknown hypothesis testing, 370–376 interval estimate, 317–320 margin of error, 317–320 two tailed testing, 372–373 Population parameters, 87 Population proportions, 22–8–22–9, 22–15–22–16, 328–331, 331n1 approximate 95% confidence interval estimate of, 22–15, 22–26 hypothesis testing and, 376–381 inferences about difference between, 429–436 interval estimate of, 329 interval estimation of p1 Ϫ p2, 429–431 normal approximation of sampling distribution of, 328 point estimator of, 22–15, 22–25 sample size for an interval estimate of, 330 test statistic for hypothesis tests about, 378 Populations, 15, 22–2 Population standard deviation (σ), 99, 310 Population total, 22–7–22–8, 22–14–22–15 approximate 95% confidence interval estimate of, 22–14, 22–25 point estimator of, 22–14, 22–24 sample size when estimating, 22–18 Population variance, 97 between-treatments estimate of, 514–515 for grouped data, 127 hypothesis testing for, 454–457 inferences about, 450–459 within-treatments estimate of, 515–516 Posterior (revised) probabilities, 178, 949 Power, 385 Power curves, 385 Precision Tree add-in to Excel, 969–974 Prediction intervals, 594 Prediction intervals for individual value of y, 596–598 Price indexes Consumer Price Index (CPI), 771 deflating a series by, 773–775 Dow Jones averages, 772 Producer Price Index (PPI), 771 quality changes, 777–778 selection of base period, 777 selection of items, 777 Price relatives, 765, 769–770 Prior probabilities, 178, 949 Probabilistic sampling, 22–4, 300n1 Probabilities, 150 classical method of assigning, 155–156, 162n1 conditional, 171–175 joint, 172 marginal, 172 posterior, 178 prior, 178 relative frequency method of assigning, 156 subjective method for assigning, 156–155 of success, 215n1, 215n2 Probability density function, 234, 237n1 Probability distributions, 197 Probability functions, 197 Probability samples, 271n2, 513n1 Procter & Gamble, 233 Producer Price Index (PPI), 771 Producer’s risk, 923 Production, Proportional allocation, 22–19n2 P-value approach, 358–360 P-values, 358, 367n1 Q Quadratic trend equation, 814–816 Quality assurance, 908 Quality control, 905–908 Quality engineering, 908 Quantitative data, 7, 8n2, 33 class limits and, 45n2 summarizing, 39–53 Quantitative variables, Quantity indexes, 778–779 Quartiles, 91–92 Questionnaires, 22–3 R Random experiments, 158n1 Randomization, 508, 513n1 Randomized block design, 530–537, 535n1 Random samples, 158n2, 270, 271n1 Random variables, 194–196, 196n1 Range, 96 Ratio scales, R charts, 910, 915–917, 920n1 Regression analysis, 562, 565n1, 618n1 adding or deleting variables, 729–735 autocorrelation and the Durbin-Watson test, 750–754 computer solutions, 600–601 general linear model, 714–725 of larger problems, 735–738 multiple regression approach to experimental design, 745–749 residuals, 793 variable selection procedures, 739–745 Find more at www.downloadslide.com 1078 Index Regression equations, 563–565, 565n2 Regression models, 562, 743n3 Regression sums of squares, 732n1 Rejectable quality level (RQL), 930n2 Rejection rule for lower tail test critical value approach, 361 Rejection rules using p-value, 360 Relative efficiency, 295–296 Relative frequency distributions, 34, 41 formula for, 65 Replication, 509 Replications, 538 “Researches on the Probability of Criminal and Civil Verdicts” (Poisson), 218 Residual analysis, 605–614, 612n2 detecting influential observations, 616–618 detecting outliers, 614–616, 678 influential observations, 679 of multiple regression model, 676–681 normal probability plots, 610–612 residual for observation i, 605 residual plot against x, 606–607 residual plot against yˆ, 607 standard deviation of residual i, 676 standardized residuals, 607–610 standard residual for observation i, 676 Residual plots, 606, 612n1 against x, 606–607 against yˆ, 607 Residuals, 793 Response variables, 508 Reynolds, Inc., 714–717 Rounding errors, 100n3 S Sample correlation coefficients, 119–120, 579–580 Sampled populations, 22–3, 267 Sample information, 949 expected value of (EVSI), 954–956 Sample mean, 126, 267, 297n1, 521n1 Sample points, 150 Samples, 15, 22–2, 271n1 Sample selection, 268–271 from infinite population, 270–271 random samples, 270 sampling withouth replacement, 269 sampling with replacement, 270 Sample size determining, 325–327 estimating population mean, 22–17 estimating population total, 22–18 for hypothesis test about a population mean, 387–390 for interval estimate of population mean, 326 outliers and, 320 of population proportion, 330 sampling distribution of x´ , 285–286 skewness and, 320 small samples, 320–322 Sample space, 150 Sample statistics, 87, 273–274 Sample surveys, 15, 22–2–22–3 Sample variance, 97, 100n4, 126 Sampling distributions, 276–286 of b1, 586 of (n-1)s2/σ2, 450 of p´, 289–293 of two population variances, 460 of x´, 278–279, 281–286 Sampling units, 22–3 Sampling without replacement, 269 Sampling with replacement, 270 Scales of measurement, 6–7 Scatter diagrams, 57–59, 565 Seasonal adjustments, 836 Seasonal indexes, calculating, 830–834, 837n1 Seasonality and trend, 820–829 models based on monthly data, 825–826 seasonalty without trend, 820–823 Seasonal patterns, 788–789 Second-order model with one predictor variable, 715 Serial correlation, 750 Shewhart, Walter A., 905 Significance testing, 585–594, 590–591 using correlation, 636–637 Sign tests, 857–861, 861n2 hypothesis test about a population median, 857–861 hypothesis test with matched sample, 862–863 Simple first-order model with one predictor variable, 715 Simple linear regression, 562, 565n2 F test for significance in, 589 Simple random samples, 22–6–22–12, 271–272n2, 271n2, 300n1 determining sample size, 22–9–22–11 finite populations, 268–270 population mean, 22–6–22–7 population proportion, 22–8–22–9 population total, 22–7–22–8 Simple regression, 692n2 Simpson’s paradox, 56–57 Single-factor experiments, 508 Single-sample plans, 930 Single-stage cluster sampling, 22–21 Six Sigma, 906–908 limits and defects per million opportunities (dpmo), 907 Skewed distributions, 256n1 Skewed populations, 323n2 Skewness, 102–103, 256n1, 323n2 ⌺ known, 310 Small Fry Design, 86 Smoothing constants, 800, 801 Software packages, 17, 18 Spearman rank-correlation coefficient, 887–889, 889n1 Spreadsheet packages, 804n1 SSE See Sum of squares due to error (SSE) SSR See Sum of squares due to regression (SSR) SST See Total sum of squares SSTR See Sum of squares due to treatments (SSTR), 515 Find more at www.downloadslide.com 1079 Index Standard deviation, 99, 204 of the ith residual, 609 of p´, 290 of x´, 280–281, 304–305 Standard error, 281 of p1 Ϫ p2, 430 of p1 Ϫ p2 when p1 ϭ p2 ϭ p, 432 two independent random samples, 409 Standard error of the estimate, 585 Standard error of the proportion, 290 Standardized residual for observation i, 610 Standardized residuals, 607–610 Standard normal probability distribution, 240–245, 245–248 Standard normal random variable, 245, 258 States of nature, 939 Stationary assumption, 209 Stationary time series, 787, 804n2 Statistical analysis, 17 Statistical experiments, 158n1 Statistical inference, 15–16 Statistical models, 18 Statistical process control, 908–922 assignable causes, 909 common causes, 909 np chart, 919–920 p chart, 917–919 R chart, 915–917 x´ chart, 909–915 Statistical significance vs practical significance, 591n2 Statistical software packages, 100n1, 272n3 Statistical studies, 11–13 Statistics, StatTools analysis of completely randomized design, 556–559 box plots, 147 control charts, 935–936 covariance and correlation, 147 descriptive statistics using, 146–147 exponential smoothing, 853 forecasting with, 852–854 getting started with, 28–30 histograms, 84 Holt’s linear exponential smoothing, 853–854 hypothesis testing with, 404–405 hypothesis tests about μ1 Ϫ μ2, 446–447 inferences about the difference betweentwo populations: matched samples, 447 inferences about two populations, 446–447 interval estimation of μ1 Ϫ μ2, 446 interval estimation of population mean: σ unknown case, 346 interval estimation with, 346–347 Mann-Whitney-Wilcoxon test, 901–902 moving averages, 852–853 multiple regression analysis with, 711 nonparametric methods with, 901–902 population mean: σ unknown case, 404–405 random sampling with, 307 regression analysis, 640–641 sample size, determining, 346–347 scatter diagrams, 84 single population standard deviation with, 471 using for tabular and graphical presentations, 75–84 variable selection procedures, 761–762 Wilcoxon signed-rank test with matched samples, 901 Stem-and-leaf display, 48–51 Stepwise regression procedure, 739–740, 743n1 Stocks and stock funds, 100n2 Stratified random sampling, 297–298, 300n1 Stratified simple random sampling, 22–19n1 advantages of, 22–19n1 population mean, 22–12–22–14 population proportion, 22–15–22–16 population total, 22–14–22–15 Studentized deleted residuals, 678–679 Sum of squares due to error (SSE), 515–516, 576 Sum of squares due to regression (SSR), 557 Sum of squares due to treatments (SSTR), 515 Sum of the squares of the deviations, 566 ⌺ unknown, 316 Survey errors, 22–5–22–6 Surveys and sampling methods, 22–3–22–4 Systematic sampling, 22–29, 298–299, 300n1 T T, 586, 658–661 Taguchi, Genichi, 905 Target populations, 22–3, 275 T distribution, 316, 317 Test for significance, 585–594, 591n1, 591n3, 636–637, 658–663, 687 Test for the equality of k population means, 517, 520–521 Test of independence, 479–483 Test statistics, 357–358 for chi-square test, 483n1 for the equality of a k population means, 516 for goodness of fit, 475 for hypothesis tests about a population variance, 454 hypothesis tests about μ1 Ϫ μ2: σ1 and σ2 known, 411 for hypothesis tests about population mean: σ known, 358 for hypothesis tests about p1 Ϫ p2, 432 for hypothesis tests about two population variances, 461 for hypothesis tests involving matched samples, 425 hypothesis tests μ1 Ϫ μ2: σ1 and σ2 unknown, 417–419 Thearling, Kurt, 17 Time intervals Poisson probability distribution and, 218–220 Time series, 786–792 Time series data, deflating by price indexes, 773–775 graphs of, 9f1.2 Time series decomposition, 829–839 additive decompostion model, 829–830 Find more at www.downloadslide.com 1080 Index calculating seasonal indexes, 830–834 cyclical components, 837 deseasonalized time series, 834 models based on monthly data, 837 multiplicative model, 830 seasonal adjustments, 836 Time series patterns, 786–792 cyclical, 789–791 horizontal pattern, 786–788 seasonal patterns, 788–789 selecting forecasting methods, 791–792 trend and seasonal pattern, 788 trend pattern, 788 Time series plots, 786–792 Time series regression, 786 Total quality (TQ), 904 Total sum of squares (SST), 577 Treatments, 508 Tree diagrams, 152 Trend and seasonal patterns, 789 Trendlines, 57–59 Trend patterns, 788 Trend projection Holt’s linear exponential smoothing, 812–814 linear trend regression, 807–812 nonlinear trend regression, 814–817 Trimmed mean, 92n1 T tests, 586 for individual significane in multiple regression models, 661–662 for significance in simple linear regression, 587 Tukey's procedure, 528 Two population variances inferences about, 460–465 one-tailed hypothesis test about, 461 sampling distribution of, 460 Two-stage sampling plans, 930 Two-tailed tests, 362–367 computation of p-value, 364 critical value approach, 364 population mean σ known, 362–365 population mean σ unknown, 372–373 p-value approach, 363 Type I errors, 353–355, 355n1 comparisonwise Type I error rate, 527 experimentwise Type I error rate, 527 Type II errors, 353–355, 355n1 probability of, 382–385 U Unbiased estimators, 295–296 Uniform probability density function, 234, 258 Uniform probability distribution, 234–237 Union of two events, 165 United Way, 473 Upper control limits (UCL), 910 Upper tail tests, 356, 361, 461 U.S Commerce Department National Institute of Standards and Technology (NIST), 906 U.S Department of Labor Bureau of Labor Statistics, 764 U.S Food and Drug Administration, 407 U.S Government Accountability Office, 449 V Variability, measures of, 95–102 Variables, 5–6 adding or deleting, 729–735 random, 194–196 use of p-values, 732 Variable selection procedures Alpha to enter, 739–740 backward elimination, 741 best-subsets regression, 741–742 forward selection, 740–741 stepwise regression, 739–740 Variables sampling plans, 930n3 Variance, 97–99, 203–204 binomial distribution and, 214–215 Poisson probability distribution and, 219 Venn diagrams, 164 W Weighted aggregate price indexes, 766 Weighted means, 124–125 Weighted moving averages forecasting method, 800 Western Electric Company, 905 West Shell Realtors, 856 Wilcoxon signed-rank test, 865–871, 868n1, 868n2 Williams, Walter, 355, 355n1 Within-treatments estimate of population variance, 515–516 Within-treatments estimate of σ2, 512 X X chart x´, 909, 920n1 process mean and standard deviation known, 910–912 process mean and standard deviation unknown, 912–915 Z Z-scores, 103–104, 106 Z test, 692n1 Find more at www.downloadslide.com Statistics for Business and Economics 11eWEBfiles Chapter Morningstar Norris Shadow02 Table 1.1 Table 1.5 Exercise 25 Chapter EAI MetAreas MutualFund Section 7.1 Appendix 7.2, 7.3 & 7.4 Exercise 14 Chapter ApTest Audit BestTV Broker CityTemp Computer Crosstab DYield DJIAPrices FedBank Fortune Frequency FuelData08 GMSales Holiday LivingArea Major Marathon Movies MutualFunds Names Networks NewSAT OffCourse PelicanStores Population Restaurant Scatter SoftDrink Stereo SuperBowl Table 2.8 Table 2.4 Exercise Exercise 26 Exercise 46 Exercise 21 Exercise 29 Exercise 41 Exercise 17 Exercise 10 Exercise 51 Exercise 11 Exercise 37 Exercise 40 Exercise 18 Exercise Exercise 39 Exercise 28 Case Problem Exercise 34 Exercise Exercise Exercise 42 Exercise 20 Case Problem Exercise 44 Table 2.9 Exercise 30 Table 2.1 Table 2.12 Exercise 43 Chapter 3Points Ages Asian BackToSchool CellService Disney Economy FairValue Homes Hotels Housing MajorSalary MLBSalaries Movies Mutual NCAA PelicanStores Penalty PropertyLevel Runners Shoppers Speakers SpringTraining StartSalary Stereo StockMarket TaxCost Travel Visa WorldTemp Exercise Exercise 59 Case Problem Exercise 22 Exercise 42 Exercise 12 Exercise 10 Exercise 67 Exercise 64 Exercise Exercise 49 Figure 3.7 Exercise 43 Case Problem Exercise 44 Exercise 34 Case Problem Exercise 62 Exercise 65 Exercise 40 Case Problem Exercise 35 Exercise 68 Table 3.1 Table 3.6 Exercise 50 Exercise Exercise 66 Exercise 58 Exercise 51 Chapter Judge Case Problem Chapter Volume Exercise 24 Chapter 12 Chemline FitTest Independence NYReform Table 12.10 Appendix 12.2 Appendix 12.2 Case Problem Chapter ActTemps Alcohol Auto Flights GulfProp Interval p JobSatisfaction JobSearch Lloyd’s Miami NewBalance Nielsen NYSEStocks Professional Program Scheer TaxReturn TeeTimes TicketSales Exercise 49 Exercise 21 Case Problem Exercise 48 Case Problem Appendix 8.2 Exercise 37 Exercise 18 Section 8.1 Exercise 17 Table 8.3 Exercise Exercise 47 Case Problem Exercise 20 Table 8.4 Exercise Section 8.4 Exercise 22 Chapter AgeGroup AirRating Bayview Coffee Diamonds Drowsy Eagle FirstBirth Fowle Gasoline GolfTest Hyp Sigma Known Hyp Sigma Unknown Hypothesis p Orders Quality UsedCars WomenGolf Exercise 39 Section 9.4 Case Problem Section 9.3 Exercise 29 Exercise 44 Exercise 43 Exercise 64 Exercise 21 Exercise 67 Section 9.3 Appendix 9.2 Appendix 9.2 Appendix 9.2 Section 9.4 Case Problem Exercise 32 Section 9.5 Chapter 10 AirFare Cargo CheckAcct Earnings2005 ExamScores Golf GolfScores HomePrices Hotel Matched Mutual Occupancy PriceChange SAT SATVerbal SoftwareTest TaxPrep TVRadio Exercise 24 Exercise 13 Section 10.2 Exercise 22 Section 10.1 Case Problem Exercise 26 Exercise 39 Exercise Table 10.2 Exercise 40 Exercise 46 Exercise 42 Exercise 18 Exercise 16 Table 10.1 Section 10.4 Exercise 25 Chapter 13 AirTraffic Assembly AudJudg Browsing Chemitech Exer6 Funds GMATStudy GrandStrand HybridTest MarketBasket Medical1 Medical2 NCP Paint RentalVacancy SalesSalary SatisJob SATScores SnowShoveling Triple-A Vitamins Chapter 14 Absent AgeCost Alumni Armand’s Beer Beta Boots Ellipticals ExecSalary HomePrices HondaAccord HoursPts Hydration1 Hydration2 IPO IRSAudit Jensen JetSki JobSat Laptop MktBeta NFLValues OnlineEdu PGATour PlasmaTV RaceHelmets Safety Sales SleepingBags SportyCars Stocks500 Suitcases Chapter 11 Bags BusTimes PriceChange Return SchoolBus Training Travel Yields Exercise 19 Section 11.1 Exercise Exercise Section 11.2 Case Problem Exercise 25 Exercise 11 Table 13.5 Exercise 38 Exercise 10 Exercise 39 Table 13.1 Exercise Exercise 36 Table 13.10 Exercise 12 Exercise 32 Exercise 41 Case Problem Case Problem Table 13.4 Exercise 11 Exercise 37 Case Problem Exercise 35 Exercise 26 Exercise 27 Exercise 20 Exercise 25 Exercise 63 Exercise 64 Case Problem Table 14.1 Exercise 52 Case Problem Exercise 27 Exercises 5, 22, & 30 Exercise 10 Exercise 49 Exercise Exercise 65 Exercise 43 Exercise 53 Exercise 58 Exercise 67 Exercise 61 Exercise 12 Exercise 68 Exercise 14 Exercise 66 Exercise 54 Exercise 60 Case Problem Exercise 20 Exercise 44 Case Problem Exercises & 19 Exercises 8, 28, & 36 Exercise 11 Exercise 59 Exercise Chapter 15 Alumni Auto2 Bank Basketball Boats Brokers Case Problem Exercise 42 Exercise 46 Exercise 24 Exercises 9, 17, & 30 Exercise 25 Butler Tables 15.1 & 15.2 Chocolate Exercise 48 Consumer Case Problem Exer2 Exercise FuelData Exercise 57 Johnson Table 15.6 Lakeland Exercise 47 Laptop Exercise LPGA Exercise 43 MLB Exercises &16 MutualFunds Exercise 56 NBA Exercises 10, 18, & 26 NFLStats Case Problem PGATour Case Problem Repair Exercise 35 RestaurantRatings Exercise 37 Sedans Exercise Showtime Exercises 5, 15, & 41 Simmons Table 15.11 & Exercise 44 SportsCar Exercise 31 Stroke Exercise 38 TireRack Exercise 54 Treadmills Exercise 55 Chapter 16 Audit Bikes Browsing Cars Chemitech ClassicCars ColorPrinter Cravens IBM Layoffs LightRail LPGATour LPGATour2 MetroAreas MLBPitching MPG PGATour Resale Reynolds Stroke Tyler Yankees Exercise 31 Exercise 30 Exercise 34 Case Problem Table 16.10 Exercise Exercise 29 Table 16.5 Exercise 27 Exercise 16 Exercise Exercises 12 & 13 Exercise 17 Exercise Exercise 15 Table 16.4 Case Problem Exercise 35 Table 16.1 Exercises 14 & 19 Table 16.2 Exercise 18 Chapter 18 AptExp Bicycle CarlsonSales CDSales Cholesterol CountySales ExchangeRate Gasoline GasolineRevised HudsonMarine Masters NFLValue Pasta PianoSales Pollution Exercises 34 & 38 Tables 18.3 & 18.12 Case Problem Exercise 45 Tables 18.4 & 18.16 Case Problem Exercise 24 Table 18.1 & Exercises 7, 8, & Table 18.2 Exercise 53 Exercise 16 Exercise 27 Exercise 26 Exercise 49 Exercises 31 & 39 Find more at www.downloadslide.com Power SouthShore TextSales TVSales Umbrella Vintage Exercises 33 & 40 Exercise 32 Exercise 37 Tables 18.6 & 18.19 Tables 18.5 & 18.17 Case Problem Chapter 19 AcctPlanners Additive ChicagoIncome CruiseShips Evaluations Exercise 19 Exercise 12 Exercise Exercise 29 Exercise 45 Exams GolfScores HomeSales Hurricanes JapanUS MatchedSample Methods Microware NielsenResearch OnTime Overnight PoliceRecords PotentialActual Exercise 46 Exercise 16 Section 19.1 Exercise 21 Exercise 22 Appendix 19.1 & 19.3 Exercise 43 Exercise 24 Exercise 47 Exercise 14 Exercise 15 Exercise 23 Table 19.16 ProductWeights Professors ProGolfers Programs Refrigerators Relaxant Student SunCoast Techs TestPrepare ThirdNational Williams WritingScore Exercise 42 Exercise 37 Exercise 36 Exercise 44 Exercise 40 Exercise 13 Exercise 34 Appendix 19.1 Exercise 35 Exercise 27 Appendix 19.1 & 19.3 Appendix 19.1 Exercise 17 Chapter 20 Coffee Jensen Tires Exercise 20 Table 20.2 Exercise Chapter 21 PDC Tree Appendix 21.1 Appendix F p-Value Appendix F ... test and the following data to test this assumption Use α ϭ 10 The sample mean is 24 .5 and the sample standard deviation is 18 25 26 27 26 25 20 22 23 25 25 28 22 27 20 19 31 26 27 25 24 21 29 28 ... test and α ϭ 05 to test this claim 17 21 23 18 22 15 24 24 19 23 23 23 18 43 22 29 20 27 13 26 11 30 21 28 18 33 20 23 21 29 Applications 22 The number of automobile accidents per day in a particular... ei )2/ ei or or more 10 10 12 18 22 22 16 12 5.17 10.78 17.97 22 .46 22 .46 18. 72 13.37 8.36 8. 72 4.83 Ϫ0.78 Ϫ5.97 Ϫ4.46 Ϫ0.46 3 .28 2. 63 3.64 ? ?2. 72 23 .28 0.61 35. 62 19.89 0 .21 10.78 6. 92 13 .28 7.38