MULTIPLE CHOICE QUESTIONS. In the following multiple choice que s tions, plea s e circle the correc t answ e r.

Transcription

1 CHAPTER ANALYSIS O VARIANCE MULTIPLE CHOICE QUESTIONS In the following multiple choice que s tions, plea s e circle the correc t answ e r.. In one way ANOVA, the amo u n t of total variation that is unexplain e d is me a s u r e d by the: a. sum of squ ar e s for treat m e n t s b. sum of squ ar e s for error c. total sum of squ ar e s d. degr e e s of freedo m b. The test statistic of the single factor ANOVA equ als a. Sum of squ ar e s for treat m e n t s / Sum of squa r e s for error b. Sum of squ ar e s for error / Sum of squ ar e s for tre a t m e n t s c. Mean squ ar e for treat m e n t s / Mean squa r e for error d. Mean squ ar e for error / Mean squa r e for tre a t m e n t s c. Which of the following stat e m e n t s is false? a. The sum of squ ar e s for tre a t m e n t s (T) explains som e of the variation. b. The sum of squ ar e s for error (E) me a s u r e s the am ou n t of variation that is unexplain e d. c. () = T + E d. () me a s u r e s the amo u n t of variation within the sa m pl e s. d

2 . Chapter ifteen In one way ANOVA, suppo s e that ther e are four tre a t m e n t s with n =, n = 6, n =, and n =. Then the rejection region for this test at the % level of significanc e is a. > 0.0,,0 b. > 0.0,,0 c. > 0.0,,6 d. > 0.0,,6 d. In an ANOVA test, the test statistic is = 6.7. The rejection region is >.97 for the % level of significanc e, >.9 for the.% level, and > 7.6 for the % level. or this test, the p value is a. great e r than 0.0 b. betw e e n 0.0 and 0.0 c. betw e e n 0.0 and 0.0 d. approxi m a t el y 0.0 c 6. In a two tail pooled varianc e t test (equ al varianc e s t test), the null and altern a tive hypot h e s e s are exactly the sa m e as in one way ANOVA with a. exactly one treat m e n t b. exactly two treat m e n t s c. exactly thre e treat m e n t s d. any num b e r of treat m e n t s b 7. Which of the following is not a require d condition for one way ANOVA? a. The sa m pl e sizes mus t be equ al b. The population s mus t all be nor m ally distribut e d c. The population varianc e s mus t be equ al. d. The sa m pl e s for each tre a t m e n t mus t be select e d rando mly indep e n d e n tl y a 8. The following equ a tion applies to which ANOVA mod el? () = (A) + (B) + (AB) + E a. One way ANOVA b. Two way ANOVA c. Complet ely rando miz e d design d. Rando miz e d block design b 9. The following equ a tion applies to which ANOVA mod el? and

3 Analysis of Varianc e () = T + B + E a. One way ANOVA b. Two way ANOVA c. Complet ely rando miz e d design d. Rando miz e d block design d 0. The prim ar y inter e s t of designing a rando miz e d block expe ri m e n t is to: a. reduc e the variation amo n g blocks b. incre a s e the betw e e n tre a t m e n t s variation to mor e ea sily det e c t differenc e s am on g the tre a t m e n t me a n s. c. reduc e the within treat m e n t s variation to mor e easily det e c t differe nc e s am on g the treat m e n t me a n s. d. Increa s e the total sum of squ ar e s c. Two indep e n d e n t sa m pl e s of 0 eac h from the male and fem al e stud e n t s of a large university have bee n select e d at rando m. To test whet h e r ther e is any differenc e in the grad e point aver a g e betw e e n male and fem al e stud e n t s, equ al varianc e s t test will be conside r e d. Another test to consid er is ANOVA. The most likely ANOVA to fit this test situa tion is the: a. one way ANOVA b. two way ANOVA c. rando miz e d block design d. chi squar e test a. The test of the rando miz e d block design of the analysis of varianc e require s that the rando m variabl e of inter e s t must be nor m ally distribut e d and the population varianc e s mus t be equ al. When the rando m variable is not nor m ally distribut e d, we can use a. one way ANOVA b. two way ANOVA c. chi squar e test d. ried m a n test d. A com pl et e x a. factor A has b. factor B has c. the num b e r d. the num b e r levels equ al c. The analysis of varianc e is a proc e d ur e that allows statisticia ns to com p a r e two or mor e popul ation factorial expe ri m e n t is called bala nc e d if thre e levels two levels of replicat e s is the sa m e for each tre a t m e n t of obs erv a tion s for each com bin a tion of factor A and factor B at least

4 Chapter ifteen a. me a n s b. proportions c. varianc e s d. stand a r d deviations a. The distribution of the test statistic for analysis of varianc e is the: a. nor m al distribution b. Stude n t t distribution c. distribution d. chi squar e d distribution c 6. The simples t experi m e n t al design has: a. a single respo n s e variabl e b. two respo n s e variabl es c. thre e respon s e variabl es d. no respon s e variabl es at all a 7. Which of the following is not true of the distribution? a. Mean and medi an are equ al b. It is skew e d to the right c. Its value s are always positive d. It is used in ANOVA test a 8. In a single factor analysis of varianc e, T is the me a n squ ar e for trea t m e n t s and E is the me a n squa r e for error. The null hypot h e si s of equ al population me a n s is reject e d if: a. T is much sm aller tha n E b. T is much larger than E c. T is equ al to E d. none of the above is correct b 9. If we want to conduct a test to det er mi n e whet h e r a popul ation me a n is great e r than anot h e r popula tion me a n, we a. can use the analysis of varianc e b. mus t use the indep e n d e n t sa m pl e s t test for differe nc e betw e e n two me a n s c. mus t use the chi squar e test d. both a and b are correct b 0. In ANOVA, error variability is com p u t e d as the sum of the squ ar e d errors, E, for all values of the respon s e variable. This variability is the: a. the total variation

5 Analysis of Varianc e 6 b. within group variation c. betw e e n groups variation d. none of the above b. In a one way ANOVA wher e ther e are k trea t m e n t s and n obs e rv a tion s, the degr e e s of freedo m for the statistic are equ al to: a. n and k b. k and n c. n k and k d. k and n k d. One way ANOVA is applied to thre e inde p e n d e n t sa m pl e s having me a n s 0,, and 8, resp e c tiv ely. If eac h obs erv a tion in the third sa m pl e were incre a s e d by 0, the value of the statistics would: a. incre a s e b. decr e a s e c. rem ain unch a n g e d d. incre a s e by 0 a. The statistic in a one way ANOVA repr e s e n t s the variation: a. betw e e n the treat m e n t s plus the variation within the tre a t m e n t s b. within the treat m e n t s minus the variation betw e e n the tre a t m e n t s c. betw e e n the treat m e n t s divide d by the variation within the trea t m e n t s d. variation within the tre a t m e n t s divide d by the variation betw e e n the treat m e n t s c. In the two way ANOVA wher e a is the num b e r of factor A levels, b is the num b e r of factor B levels, and r in the num b e r of replicat e s, the degr e e s of free do m for inter ac tion is given by: a. (a )( b ) b. abr c. (a )( r ) d. ab (r ) a. In the rando miz e d block design for ANOVA wher e k is the num b e r of trea t m e n t s, and b in the num b e r of blocks, the degr e e s of freedo m for error is given by: a. k b. b c. (k )( b ) d. kb c In the one way ANOVA wher e k is the num b e r of trea t m e n t s and n is the num b e r of obs erv a tion s in all sa m pl e s, the degr e e s of freedo m for error is given by: 6.

6 7 Chapter ifteen a. k b. n k c. n d. n k + b 7. In the one way ANOVA wher e k is the num b e r of trea t m e n t s and n is the num b e r of obs erv a tion s in all sa m pl e s, the degr e e s of freedo m for trea t m e n t s is given by: a. k b. n k c. n d. n k + a 8. Three tennis player s, a beginn e r, an inter m e di a t e, and adva n c e d, have be e n rando mly select e d from the me m b e r s hi p of a racqu e t facility club in a large city. Using the sam e tennis ball, eac h player hits ten serve s, one with eac h of thre e racqu e t mod els, with the thre e racqu e t mod el s select e d rando mly. The spe e d of each serve is me a s u r e d with a mac hin e and the result record e d. Among the ANOVA mod els liste d below, the most likely mod el to fit this situation is the: a. one way ANOVA b. two way ANOVA c. rando miz e d block design d. matc h e d pairs mod el c 9. A survey will be conduct e d to com p a r e the gra d e point aver a g e s of high school stud e n t s from four differe n t school districts. Stud e n t s are to be rando mly select e d from each of the four districts and their grad e point aver a g e s record e d. The ANOVA mod el most likely to fit this situa tion is: a. one way ANOVA b. two way ANOVA c. rando miz e d block design d. com plet e x factorial design a 0. In the two way ANOVA wher e a is the num b e r of factor A levels, b is the num b e r of factor B levels, and r in the num b e r of replicat e s, the degr e e s of free do m for error is given by: a. (a )( b ) b. abr c. (a )( r ) d. ab (r ) d. In ANOVA, the test is the ratio of two sa m pl e varianc e s. In the one way ANOVA (com pl e t ely rando miz e d design), the varianc e use d as a num e r a t o r of the ratio is:

7 Analysis of Varianc e 8 a. me a n squ ar e for treat m e n t s b. me a n squ ar e for error c. me a n squ ar e for blocks d. total sum of squ ar e s a. In the rando miz e d block design ANOVA, the sum of squa r e s for error equ al s: a. () T b. () B c. () T B d. () (A) (B) (AB) c. The value of the test statistic in a com pl e t ely rando miz e d design for ANOVA is = 6.9. The num b e r of degr e e s of free do m for the num e r a t o r and deno mi n a t or are and 0, resp e c tively. The most accur a t e stat e m e n t s to be mad e about the p value is that it is: a. great e r than 0.0 b. betw e e n 0.0 and 0.0 c. betw e e n 0.0 and 0,0 d. sm aller than 0.0 d. When the effect of a level for one factor dep e n d s on which level of anot h e r factor is pres e n t, the mos t appropria t e ANOVA design to use in this situa tion is the: a. one way ANOVA b. two way ANOVA c. rando miz e d block design d. matc h e d pairs design b. The rando miz e d block design with exactly two tre a t m e n t s is equivale nt to a two tail: a. indep e n d e n t sam pl e s z test b. indep e n d e n t sam pl e s equ al varianc e s t test c. indep e n d e n t sam pl e s une q u al varianc e s t test d. matc h e d pairs t test d 6. A rando miz e d block design with tre a t m e n t s and blocks produc e d the following sum of squ ar e s value s : () = 9, T = 9, E = 88. The value of B mus t be: a. b. 7 c. 76 d. 60 a

8 9 7. Chapter ifteen One way ANOVA is perfor m e d on indep e n d e n t sa m pl e s take n from thre e nor m ally distribut e d popul ations with equ al varianc e s. The following sum m a r y statistics wer e calculat e d: n = 7 x = 6 s =. n = 8 x = 6 s =.9 n = 9 x = 6 s =.6 The value of the test statistics,, equ al s: a. 6 b. c..7 d. 0 d 8. In a com pl et ely rando miz e d design for ANOVA, the num b e r of degr e e s of free do m for the num er a t or and deno mi n a t o r are and, resp e c tiv ely. The total num b e r of obs erv a tion s mus t equ al: a. 9 b. c. 0 d. c 9. In a two way ANOVA, ther e are levels for factor A, levels for factor B, and obs erv a tion s for each com bin a tion of factor A and factor B levels. The num b e r of treat m e n t s in this expe ri m e n t equ als: a. 60 b. c. 0 d. 6 c 0. The degr e e s of freedo m for the deno mi n a t or of a one way ANOVA test for population me a n s with obs erv a tion s sa m pl e d from each popula tion is: a. 60 b. 9 c. 6 d. c. In one way ANOVA, the ter m x refers to: a. sum of the sa m pl e me a n s b. sum of the sa m pl e me a n s divide d by the total num b e r of obs e rv a tion s c. sum of the population me a n s d. weight e d me a n of the sa m pl e me a n s d

9 Analysis of Varianc e 0. or which of the following dep a r t u r e s from the conditions require d for a com plet ely rando miz e d design is the proc e d ur e not conside r e d robus t? a. The population s are not nor m ally distribut e d. b. The population varianc e s are not equ al c. The sa m pl e s are not indep e n d e n t d. All of the above. c. One way ANOVA is perfor m e d on thre e indep e n d e n t sa m pl e s with n = 6, n = 7, and n = 8. The critical value obtain e d from the table for this test at the.% level of significanc e equ als: a.. b. 9. c..6 d. 9. c. Which of the following is a correct formul a tion for the null hypot h e si s in one way ANOVA? a. µ + µ + µ = 0 b. µ + µ + µ 0 c. µ = µ = µ d. µ µ µ c. One way ANOVA is perfor m e d on indep e n d e n t sa m pl e s take n from thre e nor m ally distribut e d popul ations with equ al varianc e s. The following sum m a r y statistics wer e calculat e d: n = 6 x = 0 s =. n = 8 x = s =.9 n = 6 x = s =. The grand me a n equ al s a. 0.0 b..0 c.. d..0 c 6. One way ANOVA is applie d to indep e n d e n t sa m pl e s take n from thre e nor m ally distribut e d popul ations with equ al varianc e s. The following sum m a r y statistics wer e calculat e d:

10 Chapter ifteen n = 8 n = 0 n = 8 x = s = x = 8 s = x = 0 s = The within treat m e n t s variation equ als a. 7 b. 60 c. d. 60 a 7. Which of the following is not true of Tukey s Multiple Comp a ris on Method? a. It is bas e d on the stud e n ti z e d range statistic q to obtain the critical value need e d to constr uc t individu al confide nc e interv als b. It require s that all sa m pl e sizes are equ al, or at least similar. c. It can be em ploye d inst e a d of the analysis of varianc e. d. All of the above stat e m e n t s are true. d 8. If four confidenc e interv al esti m a t e s for the popula tion me a n s wer e simult a n e o u s ly cons tr uct e d with 9% confide nc e for four inde p e n d e n t trea t m e n t s, the prob a bility that all four interv als would cont ain the population me a n s would be: a b. 0.8 c d b 9. A profes s or of statistics in Wayne Stat e University want s to det er mi n e whet h e r the aver a g e starting salarie s am on g gradu a t e s of the univer sitie s in Michigan are equ al. A sa m pl e of rec e n t gra du a t e s from each university was rando mly take n. The appro pri a t e critical value for the ANOVA test is obtaine d from the distribution with degr e e s of free do m equ al: a. and b. and 60 c. 60 and d. and b 0. One way ANOVA is applie d to indep e n d e n t sa m pl e s take n from thre e nor m ally distribut e d popul ations with equ al varianc e s. The following sum m a r y statistics wer e calculat e d: n = 0 x = 0 s =

11 Analysis of Varianc e n = 0 n = 0 x = 8 x = 0 s = 6 s = The betw e e n treat m e n t s variation equ als a. 60 b. 688 c. 60 d. 8 c. One way ANOVA is applied to inde p e n d e n t sa m pl e s take n from four nor m ally distribut e d population s with equ al varianc e s. If the null hypot h e si s is reject e d, then we can infer that a. all popul ation me a n s are equ al b. all popul ation me a n s differ c. at least two popul ation me a n s are equ al d. at least two popul ation me a n s differ d. Consider the following partial ANOVA table: Source of Variation The num er a t or and deno mi n a t o r degr e e s of freedo m (identified by ast erisks) are a. and b. and 6 c. and d. 6 and b. In single factor analysis of varianc e, betw e e n trea t m e n t s variation sta nd s for: a. sum of squ ar e s for error b. sum of squ ar e s for treat m e n t s c. total sum of squ ar e s d. both a and b b. Consider the following ANOVA table: Source of Variation

12 Chapter ifteen The num b e r of treat m e n t s is a. b. c. d. c. In one way analysis of varianc e, within trea t m e n t s variation sta nd s for: a. sum of squ ar e s for error b. sum of squ ar e s for treat m e n t s c. total sum of squ ar e s d. none of the above is correct a 6. Consider the following ANOVA table: Source of Variation The num b e r of obs erv a tion s in all sa m pl e s is: a. b. 9 c. 0 d. c 7. In one way analysis of varianc e, if all the sa m pl e me a n s are equ al, then a. total sum of squ ar e s is zero b. sum of squ ar e s for error is zero c. sum of squ ar e s for treat m e n t s is zero d. sum of squ ar e s for error equ al s sum of squ ar e s for tre a t m e n t s c 8. In single factor analysis of varianc e, if large differe nc e s exist amo n g the sa m pl e me a n s, it is then rea s o n a bl e to a. reject the null hypot h e si s b. reject the altern a tiv e hypot h e si s c. fail to reject the null hypot h e si s d. none of the above is correct a

13 Analysis of Varianc e 9. Which of the following is not a require d condition for one way ANOVA? a. The population s are nor m ally distribut e d b. The population varianc e s are equ al c. The sa m pl e s are select e d indep e n d e n t ly of each other d. The population me a n s are equ al d 60. In one way ANOVA, suppos e that ther e are five tre a t m e n t s with n = n = n =, and n = n = 7. Then the me a n squa r e for error, E, equ als a. E / b. E / 9 c. E / d. E / c

14 Chapter ifteen TRUE/ALSE QUESTIONS 6. Statistics practition er s use the analysis of varianc e (ANOVA) techniqu e to com p a r e two or mor e popula tions of interv al dat a. T 6. Given the significanc e level 0.0, the value for the degr e e s of freedo m, d.f. = (,) is.. T 6. Three tennis player s, a beginn e r, an experi e n c e d, and a profes sion al, have been rando mly select e d from the me m b e r s hi p of a large city tennis club. Using the sam e ball, each perso n hits four serve s with each of five racqu e t mod els, with the five racqu e t mod els select e d rando mly. Each serve is clocked with a radar gun and the result record e d. Among ANOVA mod els, this setu p is mos t like the simple regr e s sion mod el. 6. Given the significanc e level 0.0, the value for the degr e e s of freedo m, d.f. = (9,6) is T 6. Three tennis player s, a beginn e r, an experi e n c e d, and a profes sion al, have been rando mly select e d from the me m b e r s hi p of a large city tennis club. Using the sam e ball, each perso n hits four serve s with each of five racqu e t mod els, with the five racqu e t mod els select e d rando mly. Each serve is clocked with a radar gun and the result record e d. Among ANOVA mod els, this setu p is mos t like the rando miz e d block design. T 66. A balanc e d experi m e n t requir e s that the sa m pl e size for each trea t m e n t be equ al. T 67. The sum of squ ar e s for tre a t m e n t s, T, achiev e s its sm alles t value (zero) when all the sa m pl e me a n s are equ al. T 68. The analysis of varianc e (ANOVA) techniqu e analyz e s the varianc e of the dat a to det er mi n e whet h e r differe nc e s exist betw e e n the popula tion me a n s. T 69. The equ a tion: () = (A) + (B) + (AB) + E, applie s to one way ANOVA mod el. 70. In ANOVA, the betw e e n tre a t m e n t variation is denot e d by T, which sta nd s for sum of squ ar e s for treat m e n t s. T

15 Analysis of Varianc e 6 7. A study will be und er t a k e n to exa mi n e the effect of two kinds of backgro u n d music and of two ass e m bly met ho d s on the output of worker s at a fitne s s shoe factory. Two worker s will be rando mly assign e d to eac h of four groups, for a total of eight in the study. Each worker will be given a he a d p h o n e set so that the music type can be controlled. The num b e r of shoe s com pl e t e d by each worker will be record e d. Does the kind of music or the ass e m bly met h o d or a com bin a tion of music and me t h o d affect output? The ANOVA mod el most likely to fit this situa tion is the two way analysis of varianc e. T 7. The sum of squ ar e s for error is also known as the betw e e n trea t m e n t s variation. 7. Two sa m pl e s of ten each from the male and fem al e worker s of a large com p a n y have been take n. The dat a involved the wag e rate of eac h worker. To test whet h e r ther e is any differe nc e in the aver a g e wag e rat e betw e e n male and fem al e worker s a pooled varianc e s t test will be consid er e d. Another test option to conside r is ANOVA. The most likely ANOVA to fit this test situation is the rando miz e d block design. 7. We do not need the t test of µ µ, since the analysis of varianc e can be used to test the differenc e betw e e n the two popula tion me a n s. 7. Concep t u ally and mat h e m a t i c ally, the test of the indep e n d e n t single factor ANOVA is an exte n sion of the t test of µ µ. T 76. When the proble m objective is to com p a r e mor e than two popul ations, the experi m e n t al design that is the count e r p a r t of the mat c h e d pairs expe ri m e n t is called the rando miz e d block design. T 77. In em ploying the rando miz e d block design, the prim a ry inter e s t reducing sum of squ ar e s for blocks (B). 78. A rando miz e d block design ANOVA has two tre a t m e n t s. The test to be perfor m e d in this proced ur e is equivale nt to dep e n d e n t sa m pl e s t test. T 79. In a two way ANOVA, ther e are levels for factor A, and levels for factor B, and obs erv a tion s within each cell. The num b e r of tre a t m e n t s in this experi m e n t will be. 80. In ANOVA, a factor is an inde p e n d e n t variabl e. sa m pl e s lies in

16 7 Chapter ifteen T 8. When the dat a are obt ain e d throug h a controlled experi m e n t in the single factor ANOVA, we call the experi m e n t a l design the com pl e t ely rando miz e d design of the analysis of varianc e. T 8. A rando miz e d block design ANOVA has five trea t m e n t s and four blocks. The com p u t e d test statistic (value of ) is 6.. With a 0.0 significanc e level, the conclusion will be to acc e p t the null hypot h e si s. 8. In one way ANOVA, the total variation () is partition e d into two source s of variation: the sum of squ ar e s for tre a t m e n t s (T) and the sum of squ ar e s for error (E). T 8. A study will be und er t a k e n to exa mi n e the effect of two kinds of backgro u n d music and of two ass e m bly met ho d s on the output of worker s at a fitne s s shoe factory. Two worker s will be rando mly assign e d to eac h of four groups, for a total of eight in the study. Each worker will be given a he a d p h o n e set so that the music type can be controlled. The num b e r of shoe s com pl e t e d by each worker will be record e d. Does the kind of music or the ass e m bly met h o d or a com bin a tion of music and me t h o d affect output? The ANOVA mod el most likely to fit this situa tion is the simple regr e s sion mod el. 8. If we exa mi n e two or mor e inde p e n d e n t sa m pl e s to det er mi n e if their population me a n s could be equ al, we are perfor min g one way analysis of varianc e (ANOVA). T 86. The rando miz e d block design is also called the two way analysis of varianc e. T 87. The purpos e of designing a rando miz e d block expe ri m e n t is to reduc e the betw e e n treat m e n t s variation (T) to mor e ea sily det e c t differe nc e s betw e e n the treat m e n t me a n s. 88. The test of the analysis of varianc e requir e s nor m ally distribut e d with equ al varianc e s. T 89. The Bartlett s test is a statistic al proc e d u r e design e d to test for the equ ality of varianc e s. T 90. The sum of squ ar e s for tre a t m e n t s, T, achiev e s its sm alles t value (zero) when all the sa m pl e sizes are equ al. that the popul ations be

17 Analysis of Varianc e 8 9. The test of the rando miz e d block design of the analysis of varianc e has the sa m e require m e n t s as the indep e n d e n t sa m pl e s design; that is, the rando m variabl e mus t be nor m ally distribut e d and the popula tion varianc e s mus t be equ al. T 9. A rando miz e d block expe ri m e n t having five tre a t m e n t s and six blocks produc e d the following value s: T =, () =,, E = 98. The value of B mus t be 09. T 9. In a two way ANOVA, ther e are levels for factor A and levels for factor B, and two obs erv a tion s within each cell. The num b e r of trea t m e n t s in this experi m e n t will be In one way ANOVA, the test statistic is define d as the ratio of the me a n squ ar e for error (E) and the me a n squ ar e for tre a t m e n t s (T); na m ely, = E / T. 9. When the respo n s e is not norm ally distribut e d, we can replac e the rando miz e d block ANOVA with its nonp a r a m e t ri c count e r p a r t ; the ried m a n test. T 96. The Bonferroni adjust m e n t to isher s Least Significant Differenc e (LSD) multiple com p a ris on met ho d is ma d e by dividing the specified experi m e n t wi s e Type I error rat e by the num b e r of com bin a tion s of pairs of population me a n s. T 97. If we first arran g e test units into similar groups before assigning tre a t m e n t s to the m, the test design we should use is the rando miz e d block design. T 98. If we simult a n e o u sly exa mi n e the effects of two factors on the dep e n d e n t variabl e, along with the effect s of inter a c tion s betw e e n the differe nt levels of thos e factors, we are perfor mi ng Thre e way analysis of varianc e (ANOVA). 99. If the dat a are not nor m ally distribut e d, we can replac e the inde p e n d e n t sa m pl e s single factor mod el of the analysis of varianc e with its nonp a r a m e t ri c count er p a r t, which is the Kruskal Wallis test. T

18 9 Chapter ifteen 00. The sum of squ ar e s for tre a t m e n t s (T) is the variation attribut e d to the differenc e s betw e e n the tre a t m e n t me a n s, while the sum of squ ar e s for error (E) me a s u r e s the variation within the sa m pl e s. T 0. The rando miz e d block design with two tre a t m e n t s is equivale nt direction al dep e n d e n t sa m pl e s z test. The calculat e d value of in a one way analysis is The degr e e s of freedo m and deno mi n a t or degr e e s of free do m are resp e c tiv ely. The mos t accur a t e stat e m e n t to be ma d e about the that p value < 0.0. T 0. to a non num e r a t o r and 9, p value is 0. The num er a t or or T degr e e s of freedo m are and the deno mi n a t o r or E degr e e s of freedo m are 8. The total num b e r of obs e rv a tion s in the com plet ely rando miz e d design mus t equ al A survey will be conduct e d to com p a r e the United Way contribution s ma d e by em ploye e s from thre e Michiga n universitie s. Employe e s are to be rando mly select e d from each of the thre e univer sitie s and the dollar am ou n t s of their contribution recor d e d. The ANOVA mod el most likely to fit this situation is the one way analysis of varianc e. T 0. Given the significanc e level 0.0, the value for the degr e e s of freedo m, d.f. = (,8) is Three tennis player s, a beginn e r, an experi e n c e d, and a profes sion al, have been rando mly select e d from the me m b e r s hi p of a large city tennis club. Using the sam e ball, each perso n hits four serve s with each of five racqu e t mod els, with the five racqu e t mod els select e d rando mly. Each serve is clocked with a radar gun and the result record e d. Among ANOVA mod els, this setu p is mos t like the paire d sa m pl e mod el. 07. One way ANOVA is applied to thre e inde p e n d e n t sa m pl e s having me a n s,, and 0, resp e c tiv ely. If eac h obs erv a tion in the third sa m pl e were incre a s e d by 0, the value of the statistics would incre a s e by The statistic in a one way ANOVA repr e s e n t s the variation betw e e n the trea t m e n t s divided by the variation within the trea t m e n t s. T 09. The sum of squar e s for error (E) explains som e of the total variation, while the sum of squar e s for trea t m e n t s (T) me a s u r e s the amo u n t of variation that is unexplain e d.

19 Analysis of Varianc e 0 0. The distribution of the test statistic for analysis of varianc e is the distribution. T. In one way ANOVA, suppos e that ther e are five tre a t m e n t s with n = n = n = 6, and n = n = 8. Then the me a n squ ar e for error, E, equ al s E/.. A rando miz e d block design with tre a t m e n t s and blocks produc e d the following sum of squar e s value s : () = 000, T = 00, E = 00. The value of B mus t be 0. T. In a two way ANOVA, ther e are levels for factor A, levels for factor B, and obs erv a tion s for each com bin a tion of factor A and factor B levels. The num b e r of treat m e n t s in this expe ri m e n t equ als 0. T. If the dat a are not nor m ally distribut e d, we can replac e the inde p e n d e n t sa m pl e s single factor mod el of the analysis of varianc e with its nonp a r a m e t ri c count er p a r t, which is the ried m a n test.. The degr e e s of freedo m for the deno mi n a t or of a one way ANOVA test for population me a n s with obs erv a tion s sa m pl e d from each popula tion are A study will be und er t a k e n to exa mi n e the effect of two kinds of backgro u n d music and of two ass e m bly met ho d s on the output of worker s at a fitne s s shoe factory. Two worker s will be rando mly assign e d to eac h of four groups, for a total of eight in the study. Each worker will be given a he a d p h o n e set so that the music type can be controlled. The num b e r of shoe s com pl e t e d by each worker will be record e d. Does the kind of music or the ass e m bly met h o d or a com bin a tion of music and me t h o d affect output? The ANOVA mod el most likely to fit this situa tion is the one way analysis of varianc e. 7. Tukey s multiple com p ari so n me t h o d det e r mi n e s a critical num b e r, ω ; such that if any pair of sam pl e me a n s has a differe nc e gre a t e r than ω, we conclud e that the pair s two corre s p o n di n g popula tion me a n s are differe nt. T

20 Chapter ifteen TEST QUESTIONS 8. Given the following dat a drawn from thre e nor m al popula tions : Treat m e n t Set up the ANOVA table and test at the % level of significa nc e to det e r mi n e whet h e r differenc e s exist amo n g the popula tion me a n s. Source of Variation P value critical H 0 : µ = µ = µ H : At least two me a n s differ Conclusion: Don t reject the null hypot h e si s. No 9. Provide an exa m pl e for a rando miz e d block design with thre e trea t m e n t s (k = ) and four blocks (b = ), in which T is equ al to zero and B and E are not equ al to zero. Treatment Block

21 Analysis of Varianc e 0. 6 A rando miz e d block design experi m e n t produc e d the following dat a. Treat m e n t Block a. Test to det er mi n e whet h e r the trea t m e n t me a n s differ. (Use α = 0.0.) b. Test to det er mi n e whet h e r the block me a n s differ. (Use α = 0.0.) ANSWERS: Source of Variation Blocks a. H 0 : µ = µ = µ H : At least two me a n s differ Conclusion: Reject the null hypot h e si s. Yes b. H 0 : µ = µ = µ = µ = µ H : At least two me a n s differ Conclusion: Reject the null hypot h e si s. Yes P value critical.9.88

22 . Chapter ifteen The following statistics wer e calculat e d bas e d on sa m pl e s drawn from four nor m al population s: Treat m e n t Statistic j j nj x x Test at the % level of significanc e to det er mi n e whet h e r differe nc e s exist am on g the popul ation me a n s. Source of Variation critical H 0 : µ = µ = µ = µ H : At least two me a n s differ Conclusion: Don t reject the null hypot h e si s. No. The following statistics were calculat e d bas e d on sa m pl e s drawn from thre e nor m al population s: Treat m e n t Statistic n x s Set up the ANOVA table and test at the % level of significa nc e to det e r mi n e whet h e r differenc e s exist amo n g the popula tion me a n s. Source of Variation critical

23 Analysis of Varianc e H 0 : µ = µ = µ H : At least two me a n s differ Conclusion: Don t reject the null hypot h e si s. No. Provide an exa m pl e for a rando miz e d block design with thre e tre a t m e n t s ( k = ) and four blocks (b = ), in which B = 0 and T and E are not equ al to zero. Treat m e n t Block Is it possible to have a rando miz e d block design of the analysis of varianc e in which E = 0 and B is not equ al to zero? Explain No, since if ther e is no variation within the tre a t m e n t s, the block me a n s mus t be equ al.. ill in the blanks (identified by ast erisks) in the following partial ANOVA table: Source of Variation Source of Variation

24 6. Chapter ifteen A statistician em ploy e d by a television rating servic e want e d to det er mi n e if ther e wer e differenc e s in television viewing habits amo n g thre e differe nt cities in California. She took a rando m sa m pl e of five adults in each of the cities and asked each to report the num b e r of hours spe n t watc hing television in the previous week. rom the dat a shown below, can she infer at the % significanc e level that dif fere nc e s in hours of television watc hing exist am on g the thre e cities? Hours Spe n t Watc hing Television San Diego 8 7 Los Angeles San rancisco 8 7 Source of Variation P value critical H 0 : µ = µ = µ 7. H : At least two me a n s differ Conclusion: Reject the null hypot h e si s. Yes A phar m a c e u t ic al man uf a c t ur e r has be e n res e a r c hi ng new formul a s to provide quicker relief of minor pains. His labor a t orie s have produc e d thre e different formul as, which he want e d to test. iftee n peopl e who com pl ain e d of minor pains wer e recruit e d for an expe ri m e n t. ive were given formul a, five were given form ul a, and the last five wer e given formul a. Each was aske d to take the medicine and report the lengt h of time until som e relief was felt. The results are shown below. Do the s e dat a provide suffi cient evidenc e to indicat e that differe nc e s in the tim e of relief exist amo n g the thre e formul as? Use α = 0.0. Time in Minute s Until Relief is elt ormul a ormul a 7 orm ula

25 Analysis of Varianc e Source of Variation P value critical H 0 : µ = µ = µ H : At least two me a n s differ Conclusion: Reject the null hypot h e si s. Yes 8. Autom o bile insur a nc e appr ais e r s exa mi n e cars that have be e n involved in accide n t al collisions and estim a t e the cost of repairs. An insur a n c e exec utive claim s that ther e are significa nt differe nc e s in the esti m a t e s from differe n t appr ais er s. To support his claim he take s a rando m sa m pl e of six cars that have rece ntly bee n dam a g e d in accide n t s. Three appr ais e r s the n estim a t e the repair cost s of all six cars. rom the dat a shown below, can we infer at the % significanc e level that the exec utive s claim is true? Esti mat e d Repair Cost Car 6 Apprais e r Apprais e r Apprais er Source of Variation P value critical, ,8,.,. 6,9, Blocks 6, , H 0 : µ = µ = µ H : At least two me a n s differ Conclusion: Reject the null hypot h e si s. Yes 0.0.6

26 7 9. Chapter ifteen The stren g t h of a weld dep e n d s to som e exte n t on the met al alloy use d in the welding proce s s. A scientist working in the res e a r c h labor a t ory of a major auto m o bile man uf ac t u r e r has develop e d thre e new alloys. In order to test their stren g t h s each alloy is use d in seve r al welds. The stre n g t h s of the welds are then me a s u r e d with the result s shown below. Can the scientist conclud e at the % significanc e level that differe nc e s exist am on g the stre n g t h s of the welds with the different alloys? Streng t h of Welds Alloy 6 9 Alloy Alloy 7 Source of Variation P value critical H 0 : µ = µ = µ H : At least two me a n s differ Conclusion: Don t reject the null hypot h e si s. No 0. In recen t year s the irradia tion of food to reduc e bact e ria and pres e r v e the food longer has beco m e mor e com m o n. A com p a n y that perfor m s this service has develop e d four differe n t met ho d s of irradia ting food. To det er mi n e which is bes t, it conduc t s an expe ri m e n t wher e differe n t foods are irradiat e d and the bact eri a count is me a s u r e d. As part of the expe ri m e n t the following foods are irradia t e d : be ef, chicke n, turkey, eggs, and milk. The results are shown below. Can the com p a n y infer at the % significa nc e level that differenc e s in the bact e ria count exist amo n g the four irradia tion met h o d s? Bact eria Count ood Beef Method 7 Method Method 6 Method 68

27 Analysis of Varianc e Chicken Turkey Eggs Milk Source of Variation P value Blocks critical H 0 : µ = µ = µ = µ H : At least two me a n s differ Conclusion: Don t reject the null hypot h e si s. No. In recen t year s a controv e r s y has arise n in major leagu e bas e b a ll. Som e player s have been accus e d of doctoring their bats to incre a s e the dist a nc e the ball travels. Howev er, a physics profes s or claims that the effect of doctoring is negligible. A major leagu e ma n a g e r decide s to test the profes s or s claim. He doctors two bats by inserting cork into one and rubbe r into anot h e r. He then tells five player s on his tea m to hit a ball with an un doctor e d bat and with the doctor e d bat s. The dist a n c e s are me a s u r e d and listed below. Do thes e dat a provide sufficient evide n c e with the % level of significanc e to refut e the profe s s or s claim? Distanc e Ball Travels (in feet ) Player Un doctor e d Bat Bat with Cork Bat with Rubber Source of Variation Blocks 6. 67, , P value critical.9.88

28 9 Chapter ifteen 68,0. H 0 : µ = µ = µ H : At least two me a n s differ Conclusion: Don t reject the null hypot h e si s. No. The mark e ti ng man a g e r of a pizza chain is in the proc e s s of exa mi ning som e of the dem o g r a p hic char a ct e ri s tics of her custo m e r s. In particular, she would like to inves tig a t e the belief that the age s of the custo m e r s of pizza parlors, ha m b u r g e r em poriu m s, and fast food chicke n rest a ur a n t s are differe n t. As an experi m e n t, the age s of eight custo m e r s of eac h of the rest a ur a n t s are record e d and listed below. Do thes e dat a provide enoug h evide nc e at the % significanc e level to infer that ther e are differe nc e s in age s am on g the custo m e r s of the thre e rest a u r a n t s? rom previous analys e s we know that the ages are nor m ally distribut e d. Custo m e r s Ages Pizza Ham b ur g e r Chicke n ANSWERS: Source of Variation P value 0.08 critical.67 H 0 : µ = µ = µ H : At least two me a n s differ Conclusion: Don t reject the null hypot h e si s. No QUESTIONS THROUGH ARE BASED ON THE OLLOWING INORMATION: In order to exa mi n e the differenc e s in age s of teac h e r s amo n g five school districts, an educ a tion al statistician took rando m sa m pl e s of six tea c h e r s age s in eac h district. The dat a are listed below.

29 Analysis of Varianc e 60 Ages of Teach er s amon g ive School District Test at the % significanc e level to det e r mi n e if differe nc e s in teac h e r s age s exist am on g the five districts. Source of Variation P value 0.06 critical.79 H 0 : µ = µ = µ = µ = µ H : At least two me a n s differ Conclusion: Reject the null hypot h e si s. Yes. Use Tukey s multiple com p a ris on me t h o d to det er mi n e which me a n s differ. ω = District District xi x j Significant? No Yes No No Yes No No Yes Yes No It is clear that the me a n for district is significa ntly differe nt from the me a n for each of the other four districts.

30 6. Chapter ifteen Use isher s LSD proce d u r e with me a n s differ. α=.0 to det e r mi n e which popula tion LSD = 7.6 District Distric t xi x j Significant? No Yes No Yes Yes No No Yes Yes Yes It is clear that the me a n for district is significa ntly differe nt from the me a n for each of the other four districts. QUESTIONS 6 THROUGH 8 ARE BASED ON THE OLLOWING INORMATION: A recen t college grad u a t e is in the proc e s s of deciding which one of thre e gradu a t e schools he should apply to. He decide s to judge the quality of the schools on the basis of the Gradu a t e Manag e m e n t Admission Test (GMAT) score s of thos e who are accep t e d into the school. A rando m sa m pl e of six stud e n t s in each school produc e d the following GMAT score s. GMAT Score s School School School Assuming that the dat a are nor m ally distribut e d, can he infer at the 0% significanc e level that the GMAT score s differ am on g the thre e schools?

31 Analysis of Varianc e 6 Source of Variation 7, , ,7. 6, ,9. 7 P value 0.00 critical.69 H 0 : µ = µ = µ H : At least two me a n s differ Conclusion: Reject the null hypot h e si s. Yes 7. Use isher s LSD met ho d me a n s differ. with α= 0.0 to det e r mi n e which popul a tion LSD = 6.6 School School xi x j Significant? No Yes Yes It is clear that the me a n for district is significa ntly differe nt from the me a n for each of the other four schools. 8. Use Tukey s met ho d with differ. α =0. 0 to det e r mi n e which popul a tion me a n s ω = 78.7 School School xi x j Significant? No Yes Yes It is clear that the me a n for district is significa ntly differe nt from the me a n for each of the other four schools.

32 6 Chapter ifteen QUESTIONS 9 THROUGH ARE BASED ON THE OLLOWING INORMATION: The following dat a wer e gen er a t e d from a x factorial expe ri m e n t with replicat e s actor B actor A Test at the % significanc e level to det e r mi n e if differe nc e s exist am on g the four treat m e n t me a n s. Source of Variation P value critical H 0 : µ = µ = µ = µ H : At least two me a n s differ Conclusion: Reject the null hypot h e si s. Yes 0. Test at the % significanc e level to det e r mi n e if factors A and B inter a c t. Source of Variation actor A actor B Inter action P value critical H 0 : actors A and B do not inter a c t H : actors A and B do inter a c t Conclusion: Don t reject the null hypot h e si s. No

33 Analysis of Varianc e. 6 Test at the % significanc e level to det e r mi n e if differe nc e s exist am on g the levels of factor A. H 0 : No differenc e amo n g the me a n s of the a levels of factor A H : At least two me a n s differ Conclusion: Don t reject the null hypot h e si s. No. Test at the % significanc e level to det e r mi n e if differe nc e s exist am on g the levels of factor B. H 0 : No differenc e amo n g the me a n s of the b levels of factor B H : At least two me a n s differ Conclusion: Reject the null hypot h e si s. Yes. In a com pl et ely rando miz e d design, experi m e n t a l units wer e assign e d to each of four treat m e n t s. ill in the blanks (identified by ast e risks) in the partial ANOVA table shown below. Source of Variation Source of Variation. A statistics profes s or has carrie d out a study to com p a r e differe n t teac hing met h o d s used in thre e differe nt sections of an ele m e n t a r y statistics cours e. A sa m pl e of stud e n t s have bee n rando mly selec t e d form each section, and their grad e s in the final test, as shown below, are use d to det er mi n e whet h e r the teac hing met ho d s mad e any differe nc e. Method Method Method Can we infer at the % significanc e level that the popula tion me a n s of the thre e met h o d s differ?

34 6 Chapter ifteen Source of Variation ,0. 7, H 0 : µ = µ = µ H : At least two me a n s differ Rejection region: >.0,, =.98 Conclusion: Don t reject the null hypot h e si s. No QUESTIONS THROUGH 7 ARE BASED ON THE OLLOWING INORMATION: The dat a shown replicat e s. below wer e take n from a x factorial expe ri m e n t with actor B actor A Is ther e sufficient evidenc e at the % significanc e level to infer that factors A and B inter ac t? Source of Variation actor A actor B Inter action P value critical...

35 Analysis of Varianc e 66 H 0 : actors A and B do not inter a c t H : actors A and B do inter a c t Conclusion: Don t reject the null hypot h e si s. No 6. Test at the % significanc e level to det e r mi n e if differe nc e s exist am on g the levels of factor A. H 0 : No differenc e amo n g the me a n s of the a levels of factor A H : At least two me a n s differ Conclusion: Reject the null hypot h e si s. Yes 7. Test at the % significanc e level to det e r mi n e if differe nc e s exist am on g the levels of factor B. H 0 : No differenc e amo n g the me a n s of the b levels of factor B H : At least two me a n s differ Conclusion: Don t reject the null hypot h e si s. No QUESTIONS 8 AND 9 ARE BASED ON THE OLLOWING INORMATION: In a com pl et ely rando miz e d design, experi m e n t a l units wer e assign e d to the first treat m e n t, units to the secon d tre a t m e n t, and 8 units to the third tre a t m e n t. A partial ANOVA table is shown below: Source of Variation 8. 9 ill in the blanks (identified by ast erisks) in the abov e ANOVA table. Source of Variation

36 67 9. Chapter ifteen Test at the % significanc e level to det e r mi n e if differe nc e s exist am on g the thre e treat m e n t me a n s. H 0 : µ = µ = µ H : At least two me a n s differ Rejection region: > 0.0,,. Test statistics: = 9.0 Conclusion: Reject the null hypot h e si s. Yes QUESTIONS 0 AND ARE BASED ON THE OLLOWING INORMATION: In a com pl et ely rando miz e d design, 7 expe ri m e n t a l units were assign e d to the first treat m e n t, units to the secon d tre a t m e n t, and 0 units to the third tre a t m e n t. A partial ANOVA table for this experi m e n t is shown below: Source of Variation 0..0 ill in the blanks (identified by ast erisks) in the abov e ANOVA. Table. Source of Variation Test at the % significanc e level to det e r mi n e if differe nc e s exist am on g the thre e treat m e n t me a n s. H 0 : µ = µ = µ H : At least two me a n s differ Rejection region: > 0.0,,7 =. Test statistics: =.0 Conclusion: Don t reject the null hypot h e si s. No QUESTIONS THROUGH ARE BASED ON THE OLLOWING INORMATION: A partial ANOVA table in a rando miz e d block design is shown below:

37 Analysis of Varianc e Source of Variation Blocks.,0, ill in the missing value s (identified by ast erisks) in the above ANOVA Table. Source of Variation Blocks. 6,70,0,760,60 0 6, Can we infer at the % significanc e level that the tre a t m e n t me a n s differ? H 0 : µ = µ = µ = µ = µ H : At least two me a n s differ Rejection region: > 0.0,, =.78 Test statistics: = Conclusion: Reject the null hypot h e si s. Yes. Can we infer at the % significanc e level that the block me a n s differ? H 0 : µ = µ = µ = µ = µ = µ 6 = µ 7 H : At least two me a n s differ Rejection region: >.0, 6, =. Test statistics: =. 7 Conclusion: Reject the null hypot h e si s. Yes QUESTIONS THROUGH 7 ARE BASED ON THE OLLOWING INORMATION: A profes s or of statistics is trying to det e r mi n e which of thre e statistic al softwar e is the best for his stud e n t s. He believe s that the time (in hours) it take s a stud e n t to

38 69 Chapter ifteen mas t e r particular softwar e may be influenc e d by gend e r. A X factorial experi m e n t with thre e replicat e s was design e d, as shown below: Gend er Softwar e. Male em al e Is ther e sufficient evidenc e at the 0% significanc e level to infer that the time it take s a stud e n t to ma s t e r softwar e and the gend e r of the stud e n t inter ac t? Source of Variation Softwar e Gend er Inter action P value critical H 0 : Softwar e type and gend e r do not inter a c t H : Softwar e type and gend e r do inter a c t Conclusion: Don t reject the null hypot h e si s. No 6. Test at the 0% significanc e level to det er mi n e if differe nc e s exist amo n g the types of softwar e. H 0 : No differenc e amo n g the me a n s of the type s of softwar e H : At least two me a n s differ Conclusion: Don t reject the null hypot h e si s. No 7. Test at the 0% significanc e level to det er mi n e if differe nc e s exist amo n g male and fem al e stud e n t s. H 0 : No differenc e amo n g the me a n s of the male and fem al e stud e n t s

39 Analysis of Varianc e 70 H : At least two me a n s differ Conclusion: Don t reject the null hypot h e si s. No. QUESTIONS 8 THROUGH 6 ARE BASED ON THE OLLOWING INORMATION: An insur a nc e com p a n y is consid ering ope ning a new branc h in Lansing. The com p a n y will choos e the final location from two locations within the city. One of the factors in the decision is the annu al family incom e (in thous a n d s of dollars) of five families rando mly sa m pl e d from a radius of five miles from the pote nti al locations. Area Area Perfor m equ al varianc e s t test at the % significanc e whet h e r the popul ation me a n s differ. level to det er mi n e H 0 : µ = µ H : µ µ Rejection region: t > t0.0,8 =.06 Test statistics: t =.098 Conclusion: Don t reject the null hypot h e si s. No 9. Perfor m an test for one way ANOVA at the det er mi n e whet h e r the popul ation me a n s differ. % significanc e level to Source of Variation 9.6,8. 8, Rejection region: > 0.0,,8 =. Test statistic: =.06 Conclusion: Don t reject the null hypot h e si s. No 60. What is the relation betw e e n the obs erv e d t and obs e rv e d test statistics from Ques tion s 8 and 9? Does the sa m e relation hold true for the corres p o n di n g critical value s? (t ) = (.098) =.06 =