Ch12 Analysis of Variance.docx

Chapter 12 Analysis of Variance True/False 1. The F distribution's curve is positively skewed. Answer: True 2. The test statistic used in ANOVA is Student's t. Answer: False 3. There is one, unique F distribution for a F-statistic with 29 degrees of freedom in the numerator and 28 degrees of freedom in the denominator. Answer: True 4. One characteristic of the F distribution is that F cannot be negative. Answer: True 5. One characteristic of the F distribution is that the computed F can only range between -1 and +1. Answer: False 6. The F distribution is positively skewed and its values may range from 0 to plus infinity. Answer: True 7. The shape of the F distribution is determined by the degrees of freedom for the F-statistic, one for the numerator and one for the denominator. Answer: True 8. Unlike Student's t distribution, there is only one F distribution. Answer: False 9. Like Student's t distribution, a change in the degrees of freedom causes a change in the shape of the F distribution. Answer: True 10. If the computed value of F is 0.99 and the critical value is 3.89, we would not reject the null hypothesis. Answer: True 11. For the hypothesis test, , with n1 = 10 and n2 = 10, the F-test statistic is 2.56. At the 0.01 level of significance, we would reject the null hypothesis. Answer: False 12. For the hypothesis test, , with n1 = 4 and n2 = 4, the F-test statistic is 50.01. At the 0.01 level of significance, we would reject the null hypothesis. Answer: True 13. For the hypothesis test, , with n1 = 7 and n2 = 7, the F-test statistic is 2.56. At the 0.05 level of significance, we would reject the null hypothesis. Answer: False 14. For the hypothesis test, , with n1 = 9 and n2 = 9, the F-test statistic is 4.53. At the 0.05 level of significance, we would reject the null hypothesis. Answer: True 15. To employ ANOVA, the populations being studied must be approximately normally distributed. Answer: True 16. To employ ANOVA, the populations should have approximately equal standard deviations. Answer: True 17. In an ANOVA table, k represents the total number of sample observations and n represents the total number of treatments. Answer: False 18. In an ANOVA table, k represents the number of treatments, b represents the number of blocks, and n represents the total number of sample observations. Answer: True 19. The alternate hypothesis used in ANOVA is . Answer: False 20. The alternate hypothesis for ANOVA states that not all the means are equal. Answer: True 21. For an ANOVA test, rejection of the null hypothesis does not identify which treatment means differ significantly. Answer: True 22. If the computed value of F is 4.01 and the critical value is 2.67, we would conclude that all the population means are equal. Answer: False 23. If the computed value of F is 11.1 and the 0.05 level is used, we would assume that a mistake in arithmetic has been made. Answer: False 24. If we want to determine which treatment means differ, we compute a confidence interval for the difference between each pair of means. Answer: True 25. If a confidence interval for the difference between a pair of treatment means includes 0, then there is no difference in the pair of treatment means. Answer: True 26. If the endpoints of a confidence interval for the difference between a pair of treatment means are both positive numbers, then the treatment means are not different. Answer: False 27. A treatment is a specific source of variation in a set of data. Answer: True 28. A blocking effect is a specific source of variation in a set of data. Answer: True 29. When a blocking effect is included in an ANOVA, the result is a smaller error sum of squares. Answer: True 30. When a blocking effect is included in an ANOVA, two sources of variation are reported: treatment variation and block variation. Answer: False 31. When a blocking effect is included in an ANOVA, the analysis is more likely to detect differences in the treatment means. Answer: True 32. The F-statistic to test for a blocking effect is computed as the ratio of the Treatment Mean Square and the Block Mean Square. Answer: False 33. In a two-way ANOVA, the sum of the treatment, block, and error degrees of freedom equal the total degrees of freedom. Answer: True 34. In a two-way ANOVA, the sum of the treatment and block mean squares equals the error mean square. Answer: False 35. In a two-way ANOVA, the sum of the treatment, block, and error sum of squares equals the total sum of squares. Answer: True 36. In a two-way ANOVA with interaction, there are two factor effects and an interaction effect. Answer: True 37. In a two-way ANOVA with treatment and block effects, an interaction effect is also included. Answer: False 38. In an interaction plot, parallel lines are an indication that there is no interaction effect. Answer: True 39. Interaction between two factors occurs when the effect of one factor on the response variable is the same for any value of another factor. Answer: False Multiple Choice 40. An F statistic is: A) a ratio of two means. B) a ratio of two variances. C) the difference between three means. D) a population parameter. Answer: B 41. What distribution does the F distribution approach as the sample size increases? A) Binomial B) Normal C) Poisson D) Exponential Answer: B 42. Which statement is correct about the F distribution? A) Cannot be negative B) Cannot be positive C) Is the same as the t distribution D) Is the same as the z distribution Answer: A 43. Analysis of variance is used to A) compare nominal data. B) compute t test. C) compare population proportion. D) simultaneously compare several population means. Answer: D 44. A large department store examined a sample of the 18 credit card sales and recorded the amounts charged for each of three types of credit cards: MasterCard, Visa and Discover. Six MasterCard sales, seven Visa and five Discover sales were recorded. The store used ANOVA to test if the mean sales for each credit card were equal. What are the degrees of freedom for the F statistic? A) 18 in the numerator, 3 in the denominator B) 3 in the numerator, 18 in the denominator C) 2 in the numerator, 15 in the denominator D) 6 in the numerator, 15 in the denominator Answer: C 45. Suppose that an automobile manufacturer designed a radically new lightweight engine and wants to recommend the grade of gasoline that will have the best fuel economy. The four grades are: regular, below regular, premium, and super premium. The test car made three trial runs on the test track using each of the four grades and the miles per gallon recorded. At the 0.05 level, what is the critical value of F used to test the hypothesis that the miles per gallon for each fuel is the same. A) 1.96 B) 4.07 C) 2.33 D) 12.00 Answer: B 46. Three different fertilizers were applied to a field of celery. In computing F, how many degrees of freedom are there in the numerator? A) 0 B) 1 C) 2 D) 3 Answer: C 47. Suppose a package delivery company purchased 14 trucks at the same time. Five trucks were purchased from manufacturer A, four from B and five from manufacturer C. The cost of maintaining each truck was recorded. The company used ANOVA to test if the mean maintenance cost of the trucks from each manufacturer were equal. To apply the F test, how many degrees of freedom are in the denominator? A) 2 B) 3 C) 11 D) 14 Answer: C 48. In an effort to determine the most effective way to teach safety principles to a group of employees, four different methods were tried. Some employees were given programmed instruction booklets and worked through the course at their own pace. Other employees attended lectures. A third group watched a television presentation, and a fourth group was divided into small discussion groups. A high of 10 was possible. A sample of five tests was selected from each group. The test grade results were: At the 0.01 level, what is the critical value? A) 1.00 B) 1.96 C) 3.24 D) 5.29 Answer: D 49. In ANOVA, an F statistic is used to test a null hypothesis such as: A) B) C) D) Answer: C 50. An electronics company wants to compare the quality of their cell phones to the cell phones from three competitors. They sample 10 phones from each company and count the number of defects for each phone. If ANOVA were used to compare the average number of defects, the treatments would be defined as: A) the number of cell phones sampled. B) the average number of defects. C) The total number of phones D) The four companies. Answer: D 51. Several employees have submitted different methods of assembling a subassembly. Sample data for each method are: How many treatments are there? A) 3 B) 4 C) 12 D) 0 Answer: B 52. If an ANOVA test is conducted and the null hypothesis is rejected, what does this indicate? A) Too many degrees of freedom B) No difference between the population means C) A difference between at least one pair of population means D) None of the above Answer: C Scrambling: Locked 53. A preliminary study of hourly wages paid to unskilled employees in three metropolitan areas was conducted. Seven employees were included from Area A, 9 from Area B and 12 from Area C. The test statistic was computed to be 4.91. What can we conclude at the 0.05 level? A) Mean hourly wages of unskilled employees all areas are equal B) Mean hourly wages in at least 2 metropolitan areas are different C) More degrees of freedom are needed D) None of these is correct Answer: B Scrambling: Locked 54. In ANOVA analysis, when the null hypothesis is rejected, we can find which means are different by A) constructing confidence intervals. B) adding another treatment. C) doing an additional ANOVA. D) doing a t test. Answer: A 55. In a two-way ANOVA, a blocking variable is used to A) increase the error sum of squares. B) decrease the error sum of squares. C) increase the treatment sum of squares. D) decrease the treatment sum of squares. Answer: B 56. In a two-way ANOVA with interaction, a significant interaction term indicates that A) the response variable is interactive. B) a blocking factor is present. C) both factors are unrelated. D) both factors have a combined effect on the response variable. Answer: D Fill-in-the-Blank 57. The F distribution is a ______________ distribution. Answer: continuous 58. What is the shape of the F distribution? ______________________ Answer: positively skewed 59. What are the minimum and maximum of values of an F distribution? _______ and _______ Answer: zero and positive infinity 60. What kind of values can the F distribution NOT have? ______________ Answer: negative values 61. When comparing two population variances we use the ___________ distribution. Answer: F 62. What test statistic is used in ANOVA? ________________ Answer: F statistic 63. The calculated F value must be equal to or greater than _________ . Answer: zero (0) 64. What test statistic is used to compare two variances? ________________ Answer: F statistic 65. The F-distribution is useful when testing a requirement of two-sample tests of hypothesis. What is the assumption? ________________ Answer: The population variances are equal 66. What is the statistical technique used to test the equality of three or more population means called? ______________________ Answer: analysis of variance (ANOVA) 67. ANOVA requires that the populations should be _______, _______, and ______. Answer: normal or normally distributed; independent, equal standard deviations or variances 68. What statistical technique is used to test the equality of three or more population means? ____________________ Answer: analysis of variance (ANOVA) 69. What is the least number of sources of variation in ANOVA? _________ Answer: two 70. In an ANOVA without a block source of variation, what are the degrees of freedom associated with the error sum of squares? ___________ Answer: n - k 71. ANOVA, how many degrees of freedom are associated with the numerator of the F ratio? _______ Answer: k - 1 or b - 1 or (k - 1 )( b - 1 ) 72. What equals the sum of squares divided by its corresponding degrees of freedom? _________________ Answer: mean square 73. In ANOVA, what is the numerator of the F ratio called? ______________ Answer: treatment mean square, block mean square, or interaction mean square 74. Assuming that the larger of two variances is in the numerator of an F statistic, in which tail of the F distribution is the rejection region for analysis of variance? ________ Answer: upper 75. In ANOVA, when we do not reject the null hypothesis, what inference do we make about the population means? ________________ Answer: they are equal 76. What is the null hypothesis for an ANOVA? ____________________ Answer: 77. When H0 is rejected in ANOVA, _______ _______ are constructed to identify means that differ. Answer: confidence intervals 78. When a second treatment is included in the ANOVA analysis without interaction, that treatment is called a __________________. Answer: blocking variable 79. How many sources of variation are summarized in a two-way ANOVA table? _________ Answer: three 80. In a two-way ANOVA table, what are the error degrees of freedom? _________ Answer: (k - 1)(b - 1) 81. In a two-way ANOVA with interaction, table, what are the error degrees of freedom? _________ Answer: (n – kb) 82. In a two-way ANOVA with interaction, what are the interaction degrees of freedom? _________ Answer: (k - 1)(b - 1) 83. In a one-way ANOVA, what are the two sources of variation? _________ Answer: treatment and error variation Goal: 5 84. In a two-way ANOVA without interaction, what are the three sources of variation? Answer: Treatment, Block and error variation Goal: 7 85. In a two-way ANOVA with interaction, what are the four sources of variation? _________ Answer: Factor A, factor B, interaction of factors A and B, error Multiple Choice Use the following to answer questions 86-96: A manufacturer of automobile transmissions uses three different processes. The management ordered a study of the production costs to see if there is a difference among the three processes. A summary of the findings is shown below. 86. What is the sum of squares for the treatment? A) 67.80 B) 58.07 C) 149.34 D) 23.47 Answer: B 87. What is the sum of squares of the error? A) 67.80 B) 58.07 C) 149.34 D) 23.47 Answer: A 88. What is the critical value of F at the 5% level of significance? A) 19.45 B) 3.00 C) 3.35 D) 3.39 Answer: C 89. What is the critical value of F at the 1% level of significance? A) 99.46 B) 5.49 C) 5.39 D) 4.61 Answer: B 90. What are the degrees of freedom for the treatment sum of squares? A) 2 B) 3 C) 10 D) 27 Answer: A 91. What are the degrees of freedom for the error sum of squares? A) 3 B) 10 C) 27 D) 30 Answer: C 92. What are the total degrees of freedom? A) 27 B) 28 C) 29 D) 30 Answer: C 93. What is the mean square for treatments? A) 2.511 B) 2.151 C) 33.9 D) 29.035 Answer: D 94. What is the mean square for error? A) 2.511 B) 2.151 C) 33.9 D) 29.035 Answer: A 95. What is the calculated F? A) 0.086 B) 1.168 C) 11.56 D) 13.50 Answer: C 96. What is the decision? A) Reject H0 -- there is a difference in treatment means B) Fail to reject H0 -- there is a difference in treatment means C) Reject H0 -- there is a difference in errors D) Fail to reject H0 -- there is a difference in errors Answer: A Fill-in-the-Blank Use the following to answer questions 97-106: In a study of low tar cigarettes, five cigarettes from each of three brands were tested to see if the mean amount of tar per cigarette differs among the brands. 97. What are the degrees of freedom for the numerator? ______ Answer: 2 98. What are the degrees of freedom for the denominator? ______ Answer: 12 99. If the sum of squares for the brands is 0.07, what is the mean square for brands? ______ Answer: 0.035 100. If the sum of squares for the error is 0.09, what is the mean square for the error? ______ Answer: 0.0075 101. What is the F critical value for á = 0.05? ______ Answer: 3.89 102. What is the calculated value of F if the brand sum of squares is 0.07 and the error sum of squares is 0.09? ______ Answer: 4.66 103. If F calculated is 4.75 what is the decision if ? = 0.05? ___________ Answer: reject H0 104. If the calculated F is 4.74, what would the decision be if ? = 0.01? _________________ Answer: Do not reject H0 105. If the sum of squares for the brands is 0.05 and the sum of squares for the error is 0.09, what is the decision rule if ? = 0.05? ________________________ Answer: Do not reject H0 since calculated F = 3.33 and F(0.05) = 3.89 106. If the sum of squares for the brands is 0.07 and the sum of squares for the error is 0.11, what is the decision rule at ? = 0.05? __________________________ Answer: Do not reject H0 since calculated F = 3.88 and F(0.05) = 3.89 Multiple Choice Use the following to answer questions 107-111: Given the following Analysis of Variance table for three treatments each with six observations. 107. What are the degrees of freedom for the numerator and denominator? A) 3 and 18 B) 2 and 17 C) 3 and 15 D) 2 and 15 Answer: D 108. What is the critical value of F at the 5% level of significance? A) 3.29 B) 3.68 C) 3.59 D) 3.20 Answer: C 109. What is the mean square for treatments? A) 71.2 B) 71.4 C) 558 D) 534 Answer: C 110. What is the computed value of F? A) 7.48 B) 7.84 C) 8.84 D) 8.48 Answer: B 111. What is the decision? A) Reject H0 -- there is a difference in treatment means B) Fail to reject H0 -- there is a difference in treatment means C) Reject H0 -- there is a difference in errors D) Fail to reject H0 -- there is a difference in errors Answer: A Fill-in-the-Blank Use the following to answer questions 112-123: A bottle cap manufacture with four machines and six operators wants to see if variation in production is due to the machines and/or the operators. Each operator is assigned to each machine with the following Analysis of Variance table. 112. What are the degrees of freedom for the machines? ______ Answer: 3 113. What are the degrees of freedom for the operators? _____ Answer: 5 114. What are the degrees of freedom for the errors? _____ Answer: 15 115. What is the critical value of F for the machine treatment effect at the 1% level of significance? ____ Answer: 5.42 116. What is the critical value of F for the operator block effect at the 1% level of significance? ____ Answer: 4.56 117. What is the mean square for machines? _____ Answer: 111.39 118. What is the mean square for operators? _____ Answer: 47.44 119. What is the mean square for errors? _____ Answer: 24.77 120. What is the computed value of F for the machines? _____ Answer: 4.50 121. What is the computed value of F for the operators? _____ Answer: 1.92 Essay 122. Using a 1% significance level, what is the decision for the machines? __________________________ Answer: Do not reject H0; there is no difference in production based on machines 123. Using a 1% level of significance, what is the decision for the operators? __________________________ Answer: Do not reject H0; there is no difference in production based on the operators Multiple Choice Use the following to answer questions 124-132: Two accounting professors decided to compare the variation of their grading procedures. To accomplish this they each graded the same 10 exams with the following results: 124. What is H0? A) B) C) D) Answer: A 125. What is H1? A) B) C) D) Answer: B 126. What are the degrees of freedom for the numerator of the F ratio? A) 8 B) 9 C) 10 D) 18 E) 20 Answer: B 127. What are the degrees of freedom for the denominator of the F ratio? A) 20 B) 18 C) 10 D) 9 E) 8 Answer: D 128. What is the critical value of F at the 0.01 level of significance? A) 5.85 B) 5.35 C) 6.51 D) 4.03 Answer: B 129. What is the critical value of F at the 0.05 level of significance? A) 5.85 B) 5.35 C) 3.18 D) 4.03 Answer: C 130. The calculated F ratio is A) 3.484 B) 1.867 C) 3.18 D) 5.35 Answer: A 131. At the 1% level of significance, what is the decision? A) Reject the null hypothesis and conclude the variance is different. B) Fail to reject the null hypothesis and conclude the variance is different. C) Reject the null hypothesis and conclude the variance is the same. D) Fail to reject the null hypothesis and conclude the variance is the same. Answer: D 132. At the 5% level of significance, what is the decision? A) Reject the null hypothesis and conclude the variance is different. B) Fail to reject the null hypothesis and conclude no significant difference in the variance. C) Reject the null hypothesis and conclude the variance is the same. D) Fail to reject the null hypothesis and conclude the variance is the same. Answer: A Use the following to answer questions 133-136: A random sample of 30 executives from companies with assets over $1 million was selected and asked for their annual income and level of education. The ANOVA comparing the average income among three levels of education rejected the null hypothesis. The Mean Square Error (MSE) was 243.7. The following table summarized the results: 133. When comparing the mean salaries to test for differences between treatment means, the t statistic is based on: A) The treatment degrees of freedom. B) The total degrees of freedom. C) The error degrees of freedom D) The ratio of treatment and error degrees of freedom Answer: C 134. When comparing the mean annual incomes for executives with Undergraduate and Master's Degree or more, the following 95% confidence interval can be constructed: A) 2.0 ? 2.052*6.51 B) 2.0 ? 3.182*6.51 C) 2.0 ? 2.052*42.46 D) None of the above Answer: A 135. Based on the comparison between the mean annual incomes for executives with Undergraduate and Master's Degree or more, A) A confidence interval shows that the mean annual incomes are not significantly different. B) The ANOVA results show that the mean annual incomes are significantly different. C) A confidence interval shows that the mean annual incomes are significantly different. D) The ANOVA results show that the mean annual incomes are not significantly different. Answer: A 136. When comparing the mean annual incomes for executives with a High School education or less and Undergraduate Degree, the 95% confidence interval shows an interval of 11.7 to 42.7 for the difference. This result indicates that A) There is no significant difference between the two incomes. B) The interval contains a difference of zero. C) Executives with and Undergraduate Degree earn significantly more than executives with a High School education or less. D) Executives with and Undergraduate Degree earn significantly less than executives with a High School education or less. Answer: C Fill-in-the-Blank Use the following to answer questions 137-153: A bottle cap manufacture with four machines and three operators wants to see if variation in hourly production is due to the machines and/or the operators or an interaction effect of machine and operator. Each operator is assigned to each machine and the production of caps from 3 randomly selected hours is recorded. The analysis shows the following Analysis of Variance table. 137. What are the degrees of freedom for the machines? ______ Answer: 3 138. What are the degrees of freedom for the operators? _____ Answer: 2 139. What are the degrees of freedom for the interaction? _____ Answer: 6 140. What are the degrees of freedom for the error? _____ Answer: 24 141. What is the critical value of F for the machine effect at the 1% level of significance? ____ Answer: 4.72 142. What is the critical value of F for the operator effect at the 1% level of significance? ____ Answer: 5.61 143. What is the critical value of F for the interaction effect at the 1% level of significance? ____ Answer: 3.67 144. What is the mean square for machines? _____ Answer: 38 145. What is the mean square for operators? _____ Answer: 57.5 146. What is the mean square for interaction? _____ Answer: 16.67 147. What is the mean square for errors? _____ Answer: 2.25 148. What is the computed value of F for the machines? _____ Answer: 16.89 149. What is the computed value of F for the operators? _____ Answer: 25.56 150. What is the computed value of F for the interaction? _____ Answer: 7.41 151. Using a 1% significance level, what is the decision for the machines? __________________________ Answer: Reject H0; there is a difference in production based on machines Essay 152. Using a 1% level of significance, what is the decision for the operators? __________________________ Answer: Reject H0; there is a difference in production based on the operators 153. Using a 1% level of significance, what is the decision for the interaction? __________________________ Answer: Reject H0; there is an interaction effect of machine and operator