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www.downloadslide.net CHAPTER 12 Comparing Multiple Proportions, Test of Independence and Goodness of Fit CONTENTS 12.2 TEST OF INDEPENDENCE STATISTICS IN PRACTICE: UNITED WAY 12.3 GOODNESS OF FIT TEST Multinomial Probability Distribution Normal Probability Distribution 12.1 TESTING THE EQUALITY OF POPULATION PROPORTIONS FOR THREE OR MORE POPULATIONS A Multiple Comparison Procedure Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 508 Chapter 12 STATISTICS www.downloadslide.net Comparing Multiple Proportions, Test of Independence and Goodness of Fit 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 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 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 obtain 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 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 chi-square tests of independence One use of such statistical tests was to determine whether perceptions of administrative expenses were independent of the occupation of the respondent The hypotheses for the test of independence were: H0: Perception of United Way administrative expenses is independent of the occupation of the respondent Ha: Perception of United Way administrative expenses is not independent of the occupation of the respondent Two questions in the survey provided categorical data for the statistical test One question obtained data on *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 © Jim West/Alamy 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 test of independence led to rejection of the null hypothesis and to the conclusion that perception of United Way administrative expenses is not independent of the occupation of the respondent Actual administrative expenses were less than 9%, but 35% of the respondents perceived that administrative expenses were 21% or more Hence, many respondents had inaccurate perceptions of administrative expenses In this group, production-line, clerical, sales, and professional-technical employees had the more inaccurate perceptions The community perceptions study helped United Way of Rochester develop adjustments to its programs and fund-raising activities In this chapter, you will learn how tests, such as described here, are conducted In Chapters 9, 10, and 11 we introduced methods of statistical inference for hypothesis tests about the means, proportions, and variances of one and two populations In this chapter, we introduce three additional hypothesis-testing procedures that expand our capacity for making statistical inferences about populations Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 12.1 www.downloadslide.net Testing the Equality of Population Proportions for Three or More Populations 509 The test statistic used in conducting the hypothesis tests in this chapter is based on the chi-square ( 2) distribution In all cases, the data are categorical These chi-square tests are versatile and expand hypothesis testing with the following applications Testing the equality of population proportions for three or more populations Testing the independence of two categorical variables Testing whether a probability distribution for a population follows a specific historical or theoretical probability distribution We begin by considering hypothesis tests for the equality of population proportions for three or more populations 12.1 Testing the Equality of Population Proportions for Three or More Populations In Section 10.2 we introduced methods of statistical inference for population proportions with two populations where the hypothesis test conclusion was based on the standard normal (z) test statistic We now show how the chi-square ( 2) test statistic can be used to make statistical inferences about the equality of population proportions for three or more populations Using the notation and p1 ϭ population proportion for population p2 ϭ population proportion for population pk ϭ population proportion for population k the hypotheses for the equality of population proportions for k Ն populations are as follows: H0: p1 ϭ p2 ϭ ϭ pk Ha: Not all population proportions are equal If the sample data and the chi-square test computations indicate H0 cannot be rejected, we cannot detect a difference among the k population proportions However, if the sample data and the chi-square test computations indicate H0 can be rejected, we have the statistical evidence to conclude that not all k population proportions are equal; that is, one or more population proportions differ from the other population proportions Further analyses can be done to conclude which population proportion or proportions are significantly different from others Let us demonstrate this chi-square test by considering an application Organizations such as J.D Power and Associates use the proportion of owners likely to repurchase a particular automobile as an indication of customer loyalty for the automobile An automobile with a greater proportion of owners likely to repurchase is concluded to have greater customer loyalty Suppose that in a particular study we want to compare the customer loyalty for three automobiles: Chevrolet Impala, Ford Fusion, and Honda Accord The current owners of each of the three automobiles form the three populations for the study The three population proportions of interest are as follows: p1 ϭ proportion likely to repurchase an Impala for the population of Chevrolet Impala owners p2 ϭ proportion likely to repurchase a Fusion for the population of Ford Fusion owners p3 ϭ proportion likely to repurchase an Accord for the population of Honda Accord owners Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 510 Chapter 12 TABLE 12.1 WEB file AutoLoyalty www.downloadslide.net Comparing Multiple Proportions, Test of Independence and Goodness of Fit SAMPLE RESULTS OF LIKELY TO REPURCHASE FOR THREE POPULATIONS OF AUTOMOBILE OWNERS (OBSERVED FREQUENCIES) Automobile Owners Chevrolet Impala Ford Fusion Honda Accord Likely to Repurchase Yes No 69 56 Total 125 120 Total 123 312 80 52 188 200 175 500 The hypotheses are stated as follows: H0: p1 ϭ p2 ϭ p3 Ha: Not all population proportions are equal In studies such as these, we often use the same sample size for each population We have chosen different sample sizes in this example to show that the chi-square test is not restricted to equal sample sizes for each of the k populations To conduct this hypothesis test we begin by taking a sample of owners from each of the three populations Thus we will have a sample of Chevrolet Impala owners, a sample of Ford Fusion owners, and a sample of Honda Accord owners Each sample provides categorical data indicating whether the respondents are likely or not likely to repurchase the automobile The data for samples of 125 Chevrolet Impala owners, 200 Ford Fusion owners, and 175 Honda Accord owners are summarized in the tabular format shown in Table 12.1 This table has two rows for the responses Yes and No and three columns, one corresponding to each of the populations The observed frequencies are summarized in the six cells of the table corresponding to each combination of the likely to repurchase responses and the three populations Using Table 12.1, we see that 69 of the 125 Chevrolet Impala owners indicated that they were likely to repurchase a Chevrolet Impala One hundred and twenty of the 200 Ford Fusion owners and 123 of the 175 Honda Accord owners indicated that they were likely to repurchase their current automobile Also, across all three samples, 312 of the 500 owners in the study indicated that they were likely to repurchase their current automobile The question now is how we analyze the data in Table 12.1 to determine if the hypothesis H0: p1 ϭ p2 ϭ p3 should be rejected? The data in Table 12.1 are the observed frequencies for each of the six cells that represent the six combinations of the likely to repurchase response and the owner population If we can determine the expected frequencies under the assumption H0 is true, we can use the chi-square test statistic to determine whether there is a significant difference between the observed and expected frequencies If a significant difference exists between the observed and expected frequencies, the hypothesis H0 can be rejected and there is evidence that not all the population proportions are equal Expected frequencies for the six cells of the table are based on the following rationale First, we assume that the null hypothesis of equal population proportions is true Then we note that in the entire sample of 500 owners, a total of 312 owners indicated that they were likely to repurchase their current automobile Thus, 312/500 ϭ 624 is the overall sample proportion of owners indicating they are likely to repurchase their current automobile If H0: p1 ϭ p2 ϭ p3 is true, 624 would be the best estimate of the proportion responding likely to repurchase for each of the automobile owner populations So if the assumption of H0 is true, we would expect 624 of the 125 Chevrolet Impala owners, or 624(125) ϭ 78 owners to indicate they are likely to repurchase the Impala Using the 624 overall sample proportion, we would expect 624(200) ϭ 124.8 of the 200 Ford Fusion owners and 624(175) ϭ 109.2 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 12.1 www.downloadslide.net 511 Testing the Equality of Population Proportions for Three or More Populations of the Honda Accord owners to respond that they are likely to repurchase their respective model of automobile Let us generalize the approach to computing expected frequencies by letting eij denote the expected frequency for the cell in row i and column j of the table With this notation, now reconsider the expected frequency calculation for the response of likely to repurchase Yes (row 1) for Chevrolet Impala owners (column 1), that is, the expected frequency e11 Note that 312 is the total number of Yes responses (row total), 175 is the total sample size for Chevrolet Impala owners (column total), and 500 is the total sample size Following the logic in the preceding paragraph, we can show e11 ϭ Row Total 312 Total Sample Size (Column Total) ϭ 500125 ϭ (.624)125 ϭ 78 Starting with the first part of the above expression, we can write e11 ϭ (Row Total)(Column Total) Total Sample Size Generalizing this expression shows that the following formula can be used to provide the expected frequencies under the assumption H0 is true EXPECTED FREQUENCIES UNDER THE ASSUMPTION H0 IS TRUE eij ϭ (Row i Total)(Column j Total) Total Sample Size (12.1) Using equation (12.1), we see that the expected frequency of Yes responses (row 1) for Honda Accord owners (column 3) would be e13 ϭ (Row Total)(Column Total)/(Total Sample Size) ϭ (312)(175)/500 ϭ 109.2 Use equation (12.1) to verify the other expected frequencies are as shown in Table 12.2 The test procedure for comparing the observed frequencies of Table 12.1 with the expected frequencies of Table 12.2 involves the computation of the following chi-square statistic: CHI-SQUARE TEST STATISTIC ϭ ͚͚ i j ( fij Ϫ eij)2 eij (12.2) where fij ϭ observed frequency for the cell in row i and column j eij ϭ expected frequency for the cell in row i and column j under the assumption H0 is true Note: In a chi-square test involving the equality of k population proportions, the above test statistic has a chi-square distribution with k – degrees of freedom provided the expected frequency is or more for each cell Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 512 Chapter 12 TABLE 12.2 www.downloadslide.net Comparing Multiple Proportions, Test of Independence and Goodness of Fit EXPECTED FREQUENCIES FOR LIKELY TO REPURCHASE FOR THREE POPULATIONS OF AUTOMOBILE OWNERS IF H0 IS TRUE Automobile Owners Chevrolet Impala Ford Fusion Honda Accord Likely to Repurchase The chi-square test presented in this section is always a one-tailed test with the rejection of H0 occurring in the upper tail of the chi-square distribution TABLE 12.3 Yes No 78 124.8 109.2 312 47 75.2 65.8 188 Total 125 200 175 500 Reviewing the expected frequencies in Table 12.2, we see that the expected frequency is at least five for each cell in the table We therefore proceed with the computation of the chisquare test statistic The calculations necessary to compute the value of the test statistic are shown in Table 12.3 In this case, we see that the value of the test statistic is ϭ 7.89 In order to understand whether or not ϭ 7.89 leads us to reject H0: p1 ϭ p2 ϭ p3, you will need to understand and refer to values of the chi-square distribution Table 12.4 shows the general shape of the chi-square distribution, but note that the shape of a specific chi-square distribution depends upon the number of degrees of freedom The table shows the upper tail areas of 10, 05, 025, 01, and 005 for chi-square distributions with up to 15 degrees of freedom This version of the chi-square table will enable you to conduct the hypothesis tests presented in this chapter Since the expected frequencies shown in Table 12.2 are based on the assumption that H0: p1 ϭ p2 ϭ p3 is true, observed frequencies, fij, that are in agreement with expected frequencies, eij, provide small values of (fij Ϫeij)2 in equation (12.2) If this is the case, the value of the chi-square test statistic will be relatively small and H0 cannot be rejected On the other hand, if the differences between the observed and expected frequencies are large, values of (fij Ϫeij)2 and the computed value of the test statistic will be large In this case, the null hypothesis of equal population proportions can be rejected Thus a chi-square test for equal population proportions will always be an upper tail test with rejection of H0 occurring when the test statistic is in the upper tail of the chi-square distribution We can use the upper tail area of the appropriate chi-square distribution and the p-value approach to determine whether the null hypothesis can be rejected In the automobile brand loyalty study, the three owner populations indicate that the appropriate chi-square COMPUTATION OF THE CHI-SQUARE TEST STATISTIC FOR THE TEST OF EQUAL POPULATION PROPORTIONS Observed Frequency ( fi j) Expected Frequency (ei j) Impala Fusion Accord Impala Fusion Accord 69 120 123 56 80 52 78.0 124.8 109.2 47.0 75.2 65.8 Total 500 500 Likely to Automobile Repurchase? Owner Yes Yes Yes No No No Total Difference ( fij ؊ ei j) Ϫ9.0 Ϫ4.8 13.8 9.0 4.8 Ϫ13.8 Squared Difference ( fij ؊ ei j)2 Squared Difference Divided by Expected Frequency ( fij ؊ ei j)2/eij 81.00 23.04 190.44 81.00 23.04 190.44 1.04 0.18 1.74 1.72 0.31 2.89 ϭ 7.89 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 12.1 www.downloadslide.net 513 Testing the Equality of Population Proportions for Three or More Populations TABLE 12.4 SELECTED VALUES OF THE CHI-SQUARE DISTRIBUTION Area or probability χ2 Area in Upper Tail Degrees of Freedom 10 05 025 01 005 2.706 4.605 6.251 7.779 9.236 3.841 5.991 7.815 9.488 11.070 5.024 7.378 9.348 11.143 12.832 6.635 9.210 11.345 13.277 15.086 7.879 10.597 12.838 14.860 16.750 10 10.645 12.017 13.362 14.684 15.987 12.592 14.067 15.507 16.919 18.307 14.449 16.013 17.535 19.023 20.483 16.812 18.475 20.090 21.666 23.209 18.548 20.278 21.955 23.589 25.188 11 12 13 14 15 17.275 18.549 19.812 21.064 22.307 19.675 21.026 22.362 23.685 24.996 21.920 23.337 24.736 26.119 27.488 24.725 26.217 27.688 29.141 30.578 26.757 28.300 29.819 31.319 32.801 distribution has k Ϫ ϭ Ϫ ϭ degrees of freedom Using row two of the chi-square distribution table, we have 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 7.89 We see the upper tail area at ϭ 7.89 is between 025 and 01 Thus, the corresponding upper tail area or p-value must be between 025 and 01 With p-value Յ 05, we reject H0 and conclude that the three population proportions are not all equal and thus there is a difference in brand loyalties among the Chevrolet Impala, Ford Fusion, and Honda Accord owners Minitab or Excel procedures provided in Appendix F can be used to show ϭ 7.89 with degrees of freedom yields a p-value ϭ 0193 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 514 Chapter 12 www.downloadslide.net Comparing Multiple Proportions, Test of Independence and Goodness of Fit Instead of using the p-value, we could use the critical value approach to draw the same conclusion With ␣ ϭ 05 and degrees of freedom, the critical value for the chi-square test statistic is ϭ 5.991 The upper tail rejection region becomes Reject H0 if Ն 5.991 With 7.89 Ն 5.991, we reject H0 Thus, the p-value approach and the critical value approach provide the same hypothesis-testing conclusion Let us summarize the general steps that can be used to conduct a chi-square test for the equality of the population proportions for three or more populations A CHI-SQUARE TEST FOR THE EQUALITY OF POPULATION PROPORTIONS FOR k Ն POPULATIONS State the null and alternative hypotheses H0: p1 ϭ p2 ϭ ϭ pk Ha: Not all population proportions are equal Select a random sample from each of the populations and record the observed frequencies, fij, in a table with rows and k columns Assume the null hypothesis is true and compute the expected frequencies, eij If the expected frequency, eij, is or more for each cell, compute the test statistic: ϭ ͚͚ i j ( fij Ϫ eij )2 eij Rejection rule: p-value approach: Reject H0 if p-value Յ α Critical value approach: Reject H0 if Ն 2α where the chi-square distribution has k Ϫ degrees of freedom and ␣ is the level of significance for the test A Multiple Comparison Procedure We have used a chi-square test to conclude that the population proportions for the three populations of automobile owners are not all equal Thus, some differences among the population proportions exist and the study indicates that customer loyalties are not all the same for the Chevrolet Impala, Ford Fusion, and Honda Accord owners To identify where the differences between population proportions exist, we can begin by computing the three sample proportions as follows: Brand Loyalty Sample Proportions Chevrolet Impala Ford Fusion Honda Accord p¯1 ϭ 69/125 ϭ 5520 p¯2 ϭ 120/200 ϭ 6000 p¯3 ϭ 123/175 ϭ 7029 Since the chi-square test indicated that not all population proportions are equal, it is reasonable for us to proceed by attempting to determine where differences among the Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 12.1 www.downloadslide.net 515 Testing the Equality of Population Proportions for Three or More Populations population proportions exist For this we will rely on a multiple comparison procedure that can be used to conduct statistical tests between all pairs of population proportions In the following, we discuss a multiple comparison procedure known as the Marascuilo procedure This is a relatively straightforward procedure for making pairwise comparisons of all pairs of population proportions We will demonstrate the computations required by this multiple comparison test procedure for the automobile customer loyalty study We begin by computing the absolute value of the pairwise difference between sample proportions for each pair of populations in the study In the three-population automobile brand loyalty study we compare populations and 2, populations and 3, and then populations and using the sample proportions as follows: Chevrolet Impala and Ford Fusion Η¯ p1 Ϫ p¯2Η ϭ Η.5520 Ϫ 6000Η ϭ 0480 Chevrolet Impala and Honda Accord Η¯ p1 Ϫ p¯3Η ϭ Η.5520 Ϫ 7029Η ϭ 1509 Ford Fusion and Honda Accord Η¯ p2 Ϫ p¯3Η ϭ Η.6000 Ϫ 7029Η ϭ 1029 In a second step, we select a level of significance and compute the corresponding critical value for each pairwise comparison using the following expression CRITICAL VALUES FOR THE MARASCUILO PAIRWISE COMPARISON PROCEDURE FOR k POPULATION PROPORTIONS For each pairwise comparison compute a critical value as follows: CVij ϭ ͙ ͱ α p¯i(1 Ϫ p¯i) p¯ (1 Ϫ p¯j) ϩ j ni nj (12.3) where α ϭ chi-square with a level of significance ␣ and k – degrees of freedom p¯i and p¯j ϭ sample proportions for populations i and j ni and nj ϭ sample sizes for populations i and j Using the chi-square distribution in Table 12.4, k Ϫ ϭ Ϫ ϭ degrees of freedom, and a 05 level of significance, we have 2.05 ϭ 5.991 Now using the sample proportions p¯1 ϭ 5520, p¯2 ϭ 6000, and p¯3 ϭ 7029, the critical values for the three pairwise comparison tests are as follows: Chevrolet Impala and Ford Fusion ͱ CV12 ϭ ͙5.991 5520(1 Ϫ 5520) 6000(1 Ϫ 6000) ϩ ϭ 1380 125 200 Chevrolet Impala and Honda Accord ͱ CV13 ϭ ͙5.991 5520(1 Ϫ 5520) 7029(1 Ϫ 7029) ϩ ϭ 1379 125 175 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 516 TABLE 12.5 Chapter 12 www.downloadslide.net Comparing Multiple Proportions, Test of Independence and Goodness of Fit PAIRWISE COMPARISON TESTS FOR THE AUTOMOBILE BRAND LOYALTY STUDY Pairwise Comparison Η¯ pi Ϫ p¯jΗ CVij Significant if Η¯ pi Ϫ p¯jΗ Ͼ CVij 0480 1509 1029 1380 1379 1198 Not significant Significant Not significant Chevrolet Impala vs Ford Fusion Chevrolet Impala vs Honda Accord Ford Fusion vs Honda Accord Ford Fusion and Honda Accord ͱ CV23 ϭ ͙5.991 6000(1 Ϫ 6000) 7029(1 Ϫ 7029) ϭ 1198 ϩ 200 175 pi Ϫ p¯jΗ exceeds its If the absolute value of any pairwise sample proportion difference Η¯ corresponding critical value, CVij, the pairwise difference is significant at the 05 level of significance and we can conclude that the two corresponding population proportions are different The final step of the pairwise comparison procedure is summarized in Table 12.5 The conclusion from the pairwise comparison procedure is that the only significant difference in customer loyalty occurs between the Chevrolet Impala and the Honda Accord Our sample results indicate that the Honda Accord had a greater population proportion of owners who say they are likely to repurchase the Honda Accord Thus, we can conclude p3 ϭ 7029) has a greater customer loyalty than the Chevrolet that the Honda Accord (¯ Impala (¯ p1 ϭ 5520) The results of the study are inconclusive as to the comparative loyalty of the Ford Fusion While the Ford Fusion did not show significantly different results when compared to the Chevrolet Impala or Honda Accord, a larger sample may have revealed a significant difference between Ford Fusion and the other two automobiles in terms of customer loyalty It is not uncommon for a multiple comparison procedure to show significance for some pairwise comparisons and yet not show significance for other pairwise comparisons in the study NOTES AND COMMENTS In Chapter 10, we used the standard normal distribution and the z test statistic to conduct hypothesis tests about the proportions of two populations However, the chi-square test introduced in this section can also be used to conduct the hypothesis test that the proportions of two populations are equal The results will be the same under both test procedures and the value of the test statistic will be equal to the square of the value of the test statistic z An advantage of the methodology in Chapter 10 is that it can be used for either a one-tailed or a two-tailed hypothesis about the proportions of two populations whereas the chi-square test in this section can be used only for two-tailed tests Exercise 12.6 will give you a chance to use the chi-square test for the hypothesis that the proportions of two populations are equal Each of the k populations in this section had two response outcomes, Yes or No In effect, each population had a binomial distribution with parameter p the population proportion of Yes responses An extension of the chisquare procedure in this section applies when each of the k populations has three or more possible responses In this case, each population is said to have a multinomial distribution The chi-square calculations for the expected frequencies, eij, and the test statistic, 2, are the same as shown in expressions (12.1) and (12.2) The only difference is that the null hypothesis assumes that the multinomial distribution for the response variable is the same for all populations With r responses for each of the k populations, the chisquare test statistic has (r Ϫ 1)(k Ϫ 1) degrees of freedom Exercise 12.8 will give you a chance to use the chi-square test to compare three populations with multinomial distributions Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Appendix F: Computing p-Values Using Minitab and Excel Here we describe how Minitab and Excel can be used to compute p-values for the z, t, 2, and F statistics that are used in hypothesis tests As discussed in the text, only approximate p-values for the t, 2, and F statistics can be obtained by using tables This appendix is helpful to a person who has computed the test statistic by hand, or by other means, and wishes to use computer software to compute the exact p-value Using Minitab Minitab can be used to provide the cumulative probability associated with the z, t, 2, and F test statistics So the lower tail p-value is obtained directly The upper tail p-value is computed by subtracting the lower tail p-value from The two-tailed p-value is obtained by doubling the smaller of the lower and upper tail p-values 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 The Minitab steps used to compute the cumulative probability corresponding to z ϭ Ϫ2.67 follow Step Step Step Step Select the Calc menu Choose Probability Distributions Choose Normal When the Normal Distribution dialog box appears: Select Cumulative probability Enter in the Mean box Enter in the Standard deviation box Select Input Constant Enter Ϫ2.67 in the Input Constant box Click OK Minitab provides the cumulative probability of 0038 This cumulative probability is the lower tail p-value used for the Hilltop Coffee hypothesis test For an upper tail test, the p-value is computed from the cumulative probability provided by Minitab as follows: p-value ϭ Ϫ cumulative probability For instance, the upper tail p-value corresponding to a test statistic of z ϭ Ϫ2.67 is Ϫ 0038 ϭ 9962 The two-tailed p-value corresponding to a test statistic of z ϭ Ϫ2.67 is times the minimum of the upper and lower tail p-values; that is, the two-tailed p-value corresponding to z ϭ Ϫ2.67 is 2(.0038) ϭ 0076 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 The Minitab steps used to compute the cumulative probability corresponding to t ϭ 1.84 follow Step Select the Calc menu Step Choose Probability Distributions Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it Appendix F www.downloadslide.net Computing p-Values Using Minitab and Excel 1077 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 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 1078 Appendix F www.downloadslide.net 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 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it Appendix F www.downloadslide.net Computing p-Values Using Minitab and Excel 1079 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 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it Index www.downloadslide.net Note: Chapters 21 and 22 can be found with the Online Content for this book Index entries found in these chapters are denoted by the chapter number, hyphen, and page number Page numbers followed by a n indicate a footnote A Acceptable quality level (AQL), 943 Acceptance criterion, 938, 945 Acceptance sampling, 921, 936–944, 945 binomial probability function, 938, 943, 946 probability of accepting a lot, 938–940 selecting a plan for, 941–942 Accounting applications, 3–4 ACNielsen, Addition law, 186–189, 207 Additive decomposition models, 845, 856, 857 Adjusted multiple coefficient of determination, 695–696, 737, 738 Aggregate price indexes, 953–956, 967, 968 computing from price relatives, 957–959 Air traffic controller stress test, 570–571 Alliance Data Systems, 599 Alpha to enter or remove, 777–778, 781 Alternative hypothesis, 383–384, 426 developing, 384–387 American Military Standard Table (MIL-STD-105D), 942 American Society for Quality (ASQ), 917 American Statistical Association, 19–20 Analysis of variance (ANOVA), 547–552 assumptions for, 549 completely randomized designs and, 552–560 computer results for, 558–559 using Excel, 594–597 using Minitab, 592–593 See also ANOVA tables ANOVA See Analysis of variance (ANOVA) ANOVA procedures, 571–572, 578 ANOVA tables, 557–558, 584, 662 multiple regression, 701–702 simple linear regression, 628–629 Applications, statistical, 3–4 Approximate class width, 42–43 Area, as measure of probability, 268–269 Assignable causes, 922, 945 Association, measures of, 136–145 Attributes sampling plans, 943 Autocorrelation of data, 788–792, 793 Average outgoing quality limit (AOQL), 943 Average range, 927, 946 B Backward elimination procedure, 779 using Minitab, 798 Baldrige, Malcolm, 919 Baldrige National Quality Program (BNQP), 919 Bar charts, 36–37, 38, 50, 78 descriptive statistics, 15 selection of, 72 side by side, 65–67 stacked, 67–68 using Excel, 89 Barnett, Bob, 919 Bayes, Thomas, 202 Bayes’ theorem, 178, 200–205, 207, 208 branch probabilities, 21–24–21–28, 21–30 Bernoulli, Jakob, 240 Bernoulli process, 240 Best-subsets regression, 779–780 using Minitab, 799 Between-treatments estimates, 550–551, 553–554 Biased estimators, 329 Bias in selections, 304 Bimodal data, 108 Binomial experiments, 240–245, 258 Binomial probability distributions, 239–249 expected values of, 246–247, 259 normal approximation of, 283–286 and the sign test, 872, 876 tables of, 245–246, 247 variances of, 246–247, 259 Binomial probability functions, 244, 258, 259 for acceptance sampling, 938, 943, 946 Binomial random variables, 283–284 Bivariate probability distributions, 230–239, 258, 259 financial applications, 233–236 Blocking, 569, 570, 584 Bloomberg Businessweek, 2–3 Bonferroni adjustment, 566–567 Bound on sampling errors, 22–7, 22–30 Box plots, 131–135, 150 using Minitab, 164 using StatTools, 168 Branches, 21–4, 21–29 Bubble charts, 76 Burke Marketing Services, Inc., 546 C Case problems African elephant populations, 162–163 Air Force training program, 503–504 bipartisan agenda for change, 540 business schools of Asia-Pacific, 159–161 compensation for sales professionals, 592 Consumer Research, Inc., 745 ethical behavior of business students, 432–433 finding the best car value, 674–675, 747–748 forecasting food and beverage sales, 861–862 forecasting lost sales, 862–864 Gulf Real Estate Properties, 373–375 Hamilton County Judges, 212–214 Heavenly Chocolates, 161–162 lawsuit defense strategy, 21–33 measuring stock market risk, 671–672 Metropolitan Research, Inc., 375 motion picture industry, 85–86, 158–159 NASCAR drivers winnings, 746–747 Par, Inc., 475–476 Pelican Stores, 84–85, 157–158 PGA tour statistics, 796–797 point-and-shoot digital camera selection, 673–674 Quality Associates, Inc., 430–431 Specialty Toys, 294–295 U.S Department of Transportation, 672–673 Wentworth Medical Center, 591 wines from the Piedmont region of Italy, 797 Young Professional magazine, 372–373 Categorical data, 8, 22, 35, 78 Categorical variables, 8, 22 complex, multiple regression, 713–714 frequency distributions, 35–36, 38 independent, multiple regression, 709–717, 737 summarizing data for, 35–42 Census, 16, 22 Centered moving average, 847–848 Center for Drug Evaluation and Research (CDER), 442 Central limit theorem, 314–316, 320, 334 Chance events, 21–3, 21–29 Chance nodes, 21–4, 21–29 Chebyshev’s Theorem, 125–126, 128, 150 Chi-square distribution, 484 goodness of fit tests, 527–536, 537 hypothesis testing, 488–491 independence of two categorical variables, 523 interval estimation, 484–488 multiple comparison procedures, 514–516 population proportions, multiple, 509–519 test of independence, 519–526, 536 test statistic, 511–514, 537 using Excel, 542–543 using Minitab, 541–542 using StatTools, 544 Cincinnati Zoo and Botanical Gardens, 74–76 Citibank, 216 Classes of a frequency distribution, 42–43, 50 Classical method for assigning probabilities, 176–177, 183, 206 Class midpoints, 43, 78 Class width, approximate, 42, 79 Cluster sampling, 331–332, 335, 22–21–22–29, 22–30, 22–33–22–34 Coefficient of determination, 614–621, 662, 663 correlation coefficient, 618, 619 multiple regression, 694–697 sum of squares due to error (SSE), 614–615 sum of squares due to regression (SSR), 616 total sum of squares (SST), 615–616 Coefficients, for multiple regression, 688–689 Coefficients of variation, 119–120, 150, 151 Colgate-Palmolive Company, 34 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it Index Combinations, 175, 179, 206, 207 Common causes, 922, 945 Comparisonwise Type I error rate, 566–567, 584 Complements, 185–186, 206, 207 Complete block designs, 573 Completely randomized design, 584–585 analysis of variance (ANOVA), 552–560 experimental design, 547–548 using Excel, 594–595 using Minitab, 592–593 using StatTools, 597 Computers, 18 See also specific computer programs Conditional probabilities, 192–199, 207, 21–24, 21–30 Confidence coefficients, 347, 368 Confidence intervals, 347, 368, 662, 664 hypothesis testing, 400 least squares estimators, 626–627 linear regression equation, estimated, 633, 634–635 multiple regression equation, estimated, 707 for normal probability distribution, 348, 354 using Fisher’s least significant difference, 565–566 See also Interval estimation Confidence levels, 347, 368 Consequences, 21–3, 21–29 Consistency of estimators, 330, 334 Consumer Price Index (CPI), 952, 959, 960, 968 Consumer’s risk, 937, 945 Continuity correction factor, 284, 291 Continuous improvement, 922 Continuous probability distributions, 265–297 binomial, normal approximation of, 283–286 exponential distribution, 287–290 normal distribution, 271–283 uniform distribution, 267–271, 291 using Excel, 296–297 using Minitab, 295–296 Continuous random variables, 218, 258 Control charts, 923–924, 945 interpretation of, 933 np chart, 933, 934 p chart, 931–932 R chart, 929–930, 934 using Minitab, 948–949 using StatTools, 949–950 x¯ charts, 923, 924–929, 934 Control limits, 945, 946 np chart, 933 p charts, 932 x¯ charts, 925 Convenience sampling, 332–333, 335 in sample surveys, 22–4, 22–30 Cook’s distance measure, 720–722, 737, 738 Correlation coefficient, 139–143, 150, 662 of bivariate probability distributions, 232–233, 259 coefficient of determination, 618, 619 sample, 618 using Minitab, 164–165 using StatTools, 168 Counting rules for experiments, 172–176, 207 Covariance, 136–139, 150 of bivariate probability distributions, 232, 258 using Minitab, 164–165 using StatTools, 168 Cravens, David W., 773 Cravens data, 773–776 www.downloadslide.net Critical value approach, 426 one-tailed test, 394–396 rejection rule, 395, 399, 408, 413, 491, 498 two-tailed test, 398 Crosby, Philip B., 918 Cross-sectional data, 8, 22 Cross-sectional regression, 802 Crosstabulations, 55–59, 78 using Excel, 93–95 using Minitab, 88 Cumulative frequency distributions, 46–47, 50, 78 Cumulative percent frequency distribution, 47, 78 Cumulative relative frequency distribution, 47, 78 Curvilinear relationships models, 753–756 Cyclical patterns, 805–807, 856 D Dashboards, data 72–74, 78, 145–148 Data, 5, 21 categorical and quantitative, 8, 10 collection of, 548–549 company internal records of, 11 cross-sectional and time series, 8–9, 10 descriptive statistics, 14–16 elements, variables, and observations, 5–7, errors in acquisition, 14 government agencies providing, 12 scales of measurement, 7–8 sources of, 11–14 statistical inference, 16–17 statistical studies, 12–14 summarizing See Summarizing data Data dashboards, 72–74, 78, 145–148 Data mining, 18–19, 22 Data set, 5, 21 Data visualization, 35, 70–76, 78 Data warehousing, 18 Decision analysis with Bayes’ theorem, 21–24–21–28 with probabilities, 21–5–21–13 problem formulation, 21–3–21–5 with sample information, 21–13–21–24 using PrecisionTree, 21–34–31–38 Decision making, 416–417 Decision nodes, 21–4, 21–29 Decision strategies, 21–15–21–18, 21–30 Decision trees, 21–4–21–5, 21–14–21–15, 21–29 Decomposition, 845–855, 856 Deflating a series, 961–964 Degree of belief, 177 Degrees of freedom of the t distribution, 350–351, 368, 451–452, 471 Deming, W Edwards, 918 De Moivre, Abraham, 271 Dependent events, 195 Dependent variables, 600, 646–648, 661 Descriptive statistics, 14–16, 22 association, measures of, 136–145 distribution shape, measures of, 123, 124 graphical displays See Graphical displays of data location, measures of, 101–115 numerical measures, 99–168 tabular displays See Tables for summarizing data using Excel, 165–167 using Minitab, 163–164 using StatTools, 167–168 variability, measures of, 116–123 See also Summarizing data 1081 Deseasonalized time series, 849–852, 856 Deviation about the mean, 117, 118 Difference of population means hypothesis testing, 445–447, 452–454, 480 interval estimates, 443–445, 450–452, 471, 479–480 Difference of population proportions hypothesis testing, 466–467 inference about two populations, 464–470 interval estimates, 464–466, 465–466, 471 standard error, 466 Discrete probability distributions, 215–264 binomial distributions, 239–249 bivariate distributions, 230–239 developing, 220–225 hypergeometric distribution, 253–257 Poisson distribution, 250–253 random variables, 217–219 using Excel, 263–264 using Minitab, 263 Discrete probability functions, 221–222 Discrete random variables, 217–218, 225–226, 258 Dispersion, measures of, 116–123 Distance intervals, 252 Distribution-free statistical methods, 872, 905 See also Nonparametric statistical methods Distributions, sampling, 310–328 Distribution shapes, measures of, 123, 124 Dot plot graphs, 44, 72, 78, 101–102 using Minitab, 86–87 Double-blind experimental design, 552 Double exponential smoothing, 828 Dow, Charles Henry, 960 Dow Chemical Company, 917 Dow Jones averages, 960–961, 968 Dow Jones Industrial Average (DJIA), 960–961 Duke Energy, 22–2 Dummy variables, 710, 712, 713, 737 seasonal pattern forecasts, 836–841 Dunnhumby, 683 Durbin-Watson Test, 788–792, 793 E Economics applications, statistical, 4–5 Efficiency of estimators, 329–330 Electronics Associates, Inc (EAI), 300–301, 306–307, 316–320 Elements of data, 5, 9, 21, 299 in sample surveys, 22–2, 22–30 Empirical discrete distributions, 220, 258 Empirical rule, 126–127, 150, 273 Error term ⑀, 600, 619 assumptions about, 622, 630 assumptions about, multiple regression, 698 and autocorrelation, 788 Estimated logistic regression equations, 726–729, 737, 739 Estimated logit, 732, 737, 739 Estimated multiple regression equations, 684–685, 736, 737 using, 706–709 Estimated regression equations least squares method, 603–614, 619 linear regression, 601–602, 662 multiple regression, 706–709 simple linear regression, 632–640 slope, 605–606, 663 y-intercept, 605–606, 663 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 1082 Index Estimated regression line, 602 Estimated simple linear regression equation, 602, 663 Ethical guidelines for statistical practice, 19–20 Events, 206 complement of, 185–186 and probabilities, 181–185 Excel analysis of variance (ANOVA), 594–597 bar charts, 89 chi-square distribution, 542–543 completely randomized design, 594–595 continuous probability distributions, 296–297 crosstabulations, 93–95 for data presentations, 88–97 descriptive statistics, 165–167 discrete probability distributions, 263–264 exponential smoothing, 867 factorial experiments, 596–597 frequency distributions, 89, 90–93 graphical displays of data, 88–97 histograms, 90–93 hypothesis testing, 435–439 inference about two populations, 478–479 interval estimates, 377–380 moving averages, 866–867 multiple regression, 748–750 nonparametric statistical methods, 912–914 population means: σ known, 377, 435, 436 population means: σ unknown, 377–379, 435–437 population proportions, 438–439 randomized block design, 595 regression analysis, 678–680 sampling, 340 scatter diagrams and trendlines, 95–97 sign test, 912–913 Spearman rank-correlation coefficient, 913–914 tables for summarizing data, 88–97 time series forecasting, 866–867 trend projection, 867 See also StatTools Expected frequencies, 510–511, 537 Expected value approach, 21–5–21–7, 21–29 Expected value of perfect information (EVPI), 21–8–21–9, 21–29, 21–30 Expected value of sample information (EVSI), 21–18–21–20, 21–30 Expected values (EVs), 258 for the binomial distribution, 246–247, 259 decision analysis, 21–6, 21–29, 21–30 of discrete random variables, 225, 258 of the hypergeometric probability distribution, 255, 259 of a linear combination of variables, 234, 237, 259 of sample means, 312–313, 335, 337–338 sample proportion, 323, 335 Expected value without perfect information (EVwoPI), 21–8 Expected value with perfect information (EVwPI), 21–7–21–9 Experimental designs, 547–552, 584 multiple regression approach to, 783–787 Experimental statistical studies, 12–13 Experimental units, 547–548, 584 Experiments, 171–172, 179, 206 binomial, 240–245, 258 Poisson, 250 Experimentwise Type I error rate, 566, 584 Exponential probability density function, 287, 291 Exponential probability distribution, 287–290, 291 www.downloadslide.net computing probabilities for, 287–288, 291 cumulative probabilities, 288 mean, 288 and the Poisson distribution, 288–289 standard deviation, 288 Exponential smoothing, 816–820, 856, 857 using Excel, 867 using Minitab, 864–865 using StatTools, 868 Exponential trend equation, 832, 857 F Factorial experiments, 576–583, 584, 586 ANOVA procedure, 578 computations, 578–581 using Excel, 596–597 using Minitab, 593 Factorial notation, 175 Factor of interest, 570 Factors, 547, 584 Failure in trials, 240 F distribution, 494–499, 555 Federal Reserve Board, 966 Feigenbaum, A.V., 918 Fermat, Pierre de, 171 Financial applications with bivariate probability distributions, 233–236 statistical, Finite population correction factor, 313–314, 334 Finite populations probability sampling methods, 333 sample mean, standard deviation of, 313–314 sample proportion, standard deviation of, 313–314, 335 sampling from, 301–303 Fisher, Ronald Aylmer, 547 Fisher’s least significant difference (LSD), 563–566, 585 Fitness for use, 918 Five-number summaries, 131, 133, 150 Food Lion, 343 Forecast accuracy exponential smoothing, 818–820, 856 moving averages, 815–816, 856 Forecast error, 808–809, 856 Forecasting See Time series forecasting Forward selection procedure, 778–779 using Minitab, 798 Frames, 300, 334 in sample surveys, 22–3, 22–30 Frequency distributions, 78 for categorical variables, 35–36, 38 cumulative, 50 for quantitative variables, 42–44 using Excel, 89, 90–93 F Test, 555–556, 664 independent variables, adding to model, 767–770, 777, 792 least squares estimators, 627 multiple regression, 699–702, 738 simple linear regression, 627–629 G Galton, Francis, 600 Gauss, Carl Freidrich, 605 General linear model, 753–767, 792 See also Linear trend regression Geographic Information System (GIS), 76 Geometric means, 106–107, 149, 151 Goodness of fit tests, 527–536, 537 multinomial probability distribution, 527–530, 536 normal probability distribution, 530–534 test statistic for, 528–529 using Minitab, 541–542 using StatTools, 544 Gosset, William Sealy, 350 Graphical displays of data, 77 bar charts, 36–37, 50, 65–68, 78 dot plots, 44 effective use of, 70–76 histograms, 44–46, 50, 72, 78 pie charts, 37–38, 78 scatter diagrams and trendlines, 64–65, 66, 78 stem-and-leaf displays, 47–50, 78 using Excel, 88–97 using Minitab, 86–88 using StatTools, 98 Gross domestic product (GDP), 962–963 G test statistic, 728, 733 H High leverage points, 656–657, 662, 720 Histograms, 44–46, 50, 72, 78 descriptive statistics, 15 and stem-and-leaf displays, 49 using Excel, 90–93 using Minitab, 87 using StatTools, 98 Holt’s linear exponential smoothing, 828–830, 856, 857 using Minitab, 865–866 using StatTools, 868–869 Horizontal patterns, 802–804, 856 Hypergeometric probability distribution, 253–257, 258, 259 Hypergeometric probability function, 253–254, 258, 259 Hypothesis testing, 382–440 alternative hypotheses, 384–387 chi-square distribution, 488–491 confidence intervals, 400 and decision making, 416–417 of difference of population means, 445–447, 452–454, 480 of difference of population proportions, 466–467 interval estimates, 400–401 lower tail test, 390–391, 399, 408, 413 matched samples, 459–460, 472 null and alternative hypotheses, 384–387 population mean: σ known, 390–404 population mean: σ unknown, 405–411 population means, 386–387, 392 population median, 872–876 and population proportions, 386–387, 411–416 of population variance, 488–491 sample sizes, 422–425, 427 standard error of the mean, 391 two-tailed test, 399, 400, 408, 413 Type I and Type II errors, 387–390 upper tail test, 399, 408, 413 using Excel, 435–439 using Minitab, 433–434 using StatTools, 439–440 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it Index I Incomplete block designs, 573 Independence, test of, 519–526, 536 using Minitab, 541 using StatTools, 544 Independent events, 195, 207 multiplication law for, 196, 208 and mutually exclusive events, 196 Independent sample design, 458 Independent simple random samples, 443, 471 Independent variables adding or deleting from model, 767–773, 777, 792 experimental design, 547 multiple regression, 703 regression analysis, 600, 661 against residual plots, 645–646 selection procedures, 777–783 Index numbers, 951–970 Consumer Price Index (CPI), 952, 959, 960, 968 price indexes See Price indexes price relatives, 953, 957–959, 967, 968 Producer Price Index (PPI), 952, 959–960, 968 quality indexes, 965–967, 968 Index of Industrial Production, 966, 968 Indicator variables, 710 Indifference quality level (IQL), 943 Individual significance, 699 Inference about two populations, 441–481 difference between population means: matched samples, 458–464, 479, 480–481 difference between population means: σ1 and σ2 known, 443–450, 478–479 difference between population means: σ1 and σ2 unknown, 450–458, 479 of difference of population proportions, 464–470 using Excel, 478–479 using Minitab, 476–478 using StatTools, 479–481 Infinite populations, 313–314, 335 sampling from, 303–304 Influential observations in linear regression models, 656–658, 662 in multiple regression models, 720–722, 737, 738 Information Resources, Inc., Information systems applications, statistical, Interactions, 577–578, 584 second order models, 756–759, 792 International Organization of Standardization (ISO), 919 Interquartile ranges (IQRs), 117, 150, 151 outlier identification, 128–129 Intersection of events, 187, 207 Interval estimation, 342–381, 368 with chi-square distribution, 484–488 of difference of population means, 443–445, 450–452, 471, 479–480 of difference of population proportions, 464–466, 465–466, 471 and hypothesis testing, 400–401 margin of error, 344–348, 351–354, 363 of population means, 344–359, 369 and population proportion, 362–367, 369, 376–377, 379–380, 381 of population variance, 484–488, 501 procedures for, 356–357 regression equation, estimated, 633 and sample size, 348, 354, 359–362, 369, 380–381 using Excel, 377–380 www.downloadslide.net using Minitab, 375–377 using StatTools, 380–381 Interval scale of measurement, 7, 8, 21 Ishikawa, Karou, 918 ISO 9000, 919 Ith observation, 656–657, 664 standardized residuals, 649, 717, 719, 738 Ith residual, 614, 644, 662 standard deviation of, 648, 717, 738 standardized residual of, 649 J John Morrell & Company, 383 Joint probabilities, 193, 207, 21–26, 21–30 Judgment sampling, 333, 335, 22–4, 22–30 Juran, Joseph, 918 K Key performance indicators (KPIs), 72–73, 145 Kruskal-Wallis test, 895–900, 906 using Minitab, 911–912 Kruskal-Wallis test statistic, 897, 906 L Laspeyres index, 955, 968 Leaf unit, 50 Least squares criterion, 605, 663 multiple regression, 685–686, 737 Least squares estimators confidence intervals, 626–627 F Test, 627 sampling distributions, 625 standard deviations, 625, 663, 664 t Test, 624–626 Least squares formulas, 675–676 Least squares method, 662 estimated regression equation, 603–614, 619 multiple regression, 685–689, 736 Levels of significance, 388–389, 394, 426 Leverage of an observation, 649, 656–657, 717, 737 Limits of box plots, 131 Linear exponential smoothing, 828–830, 856, 857 Linear regression, simple See Simple linear regression Linear trend equation, 825, 857 Linear trend regression, 833 Holt’s linear exponential smoothing, 828–830, 856, 857 trend projection, 823–828 Location, measures of, 101–115 Logarithmic transformations, 761–763 Logistic regression, 724–735 using Minitab, 750 Logistic regression equation, 725–726, 737, 738 estimating, 726–729 interpreting, 729–732 logit transformation, 732 Logit, 732, 737, 739 Logit transformation, 732 Lots, 936, 945 Lot tolerance percent defective (LTPD), 943 Lower tail test, 395–396 critical value approach, 394–395 hypothesis testing, 390–391, 399, 408, 413 1083 for population variance, 491 p-value approach, 393 M MAE (mean absolute error), 120 time series forecasting, 809–810, 856 Magazines, use of statistics in, 2–3 Malcolm Baldrige National Quality Award, 919 Mann-Whitney-Wilcoxon (MWW) test, 885–895, 906 using Minitab, 911 using StatTools, 914–915 MAPE (mean absolute percentage error), 810, 856 Marascuilo procedures, 515–516, 536, 537 Marginal probabilities, 193, 207 Margins of error, 343, 368 difference between population means, 445 and interval estimates, 344–348, 351–354, 363 for population proportions, 365 regression equation, estimated, 634, 636 and sample size, 359–361 Market basket, 959 Marketing applications, statistical, Matched sample design, 458 Matched samples, 458, 471 hypothesis testing, 459–460, 472 sign test, 877–878 MeadWestvaco Corporation, 299 Mean, 101–103, 110, 149 of the exponential distribution, 288 for the Mann-Whitney-Wilcoxon test distribution, 891 of the normal distribution, 272 regression equation, estimated, 634 for the sign test distribution, 875, 906 weighted, 103–104 for the Wilcoxon signed-rank test distribution, 882, 906 Mean absolute error (MAE), 120 time series forecasting, 809–810, 856 Mean absolute percentage error (MAPE), 810, 856 Means deviation about the, 117 sample, 101–103, 150, 300 Mean squared error (MSE), 554–555, 560, 585 multiple regression, 700, 701, 738 simple linear regression, 624, 662, 663 time series forecasting, 810, 856 Mean square due to regression (MSR) multiple, 700, 738 simple linear, 627, 664 Mean square due to treatments (MSTR), 553–554, 560, 585 Measures of association, 136–145 Measures of distribution shapes, 123, 124 Measures of location, 101–115 Measures of variability, 116–123 Medians, 104–105, 110, 149, 272 Minitab analysis of variance (ANOVA), 592–593 backward elimination procedure, 798 best-subsets regression, 799 box plots, 164 chi-square distribution, 541–542 completely randomized design, 592–593 continuous probability distributions, 295–296 control charts, 948–949 correlation coefficient, 164–165 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 1084 Index Minitab (Continued) covariance, 164–165 crosstabulations, 88 for data presentations, 86–88 descriptive statistics, 163–164 discrete probability distributions, 263 dot plot graphs, 86–87 exponential smoothing, 864–865 factorial experiment, 593 forward selection procedure, 798 goodness of fit tests, 541–542 graphical displays of data, 86–88 histograms, 87 Holt’s linear exponential smoothing, 865–866 hypothesis testing, 433–434 independence, test of, 541 inference about two populations, 476–478 interval estimates, 375–377 Kruskal-Wallis test, 911–912 logistic regression, 750 Mann-Whitney-Wilcoxon (MWW) test, 911 moving averages, 862 multiple regression, 748 nonparametric statistical methods, 910–912 population means: σ known, 376, 433–434 population means: σ unknown, 376, 434 population proportions, 434, 541 population variances, 504–505 randomized block design, 593 regression analysis, 678 sampling, 339–340 scatter diagrams and trendlines, 87–88 sign test, 910 simple linear regression, 640–641 Spearman rank-correlation coefficient, 912 stem-and-leaf displays, 87 stepwise regression procedure, 798 tables for summarizing data, 86–88 test of independence, 541 time series decomposition, 866 time series forecasting, 864–866 trend projection, 865 variable selection procedures, 798–799 Wilcoxon signed-rank test, 910–911 Modes, 107–108, 150, 272 Monsanto Company, 752 Monthly data, 841, 852, 853 Moving averages, 813–816, 856, 857 using Excel, 866–867 using Minitab, 862 using StatTools, 867–868 MSE See Mean squared error (MSE) MSR See Mean square due to regression (MSR) MSTR (mean square due to treatments), 553–554, 560, 585 Multicollinearity, 703–704, 737 Multimodal data, 108 Multinomial probability distribution, 527–530, 536 Multiple coefficient of determination, 694–697, 736, 738 Multiple comparison procedures, 514–516, 563–569, 584, 585 Multiple regression, 682–750 categorical independent variables, 709–717 coefficient of determination, 694–697 coefficients, 688–689 experimental design for, 783–787 least squares method, 685–689, 736 logistic regression, 724–735 model of, 684–685, 698–699, 736, 737 www.downloadslide.net multiple coefficient of determination, 694–697 residual analysis, 717–724 using Excel, 748–750 using Minitab, 748 using StatTools, 750 Multiple regression equation, 684, 736–737 Multiple regression models, 720–722, 737 Multiple sampling plans, 943, 945 Multiple-step experiments, 172–175, 206 Multiplication law, 195–196, 207, 208 Multiplicative decomposition model, 845–846, 856, 857 Mutually exclusive events, 189, 196, 207 N Naive forecasting method, 808 Nevada Occupational Health Clinic, 801 Newspapers, statistics use in, 2–3 Neyman allocation, 22–17 Nodes, 21–4, 21–29 Nominal scale of measurement, 7, 8, 21 Nonlinear models curvilinear relationships models, 753–756 intrinsically linear, 763–764 Nonlinear trend regression, 830–832 Nonparametric statistical methods, 870–915 Kruskal-Wallis test, 895–900, 906 Mann-Whitney-Wilcoxon test, 885–895, 906 rank correlation, 900–904 sign test, 872–880, 905 using Excel, 912–914 using Minitab, 910–912 using StatTools, 914–915 Wilcoxon signed-rank test, 880–885, 906 Nonprobabilistic sampling, 333, 22–4, 22–30 Nonsampling errors, 22–5, 22–30 Normal curve, 271–273 Normal probability density function, 272, 291 Normal probability distribution, 271–283, 291 approximations for nonparametric methods, 875, 882, 891, 906 binomial probability estimation with, 283–286 central limit theorem, 314–316, 320, 334 computing probabilities for, 278–279, 291 confidence intervals for, 348, 354 empirical rule, 126–127, 273 goodness of fit test, 530–534 mean, 272 median, 272 mode, 272 sampling distribution approximation, 875 standard deviation, 273 Normal probability plots, 650–651, 662 Normal scores, 650 Np chart, 933, 934, 945 Null hypothesis, 383, 426 challenging, 385–386 developing, 384–387 O Observational statistical studies, 12, 13 Observations of data, 5–7, 9, 21 Observed frequencies, 510 Observed level of significance, 394 Odd ratios, 729–731, 733, 737, 739 Odds in favor of an event occurring, 729, 737 Ohio Edison Company, 21–2 One-tailed test, 426 population means: σ known, 390–396 population means: σ unknown, 405–406 sample size for, 423 Open-end classes, 50 Operating characteristic (OC) curves, 939, 945 Ordinal scale of measurement, 7, 8, 21 Outcomes, formula for, 242, 259 Outliers, 127–128, 150, 662, 737 of box plots, 131, 132 data acquisition errors, 14 detecting in regression models, 653–656, 719–720, 722 interquartile ranges (IQRs), 128–129 Overall sample means, 550, 560, 585 quality control, 927, 928, 945 Overall significance, 699 P Paasche index, 955, 968 Parameters of a sampling population, 301, 334 Parametric statistical methods, 871–872, 905 Pareto, Vilfredo, 37 Pareto diagram, 37 Partitioning total sum of squares, 557, 584, 585 Pascal, Blaise, 171 Payoffs, 21–4, 21–29 Payoff tables, 21–4, 21–29 p chart, 931–932, 945 Pearson, Karl, 600 Pearson product moment correlation coefficient, 140, 151, 900, 901–902 Percent frequency distributions, 36, 43–44, 47, 78 Percentiles, 108–109, 150 quartiles, 109–110 Perfect information, 21–7–21–8 Permutations, 206 counting rules for, 175–176, 207 Pie charts, 37–38, 72, 78 Point estimates, 307–308, 334, 633 Point estimation, 306–309 Point estimators, 101, 149, 334, 343–344 consistency of, 330, 334 difference between population means, 444, 471 difference between population proportions, 464, 471 efficiency of, 329–330 population parameters, 308 of population variance, 484 properties of, 328–330 regression equation, estimated, 633 and sample means, 103 and sample standard deviations, 119 and sample variances, 117 simple random samples, 307 unbiased, 328–329 Poisson, Siméon, 250 Poisson experiments, 250 Poisson probability distribution, 250–253, 258 Citibank ATM wait times, 216 distance intervals, 252 and the exponential distribution, 288–289 time intervals, 250–252 Poisson probability function, 250, 258, 259 Pooled estimators of population proportions, 466, 471 Pooled sample variances, 454 Pooled-treatments estimates, 551 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it Index Population correlation coefficient, 140–141 Population covariance, 137–138, 151 Population means, 101, 103, 150 cluster sampling, 22–23–22–24 difference between, estimating, 444 hypothesis testing, 386–387, 392 inference about difference between: matched samples, 458–464, 479, 480–481 inference about difference between: σ1 and σ2 known, 443–450, 478–479 inference about difference between: σ1 and σ2 unknown, 450–458, 479 interval estimates, 344–359, 369 sample sizes, 422–425, 427 simple random sampling, 22–6–22–7 standard deviation, 344–359 stratified simple random sampling, 22–12–22–14 testing for equality of, 556, 559–560 Population means: σ known, 344–350, 357 hypothesis testing, 390–404 one-tailed test, 390–396 test statistic, 392, 427 using Excel, 377, 435, 436 using Minitab, 376, 433–434 Population means: σ unknown, 350–359 hypothesis testing, 405–411 one-tailed test, 405–406 test statistic, 405, 427 using Excel, 377–379, 435–437 using Minitab, 376, 434 using StatTools, 380, 439 Population median, 872–876 Population of a study, 16, 22, 299 finite, sampling from, 301–303 infinite, sampling from, 303–304 Population parameters, 101, 149 and hypothesis testing, 386–387 and point estimators, 308 regression equation, 601 Population proportions and chi-square distribution, 509–519 cluster sampling, 22–25–22–26 and hypothesis testing, 386–387, 411–416 inference about difference between, 464–470 and interval estimation, 362–367, 369, 376–377, 379–380, 381 Marascuilo pairwise comparisons, 515–516, 536, 537 and margin of error, 365 multiple comparison procedures, 514–516 multiple population testing, 509–519 pooled estimators, 466, 471 sample, 300 sample sizes estimates, 364–365, 369 simple random sampling, 22–8–22–9 stratified simple random sampling, 22–15–22–16 test statistic, 412, 427 using Excel, 438–439 using Minitab, 434, 541 using StatTools, 439, 544 Populations, 22–2, 22–30 Population totals cluster sampling, 22–24–22–25 simple random sampling, 22–7–22–8 stratified simple random sampling, 22–14–22–15 Population variances, 117 between-treatments estimates of, 553–554 formula, 151 inferences about, 482–506 interval estimates, 484–488, 501 www.downloadslide.net lower tail test, 491 point estimators, 484 single population, 484–494 test statistic, 488, 496, 498, 501 two populations, 494–501 two-tailed test, 491, 498 upper tail test, 491, 498 using Minitab, 504–505 using StatTools, 505–506 within-treatments estimates of, 554–555 Posterior probabilities, 200, 207, 21–13, 21–29 Power, 419, 426 Power curves, 419, 426 PrecisionTree, 21–34–31–38 Prediction intervals, 662, 664 linear regression equation, estimated, 633, 635–637 multiple regression equation, estimated, 707 Predictors, 633 Price indexes aggregate, 953–956, 967 considerations, 964–965 Consumer Price Index (CPI), 952, 959, 960, 968 deflating a series by, 961–964 Dow Jones averages, 960–961 and price relatives, 957–959 weighted aggregated, 954–955, 967 Price relatives, 953, 967, 968 and aggregate price indexes, 957–959 Prior probabilities, 200, 207, 21–13, 21–29 Probabilistic sampling, 22–4, 22–30 Probabilities, 169–214 assigning, 176–179, 206 conditional, 192–199 counting rules, 172–176 decision analysis, 21–5–21–13 events and, 181–185 experiments, 171–172 measuring by area, 268–269 relationships of, 185–192 Probability density functions, 267, 291 normal, 272 standard normal, 274 Probability distributions, 258 See also Continuous probability distributions; Discrete probability distributions Probability functions, 220, 258 Probability samples, 301 Probability sampling techniques, 332 Probability tree, 201 Procter & Gamble, 266 Producer Price Index (PPI), 952, 959–960, 968 Producer's risk, 937, 945 Production applications, statistical, Proportional allocation, 22–19 Protected LSD test, 566 p-value approach, 426 independent variables, adding to model, 770, 777 interpreting, 401 lower tail test, 393 one-tailed test, 392–394, 395–396 rejection rule, 394, 408, 413, 491, 498 two-tailed test, 397–398 Q Quadratic trend equation, 830–832, 857 Quality control, 916–950 acceptance sampling, 921, 936–944 1085 history of, 918–919 ISO 9000, 919 Malcolm Baldrige National Quality Award, 919 in the service sector, 922 Six Sigma, 919–921 statistical process control, 921, 922–936 Quality indexes, 965–967, 968 Quantitative data, 8, 9, 22, 35, 78 Quantitative variables, 8, 22, 42–55 Quartiles, 109–110, 150 R Radar charts, 76 Randomization, 547, 552 Randomized block design, 569–576, 584, 586 air traffic controller stress test, 570–571 ANOVA procedure, 571–572 computations, 572–573 using Excel, 595 using Minitab, 593 Random numbers, 302 Random samples, 301–304, 334 Random variables, 217–219, 258, 283–284 Ranges, 116–117, 150 Rank correlation, 900–904 Ratio scale of measurement, 7–8, 21 R chart, 929–930, 934, 945 Reciprocal transformation, 763 Regression analysis independent variables, 600, 661 larger problems, 773–776 model building, 751–799 multiple regression, 736 simple linear regression, 600 time series See Time series analysis using Excel, 678–680 using Minitab, 678 using StatTools, 681 See also Multiple regression; Simple linear regression Regression equation, 601, 662 estimated, linear, 601–602, 632–637 estimated, multiple, 706–709 multiple regression, 684 Regression models multiple, 684–685, 698–699 simple linear, 600–601, 662 variance of error, 623–624 Rejectable quality level (RQL), 943 Relative efficiency of an estimator, 329, 334 Relative frequency distributions, 78 for categorical variables, 36, 38 cumulative, 47 for quantitative variables, 43–44 Relative frequency formula, 44, 79 Relative frequency method, 177, 206 Replications, 548, 577, 584 Research hypothesis, 384–385 Residual analysis of regression model, 662 influential observations, 656–658 multiple regression, 717–724 outliers, 653–656 validating, 644–653 Residual for observation i, 644, 664 Residual plots, 651, 662 of dependent variable, 646–648 against independent variable, 645–646 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 1086 Index Response variables, 547, 584, 699 Restricted LSD test, 566 S Sample arithmetic means, 101 Sample correlation coefficients, 140, 141–143, 618, 663 Sample covariance, 136–137, 151 Sampled populations, 300, 334, 22–3, 22–30 Sample information, 21–13, 21–29 Sample means, 101–103, 150, 300 expected value of, 312–313, 335, 337–338 sampling distribution of, 312–322, 345 standard deviation of, 313–314, 335, 338–339 for treatments, 553, 584 Sample points, 172, 206 Sample proportions, 300 expected value of, 323, 335 sampling distributions, 322–328 standard deviation of, 313–314, 323–324, 335 Sample ranges, 928 Samples, 22, 299 in auditing accounts, in sample surveys, 22–2, 22–30 statistical inference, 16 Sample sizes, 422–425, 427 cluster sampling, 22–26–22–27 hypothesis testing, 422–425, 427 and interval estimates, 348, 354, 359–362, 369, 380–381 population proportion estimates, 364–365, 369 and sampling distributions, 318–320 simple random sampling, 22–9–22–11 small, 354–356 stratified simple random sampling, 22–16–22–19 Sample space, 171–172, 183, 206 Sample statistics, 101, 149, 306–307, 334 Sample surveys, 16, 22, 22–2–22–4 cluster sampling, 22–21–22–29, 22–33–22–34 sampling methods, 22–3–22–4 simple random sampling, 22–6–22–12, 22–30–22–31 stratified simple random sampling, 22–12–22–21, 22–32–22–33 survey errors, 22–5–22–6 systematic sampling, 22–29 terminology used in, 22–2–22–3 types of, 22–3–22–4 Sample variances, 117–118, 120, 151 for treatments, 553, 584 Sampling, 300–307, 316–320 cluster, 331–332, 335 convenience, 332–333, 335 distributions, 310–328 judgment, 333, 335 point estimation, 306–309 selecting a sample, 301–305 stratified random sampling, 331, 334 systematic, 332, 335 using Excel, 340 using Minitab, 339–340 using StatTools, 341 Sampling distributions, 310–312, 334 binomial, for the sign test, 872, 876 chi-square distribution, 484 F distribution, 494 least squares estimators, 625 www.downloadslide.net for the Mann-Whitney-Wilcoxon test, 888–889, 891 normal approximation of, 875 for the rank correlation test, 902 of the sample mean, 312–322, 345 of the sample proportion, 322–328 and sample size, 318–320 for the sign test, 872, 876 for the Wilcoxon signed-rank test, 882 Sampling errors, 22–5–22–6, 22–30 bound on, 22–7, 22–30 Sampling methods, 22–3–22–4 Sampling population parameters, 301, 334 Sampling units, 22–3, 22–30 Sampling without replacement, 302, 334 Sampling with replacement, 302–303, 334 San José copper and gold mine, 170 Scatter diagrams, 603–604, 662 influential observations, 656–657 and outliers, 654 Scatter diagrams and trendlines, 64–65, 66, 72, 78 time series plots, 68 using Excel, 95–97 using Minitab, 87–88 using StatTools, 98 Seasonal adjustments, 852 Seasonal indexes, 846–849, 853 Seasonality, 836–841 Seasonal patterns, 804–805, 806–807, 856 Second order models, 754–755, 756 interactions, 756–759, 792 Serial correlation of data, 788–792 Shewhart, Walter A., 918 Side-by-side bar charts, 65–67, 72, 78 σ known, 344, 368 σ unknown, 350, 368 Significance, level of, 388–389 Significance testing interpreting, 629–630 logistic regression, 728 multiple regression, 699–706 simple linear regression, 623–632 using correlation, 677 Significance tests, 389 Sign test, 872–880, 905, 906 about a population median, 872–876 with matched samples, 877–878 using Excel, 912–913 using Minitab, 910 Simple first-order model, 753, 755 Simple linear regression, 598–681 ANOVA table, 628–629 assumptions for the model, 622–623 assumptions for the model, validating, 644–653 coefficient of determination, 614–621 computer solution, 640–644 equation for, 601–602, 662 F Test, 627–629 influential observations, 656–658 least squares method, 603–614 model of, 600–603, 662 outliers, 653–656 regression analysis, 600 residual analysis, 644–661 significance testing, 623–632 t Test, 627 using estimated regression equation, 632–640 using Minitab, 640–641 Simple random samples, 301–302, 304, 334 point estimators, 307 sample surveys, 22–6–22–12, 22–30, 22–31 See also Random samples Simpson’s paradox, 58–59, 78 Single-factor experiments, 547, 584 Single-stage cluster sampling, 22–21 Six Sigma, 919–921, 944 Skewed histograms, 45–46 Skewness of distributions, 123, 124, 150, 289, 357 Slope, 605–606, 663, 825, 857 Small Fry Design, 100 Smoothing constant, 817, 856 Spearman rank-correlation coefficient, 143, 900–904, 906 using Excel, 913–914 using Minitab, 912 SSAB (sum of squares for interaction), 579, 586 SSA (sum of squares for Factor A), 579, 586 SSBL (sum of squares due to blocks), 572, 586 SSB (sum of squares for Factor B), 579, 586 SSE See Sum of squares due to error (SSE) SSR See Sum of squares due to regression (SSR) SST See Total sum of squares (SST) SSTR (sum of squares due to treatments), 554, 585, 586 Stacked bar charts, 67–68, 72, 78 Standard deviations, 118–119, 120, 150, 258 of discrete random variables, 226 of the exponential distribution, 288 formula, 151 of the ith residual, 648, 717, 738 least squares estimators, 625, 663, 664 for the Mann-Whitney-Wilcoxon test distribution, 891 of normal approximation of the sign test, 875, 906 of the normal distribution, 273 and population means, 344–359 of sample means, 313–314, 335, 338–339 of sample proportion, 323–324 for the Wilcoxon signed-rank test distribution, 882 Standard error, 314, 334 difference between population means, 444, 471 difference between population proportions, 464, 471 of difference of population proportions, 466 Standard error of the estimate, 624, 662, 663 Standard error of the mean, 164, 314, 318 hypothesis testing, 391 quality control, 925, 945 Standard error of the proportion, 323, 931, 946 Standardized residuals, 648–650, 662 of the ith observation, 649, 664, 717, 719, 738 Standardized values, 125 Standard normal probability distribution, 273–278, 291 and the t distribution, 350–351 States of nature, 21–3, 21–29 Stationarity assumption, 241 Stationary time series, 803, 856 Statistical inference, 16–17, 22, 308 Statistical process control, 921, 922–936 control charts, 923–924, 945 Statistical studies, 12–14 Statistics, defined, 3, 21 StatTools box plots, 168 chi-square distribution, 544 completely randomized design, 597 control charts, 949–950 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it Index correlation coefficient, 168 covariance, 168 for data presentations, 98 descriptive statistics, 167–168 exponential smoothing, 868 goodness of fit tests, 544 graphical displays of data, 98 histograms, 98 Holt’s linear exponential smoothing, 868–869 hypothesis testing, 439–440 independence, test of, 544 inference about two populations, 479–481 interval estimates, 380–381 introduction to, 29–32 Mann-Whitney-Wilcoxon (MWW) test, 914–915 moving averages, 867–868 multiple regression, 750 nonparametric statistical methods, 914–915 population means: σ unknown, 380, 439 population proportions, 439, 544 population variances, 505–506 regression analysis, 681 sampling, 341 scatter diagrams and trendlines, 98 tables for summarizing data, 98 test of independence, 544 time series forecasting, 867–869 variable selection procedures, 799 Wilcoxon signed-rank test, 914 Stem-and-leaf displays, 47–50, 72, 78 using Minitab, 87 Stepwise regression procedure, 777–778 using Minitab, 798 Stratified random sampling, 331, 334 population means, 22–12–22–14 population proportions, 22–15–22–16 population totals, 22–14–22–15 sample sizes, 22–16–22–19 sample surveys, 22–12–22–21, 22–30, 22–32–22–33 Studentized deleted residuals, 719–720, 737 Subjective method for assigning probabilities, 177–178, 206 Successful trials, 240, 242, 259 Summarizing data, 35–70 bar charts and pie charts, 36–38 for categorical variables, 35–42 crosstabulation, 55–59 cumulative distributions, 46–47 dot plot graphs, 44 frequency distributions, 35–36 histograms, 44–46 for quantitative variables, 42–55 stem-and-leaf displays, 47–50 using graphical displays See Graphical displays of data using tables, 55–64, 77 Summation sign (⌺), 101 Sum of squares due to blocks (SSBL), 572, 586 Sum of squares due to error (SSE), 554–555, 585, 586, 663 coefficient of determination, 614–615 and sum of squares due to regression or total sum of squares, 614–615, 694–695, 738 Sum of squares due to regression (SSR), 663 coefficient of determination, 616 multiple regression, 700 and sum of squares due to error or total sum of squares, 616, 694–695, 738 www.downloadslide.net Sum of squares due to treatments (SSTR), 554, 585, 586 Sum of squares for Factor A (SSA), 579, 586 Sum of squares for Factor B (SSB), 579, 586 Sum of squares for interaction (SSAB), 579, 586 Sum of squares of the deviations, 604 Survey errors, 22–5–22–6 Surveys, 13 See also Sample surveys Symmetric histograms, 45–46 Systematic sampling, 332, 335, 22–29, 22–30 T Tables for summarizing data, 55–64, 77 crosstabulation, 55–59, 78 using Excel, 88–97 using Minitab, 86–88 using StatTools, 98 Taguchi, Genichi, 918 Target populations, 308, 334, 22–3, 22–30 t distribution, 350–352, 368 degrees of freedom, calculating, 451–452, 471 Test of independence, 519–526, 536 using Minitab, 541 using StatTools, 544 Test statistics, 399, 408, 413, 426 chi-square distribution, 511–514, 537 difference in population means, 446, 452, 471 difference in population proportions, 467, 471 Durbin-Watson, 789 for equality of population means, 555, 585 Fisher’s LSD procedure, 564, 585 for goodness of fit, 528–529 one-tailed test, 391–392 population mean: σ known, 392, 427 population mean: σ unknown, 405, 427 and population proportions, 412, 427 and population variances, 488, 496, 498, 501 Thearling, Kurt, 18 Time series, 802, 856, 961–964 Time series analysis, 802 See also Time series forecasting Time series data, 8–9, 10, 22, 68 Time series decomposition, 845–855, 856 cyclical component, 852 deseasonalizing the time series, 849–852, 856 monthly data, models based on, 852, 853 seasonal adjustments, 852 seasonal indexes, 846–849, 853 using Minitab, 866 Time series forecasting, 800–869 accuracy of, 808–813 decomposition, 845–855, 856 patterns, 802–808 seasonality and trend, 836–844 trend projection, 823–836 using Excel, 866–867 using Minitab, 864–866 using StatTools, 867–869 Time series method, 802 Time series patterns, 802–808 cyclical patterns, 805–807, 856 exponential smoothing, 816–820, 856, 857 forecasting method, selecting, 807–808 horizontal patterns, 802–804, 856 moving averages, 813–816 seasonal patterns, 804–805, 806–807, 856 trend patterns, 804, 805, 806–807, 856 1087 Time series plots, 68, 802, 856 Time series regression, 802 Total quality (TQ), 917–918, 944 See also Quality control Total sum of squares (SST), 557, 585, 586, 663 coefficient of determination, 615–616 and sums of squares due to regression or error, 617, 694–695, 738 Transformations of dependent variables, 760–763 Treatments, 547, 553, 584 Tree diagrams, 173, 206 Trendlines and scatter diagrams, 64–65, 66, 72, 78 Trend patterns, 804, 805, 806–807, 856 seasonality, 838–841 Trend projection Holt’s linear exponential smoothing, 828–830, 856, 857 linear trend regression, 823–828 nonlinear trend regression, 830–832 time series forecasting, 823–836 using Excel, 867 using Minitab, 865 Trials, experimental, 240 Trimmed means, 110 t Tests, 664 least squares estimators, 624–626 multiple regression, 702–703, 738 simple linear regression, 627 Two-stage cluster sampling, 22–21 Two-tailed tests, 426 critical value approach, 398 hypothesis testing, 399, 408, 413 of the null hypothesis, 400 population means: σ known, 396–398 population means: σ unknown, 406–408 for population variance, 491, 498 p-value approach, 397–398 Type I errors, 426 comparison procedures, 566–567 sample size, determining, 422–424 and Type II errors, 387–390 Type II errors, 426 probability of, 417–421 sample size, determining, 422–424 and Type I errors, 387–390 U Unbiased estimators, 313, 328–329, 334 Uniform probability density function, 267, 291 Uniform probability distributions, 222, 267–271, 291 Uniform probability functions continuous, 267–268 discrete, 222, 258 Union of events, 186–187, 206 United Way, 508 Unweighted aggregated price index, 954, 968 Upper tail tests, 390, 395–396 hypothesis testing, 399, 408, 413 for population variance, 491, 498 U.S Commerce Department’s National Institute of Standards and Technology (NIST), 919 U.S Department of Labor, Bureau of Labor Statistics, 952, 959 U.S Food and Drug Administration (FDA), 389, 442 U.S Government Accountability Office (GAO), 483 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it 1088 Index www.downloadslide.net V W X Variability, measures of, 116–123 Variable selection procedures, 777–783, 792 using Minitab, 798–799 using StatTools, 799 Variables in data, 5, 21 Variables sampling plans, 943 Variances, 117–118, 150, 258 for the binomial distribution, 246–247, 259 of discrete random variables, 225–226, 258 of the hypergeometric probability distribution, 255, 259 of a linear combination of variables, 235, 237, 259 regression model error, 623–624 Venn diagrams, 185–186, 206 Warehousing, data, 18 Weighted aggregated price indexes, 954–955, 967, 968 Weighted aggregated quantity index, 966, 968 Weighted average of price relatives, 957, 968 Weighted means, 103–104, 149, 150 Weighted moving averages, 816, 856 West Shell Realtors, 871 Whiskers of box plots, 132 Wilcoxon signed-rank test, 880–885, 906 using Minitab, 910–911 using StatTools, 914 Williams, Walter, 389 Within-treatments estimates, 551, 554–555 World Trade Organization (WTO), 5, 6–7 x¯ charts, 923, 924–929, 934, 945 Y y-intercept estimated regression equation, 605–606, 663 linear trend equation, 825, 857 Z z-scores, 123–125, 150, 151 outlier identification, 127 z transformation, 125 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Statistics for Business and Economics 12e WEBfiles Chapter Nations Norris Shadow02 Table 1.1 Table 1.5 Exercise 25 Chapter 2011Movies 2012Names 2012Networks 2012Population AirSurvey ApTest Audit BaseballHall BBB Careers CEOTime Colleges Computer Crosstab Crosstab2 DJIAPrices DYield FedBank Franchise Frequency FuelData2012 Hypertension LargeCorp LivingArea ManagerTime Marathon MPG MutualFunds NewSAT PelicanStores Restaurant Scatter SmartPhones Snow SoftDrink StartUps Stereo SuperBowl Syndicated Zoo Case Exercise Exercise Exercise 46 Exercise Table 2.8 Table 2.4 Exercise Exercise 48 Exercise 24 Exercise 20 Exercises 53, 54, 55, & 56 Exercise 23 Exercise 27 Exercise 28 Exercise 19 Exercise 49 Exercise 10 Exercise 22 Exercise 11 Exercise 35 Exercise 41 Exercise 21 Exercise Exercise 43 Exercise 26 Exercise 39 Exercises 32, 33, & 34 Exercise 44 Case Table 2.9 Exercise 36 Exercise 42 Exercise 40 Table 2.1 Exercise 47 Table 2.14 Exercise 45 Exercise Exercise 58 Chapter 2011Movies 2012Start Salary 3Points AfricanElephants Ages Asian AustralianOpen BackToSchool BigBangTheory BowlGames CellService Coffee CommuteTime Case Problem Table 3.1 Exercise Case Problem Exercise 63 Case Problem Exercise 28 Exercise 32 Exercise 12 Exercise 57 Exercise 52 Exercise 31 Exercise FairValue Flights Homes Housing MajorSalary MLBSalaries Mutual NCAA NFLTickets PanamaRailroad PelicanStores Penalty PovertyLevel Runners Shoppers SpringTraining StateUnemp Stereo StockMarket TaxCost Travel WaitTracking WorldTemp Exercise 71 Exercise 27 Exercise 68 Exercise 59 Figure 3.8 Exercise 53 Exercise 54 Exercise 44 Exercise 45 Exercise 75 Case Problem Exercise 66 Exercise 69 Exercise 50 Case Problem Exercise 72 Exercise 14 Table 3.6 Exercise 60 Exercise 10 Exercise 70 Exercise 64 Exercise 61 Chapter Judge Case Problem Chapter Volume Exercise 24 Chapter EAI MetAreas Section 7.1 Appendix 7.2, 7.3, & 7.4 Exercise 14 MutualFund Chapter Alcohol Auto CorporateBonds Flights GulfProp Houston Interval p JobSatisfaction JobSearch Lloyd’s Miami NewBalance Nielsen NYSEStocks Professional Program Scheer Standing TaxReturn TeeTimes TicketSales Exercise 21 Case Problem Exercise 16 Exercise 48 Case Problem Exercise 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 49 Exercise Section 8.4 Exercise 22 Chapter Administrator AgeGroup AirRating Exercise 29 Exercise 39 Section 9.4 Bayview ChildCare Coffee Drowsy Eagle FirstBirth Fowle GolfTest Hyp Sigma Known Hyp Sigma Unknown Hypothesis p Orders Quality UsedCars WeeklyPay WomenGolf Case Problem Exercise 30 Section 9.3 Exercise 44 Exercise 43 Exercise 64 Exercise 21 Section 9.3 Appendix 9.2 Appendix 9.2 Appendix 9.2 Section 9.4 Case Problem Exercise 32 Exercise 65 Section 9.5 Chapter 10 AirDelay BusinessTravel CheckAcct CollegeCosts ExamScores Golf GolfScores HomePrices Hotel Matched Mutual Occupancy PriceChange SATMath SoftwareTest StockPrices TaxPrep TestScores Exercise 18 Exercise 24 Section 10.2 Exercise 13 Section 10.1 Case Problem Exercise 26 Exercise 39 Exercise Table 10.2 Exercise 40 Exercise 46 Exercise 42 Exercise 16 Table 10.1 Exercise 22 Section 10.4 Exercise 25 Chapter 11 Bags BatteryTime BusTimes Costco Halloween PriceChange SchoolBus Training Travel Yields Exercise 19 Exercise 21 Section 11.1 Exercise 10 Exercise Exercise Section 11.2 Case Problem Exercise 25 Exercise 11 Chapter 12 Ambulance AutoLoyalty AutoQuality BeerPreference BothWork Chemline ChiSquare Demand Grades M&M NYReform WorkforcePlan Exercise 32 Table 12.1 Exercise 14 Table 12.6 Exercise 29 Table 12.10 Appendix 12.2 Exercise 26 Exercise 34 Exercise 22 Case Problem Exercise 12 Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Chapter 13 AirTraffic Assembly AudJudg Browsing Chemitech CompactCars Exer6 GMATStudy GrandStand HoustonAstros HybridTest Medical1 Medical2 MidwestGas NCP OzoneLevels Paint RentalVacancy SalesSalary SatisJob SATScores SnowShoveling Triple-A Table 13.5 Exercise 38 Exercise 10 Exercise 39 Table 13.1 Exercise 41 Exercise Table 13.10 Exercise 12 Exercise 42 Exercise 32 Case Problem Case Problem Exercise 25 Table 13.4 Exercise 36 Exercise 11 Exercise 37 Case Problem Exercise 35 Exercise 26 Exercise 27 Exercise 20 Chapter 14 Absent AgeCost Armand’s Beta BrokerRatings BusinessSchools BusinessTravel Cameras Camry CEOGrants Charities DigitalCameras DJIAS&P500 Ellipticals FamilySedans GoldHoldings GPS HomePrices HoursPts IRSAudit Jensen Laptop MktBeta NFLPassing NFLValues OnlineEdu RaceHelmets RacingBicycles Safety Sales SportyCars Stocks500 Exercise 63 Exercise 64 Table 14.1 Case Problem Exercises & 28 Exercise 43 Exercise 12 Case Problem Exercise 68 Exercise 10 Exercise 52 Exercise 27 Exercise 58 Exercises 5, 22, & 30 Case Problem Exercise 53 Exercise Exercise 49 Exercise 65 Exercise 67 Exercise 61 Exercise 14 Exercise 66 Exercise Exercise 54 Exercise 60 Exercise 44 Exercises 20 & 31 Case Problem Exercises 7, 19, & 36 Exercise 11 Exercise 59 Chapter 15 2012FuelEcon Exercise 55 Auto2 Bank Basketball Boats Broker Butler CarValues Consumer CruiseShips Exer2 FortuneBest Johnson Lakeland Laptop LPGA MLBPitching MutualFunds NASCAR NFLPassing Repair RestaurantRatings Ships Showtime Simmons Stroke TireRack TireRatings Exercise 42 Exercise 46 Exercise 24 Exercises 9, 17, & 30 Exercise 31 Tables 15.1 & 15.2 Case Problem Case Problem Exercise 25 Exercise Exercise 57 Table 15.6 Exercise 47 Exercise Exercise 43 Exercises 10, 18, & 26 Exercise 56 Case Problem Exercises & 16 Exercise 35 Exercise 37 Exercise Exercises 5, 15, & 41 Table 15.11 & Exercise 44 Exercise 38 Exercise 54 Exercise 48 Chapter 16 Audit Bikes Browsing CarMileage Chemitech ClassicCars ClosingPrice CorporateBonds Cravens Layoffs LPGATour LPGATour2 MetroAreas MLBPitching MPG PGATour RentMortagage Reynolds Stroke Tyler WineRatings Yankees Exercise 31 Exercise 30 Exercise 34 Exercise 35 Table 16.10 Exercise Exercise 27 Exercise 29 Table 16.5 Exercise 16 Exercises 12 &13 Exercise 17 Exercise Exercise 15 Table 16.4 Case Problem Exercise Table 16.1 Exercises 14 & 19 Table 16.2 Case Problem Exercise 18 ExchangeRate Facebook Gasoline GasolineRevised HomePrices HudsonMarine KYBudget Pasta PianoSales Pollution Portfolio Power ProductSales SouthShore Textbooks TextSales TVSales UDFMilk Umbrella Vintage Chapter 18 AcctPlanners Additive ChicagoIncome CruiseShips Evaluations Exams GolfScores HomeSales Hurricanes JapanUS MatchedSamples Methods Microwave NielsenResearch OnTime Overnight PoliceRecords PotentialActual ProductWeights Professors ProGolfers Programs Refrigerators Relaxant Student Techs TestPrepare ThirdNational Writing Score Chapter 17 AptExp Bicycle CarlsonSales CDSales Cholesterol CountySales Dishwasher Exercises 34 & 38 Table 17.3 & 17.12 Case Problem Exercise 45 Table 17.4 & 17.16 Case Problem Exercise 41 Exercise 24 Exercise 27 Table 17.1 Table 17.2 Exercise 16 Exercise 53 Exercise 21 Exercise 26 Exercise 49 Exercises 31 & 39 Exercise 42 Exercises 33 & 40 Exercise 14 Exercise 32 Exercise 30 Exercise 37 Table 17.6 & 17.19 Exercise 43 Table17.5 & 17.17 Table 17.26 Exercise 19 Exercise 12 Exercise Exercise 29 Exercise 45 Exercise 46 Exercise 16 Section 18.1 Exercise 21 Exercise 22 Appendix 18.1 & 18.3 Exercise 43 Exercise 24 Exercise 47 Exercise 14 Exercise 15 Exercise 23 Table 18.16 Exercise 42 Exercise 37 Exercise 36 Exercise 44 Exercise 40 Exercise 13 Exercise 34 Exercise 35 Exercise 27 Appendix 18.1 & 18.3 Exercise 17 Chapter 19 Jensen Tires Coffee Table 19.2 Exercise Exercise 20 Appendix F p-Value Appendix F Copyright 2014 Nelson Education Ltd All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Nelson Education reserves the right to remove additional content at any time if subsequent rights restrictions require it ... Use α ϭ 10 The sample mean is 24 .5 and the sample standard deviation is Demand 18 25 26 27 26 25 20 22 23 25 25 28 22 27 20 19 31 26 27 25 24 21 29 28 22 24 26 25 25 24 Summary In this chapter we... 17 .27 5 18.549 19.8 12 21.064 22 .307 19.675 21 . 026 22 .3 62 23.685 24 .996 21 . 920 23 .337 24 .736 26 .119 27 .488 24 . 725 26 .21 7 27 .688 29 .141 30.578 26 .757 28 .300 29 .819 31.319 32. 801 distribution has... Ϫ .25 00 ϩ .25 ϩ. 52 ϩ.84 ϩ1 .28 68. 42 Ϫ 1 .28 (10.41) ϭ 55.10 68. 42 Ϫ 84(10.41) ϭ 59.68 68. 42 Ϫ 52( 10.41) ϭ 63.01 68. 42 Ϫ 25 (10.41) ϭ 65. 82 68. 42 ϩ 0(10.41) ϭ 68. 42 68. 42 ϩ 25 (10.41) ϭ 71. 02 68.42