Solution manual statistics for business and economics

www.freebookslide.com INSTRUCTOR'S SOLUTIONS MANUAL to Accompany James T McClave P George Benson and Terry Sincich's STATISTICS FOR BUSINESS AND ECONOMICS Tenth Edition Nancy S Boudreau Bowling Green State University Upper Saddle River, New Jersey Columbus, Ohio www.freebookslide.com www.freebookslide.com Contents Preface v Chapter Statistics, Data, and Statistical Thinking Chapter Methods for Describing Sets of Data The Kentucky Milk Case 46 Chapter Probability 55 Chapter Random Variables and Probability Distributions The Furniture Fire Case 82 136 Chapter Inferences Based on a Single Sample: Estimation with Confidence Intervals 137 Chapter Inferences Based on a Single Sample: Tests of Hypothesis 161 Chapter Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses The Kentucky Milk Case – Part II 201 243 Chapter Design of Experiments and Analysis of Variance 256 Chapter Categorical Data Analysis Discrimination in the Work Place 300 328 Chapter 10 Simple Linear Regression 332 Chapter 11 Multiple Regression and Model Building The Condo Sales Case 379 444 Chapter 12 Methods for Quality Improvement 448 Chapter 13 Time Series: Descriptive Analyses, Models, and Forecasting The Gasket Manufacturing Case 476 522 Chapter 14 Nonparametric Statistics 529 iii www.freebookslide.com iv www.freebookslide.com Preface This solutions manual is designed to accompany the text, Statistics for Business and Economics, Tenth Edition, by James T McClave, P George Benson, and Terry Sincich It provides answers to most evennumbered exercises for each chapter in the text Other methods of solution may also be appropriate; however, the author has presented one that she believes to be most instructive to the beginning Statistics student This manual is provided to help instructors save time in preparing presentations of the solutions and to possibly provide another point of view regarding their meaning Some of the exercises are subjective in nature Subjective decisions regarding these exercises have been made and are explained by the author Solutions based on these decisions are presented; the solution to this type of exercise is often most instructive When an alternative interpretation of an exercise may occur, the author has often addressed it and given justification for the approach taken I would like to thank Kelly Barber for creating the art work and for typing this work Nancy S Boudreau Bowling Green State University Bowling Green, Ohio v www.freebookslide.com www.freebookslide.com Statistics, Data, and Statistical Thinking Chapter 1.2 Descriptive statistics utilizes numerical and graphical methods to look for patterns, to summarize, and to present the information in a set of data Inferential statistics utilizes sample data to make estimates, decisions, predictions, or other generalizations about a larger set of data 1.4 The first element of inferential statistics is the population of interest The population is a set of existing units The second element is one or more variables that are to be investigated A variable is a characteristic or property of an individual population unit The third element is the sample A sample is a subset of the units of a population The fourth element is the inference about the population based on information contained in the sample A statistical inference is an estimate, prediction, or generalization about a population based on information contained in a sample The fifth and final element of inferential statistics is the measure of reliability for the inference The reliability of an inference is how confident one is that the inference is correct 1.6 Quantitative data are measurements that are recorded on a meaningful numerical scale Qualitative data are measurements that are not numerical in nature; they can only be classified into one of a group of categories 1.8 A population is a set of existing units such as people, objects, transactions, or events A sample is a subset of the units of a population 1.10 An inference without a measure of reliability is nothing more than a guess A measure of reliability separates statistical inference from fortune telling or guessing Reliability gives a measure of how confident one is that the inference is correct 1.12 Statistical thinking involves applying rational thought processes to critically assess data and inferences made from the data It involves not taking all data and inferences presented at face value, but rather making sure the inferences and data are valid 1.14 a The two variables measured are ‘type of credit card used’ and ‘amount of purchase.’ ‘Type of credit card used’ is qualitative It has no meaningful number associated with it, only the name of the card used ‘Amount of purchase’ is quantitative It has a meaningful number associated with it b In Study 1, it says that all purchases were tracked Thus, the data represent a population a High school GPA is a number usually between 0.0 and 4.0 Therefore, it is quantitative b Honors/awards would have responses that name things Therefore, it would be qualitative 1.16 Statistics, Data, and Statistical Thinking www.freebookslide.com 1.18 1.20 1.22 c The scores on the SAT's are numbers between 200 and 800 Therefore, it is quantitative d Gender is either male or female Therefore, it is qualitative e Parent's income is a number: $25,000, $45,000, etc Therefore, it is quantitative f Age is a number: 17, 18, etc Therefore, it is quantitative a The variable of interest is the status of a company’s e-commerce strategy Since a company either has an e-commerce strategy or not, the variable is qualitative The variable of interest is when the company will implement an e-commerce plan Since the time of implementation will be a date, this variable will be qualitative The variable of interest is whether the company is delivering products over the internet or not Since the company is either delivering products or not, the variable is qualitative The variable of interest is the company’s total revenue in the last fiscal year Since this is a meaningful number, this variable is quantitative b Since there are many more that 154 companies in the U.S., this represents a sample rather than a population a The population of interest is the collection of computer security personnel at all U.S corporations and government agencies b Surveys were sent to computer security personnel at all U S corporations and government agencies However, in 2006, only 616 organizations responded to the survey There could be nonresponse bias Often, only those subjects with strong opinions will respond to a survey Thus, the responses may not reflect what the population as a whole thinks c The variable measured in the survey is whether or not there was unauthorized use of computer systems at the firms during the year Since the responses will be either ‘Yes’ or “No’, the variable is qualitative d If we assume that the responses were a random sample from the population, we could infer that about 52% of all computer security personnel will admit to unauthorized use of computer systems at their firms during the year a The data collection method used is a designed experiment b The experimental units in the study are the 50,000 smokers c The variable of interest is the age at which the scanning method first detects a tumor Since this is a meaningful number, this variable is quantitative Chapter www.freebookslide.com 1.24 1.26 1.28 1.30 d The population of interest is the set of all smokers in the U.S The sample of interest is the set of 50,000 smokers surveyed e The researchers want to compare the age at first detection for the methods to see if one is more sensitive than the other a The variable of interest to the researchers is the rating of highway bridges b Since the rating of a bridge can be categorized as one of three possible values, it is qualitative c The data set analyzed is a population since all highway bridges in the U.S were categorized d The data were collected observationally Each bridge was observed in its natural setting a The population of interest is the set of all New York accounting firms employing two or more professionals There are two variables of interest: Whether or not the firm uses audit sampling methods, and if so, whether or not it uses random sampling The sample is the set of 163 firms whose responses were useable The inference of interest to the New York Society of CPAs is the proportion of all New York accounting firms employing two or more professionals that use sampling methods in auditing their clients b The four responses that were unusable could have been returned blank or could have been filled out incorrectly c Any time a survey is mailed it is questionable whether the returned questionnaires represent a random sample Often times, only those with very strong opinions return the surveys In such a case, the returned surveys would not be representative of the entire population a The experimental units in this study are the 24 projects b The population from which the sample was selected is the set of all new software development projects c The variable of interest in this project is the outcome of reusing previously developed software for the new software development projects d In the sample, of the 24 projects were judged failures This is (9 / 24)*100% = 37.5% We could infer that approximately 37.5% of all projects would be judged failures a The process being studied is the process of filling beverage cans with softdrink at CCSB's Wakefield plant b The variable of interest is the amount of carbon dioxide added to each can of beverage c The sampling plan was to monitor five filled cans every 15 minutes The sample is the total number of cans selected Statistics, Data, and Statistical Thinking www.freebookslide.com d The company's immediate interest is learning about the process of filling beverage cans with softdrink at CCSB's Wakefield plant To this, they are measuring the amount of carbon dioxide added to a can of beverage to make an inference about the process of filling beverage cans In particular, they might use the mean amount of carbon dioxide added to the sampled cans of beverage to estimate the mean amount of carbon dioxide added to all the cans on the process line e The technician would then be dealing with a population The cans of beverage have already been processed He/she is now interested in the outputs Chapter www.freebookslide.com 14.64 a Some preliminary calculations are: x u y v 5.2 220 4.5 5.5 227 7.5 6.0 23.5 259 15.5 5.9 20.5 210 5.8 16 224 6.0 23.5 215 5.8 16 231 5.6 10 268 19 5.6 10 239 11 5.9 20.5 212 5.4 410 24 5.6 10 256 14 5.8 16 306 22 5.5 259 15.5 5.3 284 21 5.3 383 23 5.7 12.5 271 20 5.5 264 18 5.7 12.5 227 7.5 5.3 263 17 5.9 20.5 232 10 5.8 16 220 4.5 5.8 16 246 13 5.9 20.5 241 12 ∑ u =300 ∑ v = 300 SSuv = ∑ uv − SSuu = ∑u SSvv = ∑v rs = 2 u-sq 49 552.25 420.25 256 552.25 256 100 100 420.25 25 100 256 49 9 156.25 49 156.25 420.25 256 256 420.25 ∑ u =4878 ( ∑ u )( ∑ v ) = 3197.5 − 300(300) n (∑u ) − n (∑v) − SSuv SSuuSSvv n = 24 = 4878 − v-sq 20.25 56.25 240.25 36 81 361 121 576 196 484 240.25 441 529 400 324 56.25 289 100 20.25 169 144 ∑ v =4898.5 uv 4.5 52.5 364.25 20.5 96 70.5 144 190 110 41 120 140 352 108.5 63 69 250 126 93.75 51 205 72 208 246 ∑ uv =3197.5 = −552.5 3002 = 1128 24 = 4898.5 − −552.5 1128(1148.5) 3002 = 1148.5 24 = −.4854 Since the magnitude of the correlation coefficient is not particularly large, there is a fairly weak negative relationship between sweetness index and pectin 550 Chapter 14 www.freebookslide.com b To determine if there is a negative association between the sweetness index and the amount of pectin, we test: H0: ρs = Ha: ρs < The test statistic is rs = −.4854 Reject H0 if rs < −rs,α where α = 01 and n = 24 Reject H0 if rs < −.485 (from Table XVII, Appendix B) Since the observed value of the test statistic falls in the rejection region (rs = −.4854 < −.485), H0 is rejected There is sufficient evidence to indicate there is a negative association between the sweetness index and the amount of pectin at α = 01 14.66 a Some preliminary calculations are: Parent 643 381 342 251 216 208 192 141 131 128 124 Rank, u 11 10 rs = − Subsid 2,617 1,724 1,867 1,238 890 681 1,534 899 492 579 672 6∑ di2 n( n − 1) =1− Rank, v 11 10 Difference di -1 2 -3 -2 -2 di2 1 4 4 ∑ di = 32 6(32) = − 145 = 855 11(112 − 1) Since this correlation coefficient is fairly close to 1, it indicates that there is a relatively strong positive relationship between the number of parent companies and the number of subsidiaries To determine if the number of parent companies is positively related to the number of subsidiaries, we test: H0: ρs = Ha: ρs > The test statistic is rs = 855 Nonparametric Statistics 551 www.freebookslide.com From Table XVI, Appendix B, rs,.05 = 523, with n = 11 The rejection region is rs > 523 Since the observed value of the test statistic falls in the rejection region (rs = 855 > 523), H0 is rejected There is sufficient evidence to indicate that the number of parent companies is positively related to the number of subsidiaries at α = 05 b We must assume: The sample is randomly selected The probability distributions of both of the variables are continuous The actual number of companies and subsidiaries are not continuous However, since the numbers of companies/subsidiaries are very large, this assumption is basically met From the information given, we cannot tell whether the sample was random or not 14.68 b Some preliminary calculations: Involvement 10 11 rs = − 6∑ d i2 n(n − 1) ui vi Differences di = ui − vi 10 11 10 11 −1 −1 1 −1 0 =1− d i2 ∑ di2 1 1 1 =6 6(6) = 972 11(112 − 1) To determine if a positive relationship exists between participation rates and cost savings rates, we test: H0: ρs = Ha: ρs > The test statistic is rs = 972 From Table XVII, Appendix B, rs,.01 = 736, with n = 11 The rejection region is rs > 736 552 Chapter 14 www.freebookslide.com Since the observed value of the test statistic does falls in the rejection region (rs = 972 > 736), H0 is rejected There is sufficient evidence to indicate that a positive relationship exists between participation rates and cost savings rates at α = 01 c In order for the above test to be valid, we must assume: The sample is randomly selected The probability distributions of both of the variables are continuous In order to use the Pearson correlation coefficient, we must assume that both populations are normally distributed It is very unlikely that the data are normally distributed 14.70 The appropriate test for this completely randomized design is the Kruskal-Wallis H-test Some preliminary calculations are: Sample 18 32 43 15 63 Rank 4.5 12 Sample 12 33 10 34 18 Rank Sample 12 87 53 65 50 4.5 64 77 R2 = 22.5 R1 = 34.5 Rank 16 11 14 10 13 15 R3 = 79 To determine whether at least two of the populations differ in location, we test: H0: The three probability distributions are identical Ha: At least two of the three probability distributions differ in location Rj 12 The test statistic is H = − 3( n + 1) ∑ n( n + 1) nj = ⎡ (34.5) (22.5) (79) ⎤ 12 + + ⎢ ⎥ − 3(16 + 1) 16(16 + 1) ⎣ 5 ⎦ = 60.859 − 51 = 9.859 The rejection region requires α = 05 in the upper tail of the χ2 distribution with df = p − = − = From Table VII, Appendix B, χ.05 = 5.99147 The rejection region is H > 5.99147 Since the observed value of the test statistic falls in the rejection region (H = 9.859 > 5.99147), reject H0 There is sufficient evidence to indicate a difference in location for at least two of the three probability distributions at α = 05 Nonparametric Statistics 553 www.freebookslide.com 14.72 The appropriate test for two independent samples is the Wilcoxon rank sum test Some preliminary calculations are: Sample 1.2 1.9 2.5 1.0 1.8 1.1 Rank 8.5 10 T1 = 35.5 Sample 1.5 1.3 2.9 1.9 2.7 3.5 Rank 12 8.5 11 13 T2 = 55.5 To determine if there is a difference between the locations of the probability distributions, we test: H0: The two sampled populations have identical probability distributions Ha: The probability distribution for population is shifted to the left or right of that for The test statistic is T2 = 55.5 Reject H0 if T2 ≤ TL or T2 ≥ TU where α = 05 (two-tailed), n1 = and n2 = 6: Reject H0 if T2 ≤ 28 or T2 ≥ 56 (from Table XV, Appendix B) Since T2 = 55.5 ≤/ 28 and T2 = 55.5 ≥/ 56, not reject H0 There is insufficient evidence to indicate a difference between the locations of the probability distributions for the sampled populations at α = 05 14.74 a To determine whether the median biting rate is higher in bright, sunny weather, we test: H0: η = Ha: η > b ( S − 5) − 5n (95 − 5) − 5(122) = = 6.07 n 122 (where S = number of observations greater than 5) The test statistic is z = The p-value is p = P(z ≥ 6.07) From Table IV, Appendix B, p = P(z ≥ 6.07) ≈ 0.0000 c 554 Since the observed p-value is less than α (p = 0.0000 < 01), H0 is rejected There is sufficient evidence to indicate that the median biting rate in bright, sunny weather is greater than at α = 01 Chapter 14 www.freebookslide.com 14.76 Some preliminary calculations are: Difference Highway − Highway −25 −23 −16 −16 Rank of Absolute Differences 2.5 2.5 T+ = To determine if the heavily patrolled highway tends to have fewer speeders per 100 cars than the occasionally patrolled highway, we test: H0: The two sampled populations have identical probability distributions Ha: The probability distribution for highway is shifted to the left of that for highway The test statistic is T+ = The rejection region is T+ ≤ from Table XVI, Appendix B, with n = and α = 05 Since the observed value of the test statistic falls in the rejection region (T+ = ≤ 1), H0 is rejected There is sufficient evidence to indicate the probability distribution for highway is shifted to the left of that for highway at α = 05 b Some preliminary calculations are: Day Difference Highway − Highway 25 −23 −16 −16 d= ∑ di = −76 n ∑ di2 = −15.2 ( ∑ di ) − n = n −1 sd = 131.7 = 11.4761 sd2 = (−76) 5 −1 1682 − To determine if the mean number of speeders per 100 cars differ for the two highways, we test: H0: μ1 = μ2 Ha: μ1 ≠ μ2 The test statistic is t = Nonparametric Statistics d −0 −15.2 = = − 2.96 s d / n 11.4761 555 www.freebookslide.com The rejection region requires α/2 = 05/2 = 025 in each tail of the t-distribution with df = n − = − = From Table VI, Appendix B, t.025 = 2.776 The rejection region is t > 2.776 and t < −2.776 Since the observed value of the test statistic falls in the rejection region (t = −2.96 < −2.776), H0 is rejected There is sufficient evidence to indicate the mean number of speeders per 100 cars differ for the two highways at α = 05 We must assume that the population of differences is normally distributed and that a random sample of differences was selected 14.78 a Since only 70 of the 80 customers responded to the question, only the 70 will be included To determine if the median amount spent on hamburgers at lunch at McDonald's is less than $2.25, we test: H0: η = 2.25 Ha: η < 2.25 S = number of measurements less than 2.25 = 20 The test statistic is z = ( S − 5) − 5n n = (20 − 5) − 5(70) 70 = −3.71 No α was given in the exercise We will use α = 05 The rejection region requires α = 05 in the lower tail of the z-distribution From Table IV, Appendix B, z.05 = 1.645 The rejection region is z > 1.645 Since the observed value of the test statistic does not fall in the rejection region (z = −3.71 >/ 1.645), H0 is not rejected There is insufficient evidence to indicate that the median amount spent on hamburgers at lunch at McDonald's is less than $2.25 at α = 05 556 b No The survey was done in Boston only The eating habits of those living in Boston are probably not representative of all Americans c We must assume that the sample is randomly selected from a continuous probability distribution Chapter 14 www.freebookslide.com 14.80 Some preliminary calculations: Urban 4.3 5.2 6.2 5.6 3.8 5.8 4.7 Rank Suburban Rank Rural Rank 4.5 5.9 14 5.1 10.5 6.7 17 4.8 15.5 7.6 19 3.9 12 4.9 6.2 15.5 5.2 10.5 4.2 13 6.8 18 4.3 4.5 R1 = 62.5 R2 = 86.5 R3 = 41 To determine if there is a difference in the level of property taxes among the three types of school districts, we test: H0: The three probability distributions are identical Ha: At least two of the three probability distributions differ in location The test statistic is H = Rj 12 − 3( n + 1) ∑ n( n + 1) nj ⎛ 62.52 86.52 412 ⎞ 12 + + ⎜ ⎟ − 3(20) = 65.8498 − 60 19(19 + 1) ⎝ 6 ⎠ = 5.8498 = The rejection region requires α = 05 in the upper tail of the χ2 distribution with df = p − = = 5.99147 The rejection region is H > 5.99147 − = From Table VII, Appendix B, χ.05 Since the observed value of the test statistic does not fall in the rejection region (H = 5.8498 >/ 5.99147), H0 is not rejected There is insufficient evidence to indicate that there is a difference in the level of property taxes among the three types of school districts at α = 05 14.82 a Some preliminary calculations are: Truck Static Weight of Truck (ui) 10 6 8 10 55 Nonparametric Statistics Weigh-in-Motion Prior (vi) 1.5 1.5 10 55 Weigh-in-Motion After (wi) 10 55 uivi 16 90 1.5 36 64 25 49 90 383.5 uiwi 16 100 36 64 25 49 81 384 557 www.freebookslide.com ∑ ui ∑ vi = 383.5 − 55(55) SSuv = ∑ ui vI − SSuw = ∑ u i wi − SSuu = ∑ ui2 − SSvv = ∑ SSww = ∑ rs1 = rs2 = vi2 n ( ∑ ui ∑ wi ) ( ∑ ui ) n = 385 − SSuu SSvv n = = 384.5 − SSuw SSuu SSww = = 385 − 81 82.5(82) = 81 55(55) = 81.5 10 552 = 81.5 10 ( ∑ wi ) − SSuv = 384 − n ( ∑ vi ) − wi2 n 10 552 = 82 10 552 = 82.5 10 = 9848 81.5 = 9879 82.5(82.5) The correlation coefficient for x and y1 is rs1 = 9848 Since rs1 > 0, the relationship between static weight and weigh-in-motion prior to adjustment is positive Because the value is close to 1, the relationship is very strong It is larger than r1 = 965 found in Exercise 10.89 The correlation coefficient for x and y2 is rs2 = 9879 Since rs2 > 0, the relationship between static weight and weigh-in-motion after the adjustment is positive Because the value is close to 1, the relationship is very strong It is smaller than r2 = 996 found in Exercise 10.89 b In order for rs to be exactly 1, the rankings for the static weight and the weigh-in-motion must be the same for each truck In order for rs to be exactly 0, the rankings for one of the variables (static weight) must be equal to 11 minus ranking of the other variable (weigh-in-motion) for each truck 14.84 a To determine if the median level differs from the target, we test: H0: η = 75 Ha: η ≠ 75 b S1 = number of observations less than 75 and S2 = number of observations greater than 75 The test statistic is S = larger of S1 and S2 The p-value = 2P(x ≥ S) where x is a binomial random variable with n = 25 and p = If the p-value is less than α = 10, reject H0 558 Chapter 14 www.freebookslide.com c A Type I error would be concluding the median level is not 75 when it is If a Type I error were committed, the supervisor would correct the fluoridation process when it was not necessary A Type II error would be concluding the median level is 75 when it is not If a Type II error were committed, the supervisor would not correct the fluoridation process when it was necessary d S1 = number of observations less than 75 = and S2 = number of observations greater than 75 = 18 The test statistic is S = larger of S1 and S2 = 18 The p-value = 2P(x ≥ 18) where x is a binomial random variable with n = 25 and p = From Table II, p-value = 2P(x ≥ 18) = 2(1 − P(x ≤ 17)) = 2(1 − 978) = 2(.022) = 044 Since the p-value = 044 < α = 10, H0 is rejected There is sufficient evidence to indicate the median level of fluoridation differs from the target of 75 at α = 10 e A distribution heavily skewed to the right might look something like the following: One assumption necessary for the t-test is that the distribution from which the sample is drawn is normal A distribution which is heavily skewed in one direction is not normal Thus, the sign test would be preferred 14.86 Some preliminary calculations are: Hours Rank 8 Nonparametric Statistics Fraction Defective 02 05 03 08 06 09 11 10 Rank di −1 −1 −1 d i2 ∑ di2 1 1 1 =6 559 www.freebookslide.com To determine if the fraction defective increases as the day progresses, we test: H0: ρs = Ha: ρs > The test statistic is rs = − 6∑ di2 n(n − 1) =1− 6(6) = − 071 = 929 8(82 − 1) Reject H0 if rs > rs,α where α = 05 and n = 8: Reject H0 if rs > 643 (from Table XVII, Appendix B) Since rs = 929 > 643, reject H0 There is sufficient evidence to indicate that the fraction defective increases as the day progresses at α = 05 14.88 a The design utilized was a completely randomized design b Some preliminary calculations are: Site 34.3 35.5 32.1 28.3 40.5 36.2 43.5 34.7 38.0 35.1 Rank 11 19 12 23 15 R1 = 107 Site 39.3 45.5 50.2 72.1 48.6 42.2 103.5 47.9 41.2 44.0 Rank 17 25 28 29 27 21 30 26 20 24 R2 = 247 Site 34.5 29.3 37.2 33.2 32.6 38.3 43.3 36.7 40.0 35.2 Rank 14 16 22 13 18 10 R3 = 111 To determine if the probability distributions for the three sites differ, we test: H0: The three sampled population probability distributions are identical Ha: At least two of the three sampled population probability distributions differ in location Rj 12 The test statistic is H = − 3( n + 1) − 3(n + 1) ∑ n( n + 1) nj = 560 12 ⎡107 247 1112 ⎤ + + ⎢ ⎥ − 3(31) = 109.3923 − 93 30(31) ⎣ 10 10 10 ⎦ = 16.3923 Chapter 14 www.freebookslide.com The rejection region requires α = 05 in the upper tail of the χ2 distribution with df = = 5.99147 The rejection region is p − = − = From Table VII, Appendix B, χ.05 H > 5.99147 Since the observed value of the test statistic falls in the rejection region (H = 16.3923 > 5.99147), H0 is rejected There is sufficient evidence to indicate the probability distributions for at least two of the three sites differ at α = 05 c Since H0 was rejected, we need to compare all pairs of sites Some preliminary calculations are: Site 34.3 35.5 32.1 28.3 40.5 36.2 43.5 34.7 38.0 35.1 Site 39.3 45.5 50.2 72.1 48.6 42.2 103.5 47.9 41.2 44.0 Rank 10 13 T1 = 59 Rank 15 18 19 17 12 20 16 11 14 T2 = 151 Site 39.3 45.5 50.2 72.1 48.6 42.2 103.5 47.9 41.2 44.0 Rank 15 18 19 17 12 20 16 11 14 T2 = 151 Site 34.5 29.3 37.2 33.2 32.6 38.3 43.3 36.7 40.0 35.2 Site 34.3 35.5 32.1 28.3 40.5 36.2 43.5 34.7 38.0 35.1 Rank 11 18 12 20 15 T1 = 103 Site 34.3 29.3 37.2 33.2 32.6 38.3 43.3 36.7 40.0 35.2 Rank 14 16 19 13 17 10 T3 = 107 Rank 13 10 T3 = 59 For each pair, we test: H0: The two sampled population probability distributions are identical Ha: The probability distribution for one site is shifted to the right or left of the other The rejection region for each pair is T ≤ 79 or T ≥ 131 from Table XV, Appendix B, with n1 = n2 = 10 and α = 05 Nonparametric Statistics 561 www.freebookslide.com For sites and 2: The test statistic is T1 = 59 Since the observed value of the test statistic falls in the rejection region, (TA = 59 ≤ 79), H0 is rejected There is sufficient evidence to indicate the probability distribution for site is shifted to the left of that for site at α = 05 For sites and 3: The test statistic is T1 = 103 Since the observed value of the test statistic does not fall in the rejection region (T1 = 103 / 131), H0 is not rejected There is insufficient evidence to indicate the probability distribution for site is shifted to the right or left of that for site at α = 05 For sites and 3: The test statistic is T2 = 151 Since the observed value of the test statistic falls in the rejection region (T2 = 151 ≥ 131), H0 is rejected There is sufficient evidence to indicate the probability distribution for site is shifted to the right of that for site at α = 05 Thus, the income for those at site is significantly higher than at the other two sites d The necessary assumptions are: The three samples are random and independent There are five or more measurements in each sample The three probability distributions from which the samples are drawn are continuous For parametric tests, the assumptions are: 562 The three populations are normal The samples are random and independent The three population variances are equal Chapter 14 www.freebookslide.com 14.90 Using MINITAB, the results of the Wilcoxon Rank Sum Test (Mann-Whitney Test) for each of the Variables are: Mann-Whitney Test and CI: CREATIVE-S, CREATIVE-NS CREATIVE-S CREATIVE-NS N 47 67 Median 5.0000 4.0000 Point estimate for ETA1-ETA2 is 1.0000 95.0 Percent CI for ETA1-ETA2 is (0.9999,1.0000) W = 3734.5 Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.0000 The test is significant at 0.0000 (adjusted for ties) Mann-Whitney Test and CI: INFO-S, INFO-NS INFO-S INFO-NS N 47 67 Median 5.000 5.000 Point estimate for ETA1-ETA2 is 0.000 95.0 Percent CI for ETA1-ETA2 is (-0.000,1.000) W = 2888.5 Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.2856 The test is significant at 0.2743 (adjusted for ties) Mann-Whitney Test and CI: DECPERS-S, DECPERS-NS DECPERS-S DECPERS-NS N 47 67 Median 3.000 2.000 Point estimate for ETA1-ETA2 is -0.000 95.0 Percent CI for ETA1-ETA2 is (-0.000,1.000) W = 2963.5 Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.1337 The test is significant at 0.1228 (adjusted for ties) Mann-Whitney Test and CI: SKILLS-S, SKILLS-NS SKILLS-S SKILLS-NS N 47 67 Median 6.0000 5.0000 Point estimate for ETA1-ETA2 is 1.0000 95.0 Percent CI for ETA1-ETA2 is (0.9999,1.9999) W = 3498.5 Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.0000 The test is significant at 0.0000 (adjusted for ties) Nonparametric Statistics 563 www.freebookslide.com Mann-Whitney Test and CI: TASKID-S, TASKID-NS N 47 67 TASKID-S TASKID-NS Median 5.000 4.000 Point estimate for ETA1-ETA2 is 1.000 95.0 Percent CI for ETA1-ETA2 is (-0.000,1.000) W = 3028.0 Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.0614 The test is significant at 0.0566 (adjusted for ties) Mann-Whitney Test and CI: AGE-S, AGE-NS AGE-S AGE-NS N 47 67 Median 47.000 45.000 Point estimate for ETA1-ETA2 is 1.000 95.0 Percent CI for ETA1-ETA2 is (-1.000,4.001) W = 2891.5 Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.2779 The test is significant at 0.2771 (adjusted for ties) Mann-Whitney Test and CI: EDYRS-S, EDYRS-NS EDYRS-S EDYRS-NS N 47 67 Median 13.000 13.000 Point estimate for ETA1-ETA2 is -0.000 95.0 Percent CI for ETA1-ETA2 is (0.000,-0.000) W = 2664.0 Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.8268 The test is significant at 0.8191 (adjusted for ties) A summary of the tests above and the t-tests from Chapter are listed in the table: Variable CREATIVE INFO DECPERS SKILLS TASKID AGE EDYRS Wilcoxon Test Statistic, T2 3734.5 2888.5 2963.5 3498.5 3028.0 2891.5 2664.0 p-value 0.000 0.274 0.123 0.000 0.057 0.277 0.819 t 8.847 1.503 1.506 4.766 1.738 0.742 -0.623 p-value 0.000 0.136 0.135 0.000 0.087 0.460 0.534 The p-values for the Wilcoxon Rank Sum Tests and the t-tests are similar and the decisions are the same Since the sample sizes are large (n = 47 and n = 67), the Central Limit Theorem applies Thus, the t-tests (or z-tests) are valid One assumption for the Wilcoxon Rank Sum test is that the distributions are continuous Obviously, this is not true There are many ties in the data, so the Wilcoxon Rank Sum tests may not be valid 564 Chapter 14 ... 14 Nonparametric Statistics 529 iii www.freebookslide.com iv www.freebookslide.com Preface This solutions manual is designed to accompany the text, Statistics for Business and Economics, Tenth... www.freebookslide.com Statistics, Data, and Statistical Thinking Chapter 1.2 Descriptive statistics utilizes numerical and graphical methods to look for patterns, to summarize, and to present the information... have values between $100 and $300, has a value between $300 and $500, has a value between $500 and $700, have values between $700 and $900, and has a value between $1900 and $2100 c A plot of the