7 ANALYSIS OF VARIANCE (ANOVA)

Transcription

1 7 ANALYSIS OF VARIANCE (ANOVA) Chapter 7 Analyss of Varance (Anova) Objectves After studyng ths chapter you should apprecate the need for analysng data from more than two samples; understand the underlyng models to analyss of varance; understand when, and be able, to carry out a one way analyss of varance; understand when, and be able, to carry out a two way analyss of varance. 7.0 Introducton What s the common characterstc of all tests descrbed n Chapter 4? Consder the followng two nvestgatons. (a) A car magazne wshes to compare the average petrol consumpton of THREE smlar models of car and has avalable sx vehcles of each model. (b) A teacher s nterested n a comparson of the average percentage marks attaned n the examnatons of FIVE dfferent subjects and has avalable the marks of eght students who all completed each examnaton. In both these nvestgatons, nterest s centred on a comparson of more than two populatons; THREE models of car, FIVE examnatons. In (a), sx vehcles of each of the three models are avalable so there are three ndependent samples, each of sze sx. Ths example requres an extenson of the test consdered n Secton 4.3, whch was for two normal populaton means usng ndependent samples and a pooled estmate of varance. 13

2 In (b) however, there s the addtonal feature that the same eght students each completed the fve examnatons, so there are fve dependent samples each of sze eght. Ths example requres an extenson of the test consdered n Secton 4.4, whch was for two normal populaton means usng dependent (pared) samples. Ths chapter wll show that an approprate method for nvestgaton (a) s a one way anova to test for dfferences between the three models of car. For (b), an approprate method s a two way anova to test for dfferences between the fve subjects and, f requred, for dfferences between the eght students. 7.1 Ideas for data collecton Undertake at least one of Actvtes 1 and AND at least one of Actvtes 3 and 4. You wll requre your data for subsequent analyss later n ths chapter. Actvty 1 Estmatng length Draw a straght lne of between 0 cm and 5 cm on a sheet of plan whte card. (Only you should know ts exact length.) Collect 6 to 10 volunteers from each of school years 7, 10 and 13. Ask each volunteer to estmate ndependently the length of the lne. Do dfferences n year means appear to outwegh dfferences wthn years? Actvty Apples Obtan random samples of each of at least three varetes of apple. The samples should be of at least 5 apples but need not be of the same sze. Wegh, as accurately as you are able, each apple. Compare varaton wthn varetes wth varablty between varetes. 14

3 Actvty 3 Shop prces Make a lst of 10 food/household tems purchased regularly by your famly. Obtan the current prces of the tems n three dfferent shops; preferably a small 'corner' shop, a small supermarket and a large supermarket or hyper market. Compare total shop prces. Actvty 4 Weghng scales Obtan the use of at least three dfferent models of bathroom scales, preferably one of whch s electronc. Collect about 10 volunteers and record ther weghts on each scale. If possble, also have each volunteer weghed on more accurate scales such as those found n health centres or large pharmaces. You wll need to ensure that your volunteers are each wearng, as far as s possble, the same apparel at every weghng. Assess the dfferences n total weghts between the weghng devces used. Whch model of bathroom scales appears the most accurate? 7. Factors and factor levels Two new terms for analyss of varance need to be ntroduced at ths stage. Factor a characterstc under consderaton, thought to nfluence the measured observatons. Level a value of the factor. In Actvty 1, there s one factor (school year) at three levels (7, 10 and 13). In Actvty 3, there are two factors (tem and shop) at 10 and 3 levels, respectvely. What are the factors and levels n Actvtes and 4? 15

4 7.3 One way (factor) anova In general, one way anova technques can be used to study the effect of k > ( ) levels of a sngle factor. To determne f dfferent levels of the factor affect measured observatons dfferently, the followng hypotheses are tested. H 0 : µ = µ H 1 : µ µ all = 1,, K, k some = 1,, K, k where µ s the populaton mean for level. Assumptons When applyng one way analyss of varance there are three key assumptons that should be satsfed. They are essentally the same as those assumed n Secton 4.3 for k = levels, and are as follows. 1. The observatons are obtaned ndependently and randomly from the populatons defned by the factor levels.. The populaton at each factor level s (approxmately) normally dstrbuted. 3. These normal populatons have a common varance, σ. Thus for factor level, the populaton s assumed to have a dstrbuton whch s N ( µ, σ ). Example The table below shows the lfetmes under controlled condtons, n hours n excess of 1000 hours, of samples of 60W electrc lght bulbs of three dfferent brands. 16 Brand Assumng all lfetmes to be normally dstrbuted wth common varance, test, at the 1% sgnfcance level, the hypothess that there s no dfference between the three brands wth respect to mean lfetme.

5 Soluton Here there s one factor (brand) at three levels (1, and 3). Also the sample szes are all equal (to 5), though as you wll see later ths s not necessary. H 0 : µ = µ all H 1 : µ µ some = 1,, 3 = 1,, 3 The sample mean and varance (dvsor ( n 1)) for each level are as follows. Brand 1 3 Sample sze Sum Sum of squares Mean Varance Snce each of these three sample varances s an estmate of the common populaton varance, σ, a pooled estmate may be calculated n the usual way as follows. ˆσ W ( = 5 1 ) 9+( 5 1) 10 + ( 5 1 ) 11 = Ths quantty s called the varance wthn samples. It s an estmate of σ based on v = = 1 degrees of freedom. Ths s rrespectve of whether or not the null hypothess s true, snce dfferences between levels (brands) wll have no effect on the wthn sample varances. The varablty between samples may be estmated from the three sample means as follows. Brand 1 3 Sample mean Sum 63 Sum of squares 1385 Mean 1 Varance 31 17

6 Ths varance (dvsor ( n 1)), denoted by ˆσ B s called the varance between sample means. Snce t calculated usng sample means, t s an estmate of σ 5 σ (that s n n general) based upon ( 3 1)= degrees of freedom, but only f the null hypothess s true. If H 0 s false, then the subsequent 'large' dfferences between the sample means wll result n 5 ˆσ B beng an nflated estmate of σ. The two estmates of σ, ˆσ W and 5 ˆσ B, may be tested for equalty usng the F-test of Secton 4.1 wth F = 5 ˆσ B ˆσ W as lfetmes may be assumed to be normally dstrbuted. Recall that the F-test requres the two varances to be ndependently dstrbuted (from ndependent samples). Although ths s by no means obvous here (both were calculated from the same data), ˆσ W and ˆσ B are n fact ndependently dstrbuted. The test s always one-sded, upper-tal, snce f H 0 s false, 5 ˆσ B nflated whereas ˆσ W s unaffected. s Thus n analyss of varance, the conventon of placng the larger sample varance n the numerator of the F statstc s NOT appled. The soluton s thus summarsed and completed as follows. H 0 : µ = µ all = 1,, 3 H 1 : µ µ some = 1,, 3 Sgnfcance level, α = 0.01 Degrees of freedom, v 1 =, v = 1 Accept H 0 Crtcal regon reject H 0 1% Crtcal regon s F > F Test statstc s F = 5 ˆσ B ˆσ W = = 15.5 Ths value does le n the crtcal regon. There s evdence, at the 1% sgnfcance level, that the true mean lfetmes of the three brands of bulb do dffer. 18

7 What s the value of the varance (dvsor ( n 1)), ˆσ T, of the lfetmes, f these are consdered smply as one sample of sze 15? What s the value of 14 ˆσ T? What s the value of 1 ˆσ W +10 ˆσ B? At ths pont t s useful to note that, although the above calculatons were based on (actual lfetmes ), the same value would have been obtaned for the test statstc (F) usng actual lfetmes. Ths s because F s the rato of two varances, both of whch are unaffected by subtractng a workng mean from all the data values. Addtonally, n analyss of varance, data values may also be scaled by multplyng or dvdng by a constant wthout affectng the value of the F rato. Ths s because each varance nvolves the square of the constant whch then cancels n the rato. Scalng of data values can make the subsequent analyss of varance less cumbersome and, sometmes, even more accurate. Notaton and computatonal formulae The calculatons undertaken n the prevous example are somewhat cumbersome, and are prone to naccuracy wth non-nteger sample means. They also requre consderable changes when the sample szes are unequal. Equvalent computatonal formulae are avalable whch cater for both equal and unequal sample szes. Frst, some notaton. Number of samples (or levels) = k Number of observatons n th sample = n, = 1,, K, k Total number of observatons = n = n Observaton j n th sample = x j, j = 1,, K, n Sum of n observatons n th sample = T = x j j Sum of all n observatons = T = T = x j j The computatonal formulae now follow. Total sum of squares, SS T = x j T n Between samples sum of squares, SS B = j T T n n Wthn samples sum of squares, SS W = SS T SS B 19

8 A mean square (or unbased varance estmate) s gven by (sum of squares) (degrees of freedom) e.g. ˆσ = ( x x) n 1 Hence Total mean square, MS T = SS T n 1 Between samples mean square, MS B = SS B k 1 Wthn samples mean square, MS W = SS W n k Note that for the degrees of freedom: ( k 1)+ ( n k)= ( n 1) Actvty 5 For the prevous example on 60W electrc lght bulbs, use these computatonal formulae to show the followng. (a) SS T = 430 (b) SS B = 310 (c) MS B = 155 ( 5 ˆσ B ) (d) MS W = 10 ( ˆσ W ) Note that F = MS B = 155 = 15.5 as prevously. MS W 10 Anova table It s convenent to summarse the results of an analyss of varance n a table. For a one factor analyss ths takes the followng form. Source of Sum of Degrees of Mean F rato varaton squares freedom square Between samples SS B k 1 MS B MS B MS W Wthn samples SS W n k MS W Total SS T n 1 130

9 Example In a comparson of the cleanng acton of four detergents, 0 peces of whte cloth were frst soled wth Inda nk. The cloths were then washed under controlled condtons wth 5 peces washed by each of the detergents. Unfortunately three peces of cloth were 'lost' n the course of the experment. Whteness readngs, made on the 17 remanng peces of cloth, are shown below. Detergent A B C D Assumng all whteness readngs to be normally dstrbuted wth common varance, test the hypothess of no dfference between the four brands as regards mean whteness readngs after washng. Soluton H 0 : no dfference n mean readngs µ = µ all H 1 : a dfference n mean readngs µ µ some Sgnfcance level, α = 0.05 (say) Degrees of freedom, v 1 = k 1 = 3 and v = n k = 17 4 = 13 Crtcal regon reject H 00 5% Accept H 0 Crtcal regon s F > F A B C D Total n = n T = T x j = 8636 SS T = = SS B = = SS W = =

10 The anova table s now as follows. Source of Sum of Degrees of Mean F rato varaton squares freedom square Between detergents Wthn detergents Total The F rato of 1.07 does not le n the crtcal regon. Thus there s no evdence, at the 5% sgnfcance level, to suggest a dfference between the four brands as regards mean whteness after washng. Actvty 6 Carry out a one factor analyss of varance for the data you collected n ether or both of Actvtes 1 and. Model From the three assumptons for one factor anova, lsted prevously, x j ~ N ( µ,σ ) for j = 1,, K, n and = 1,, K, k Hence x j µ = ε j ~ N( 0, σ ) where ε j denotes the varaton of x j about ts mean µ and so represents the nherent random varaton n the observatons. k If µ = 1 µ, then µ µ =0. k =1 ( ) Wrtng µ µ = L results n µ = µ + L where = 0. L Hence L can be nterpreted as the mean effect of factor level relatve to an overall mean µ. Combnng x j µ = ε j wth µ µ = L results n x j = µ + L +ε j for j = 1,, K, n and = 1,, K, k Ths formally defnes a model for one way (factor) analyss of varance, where 13

11 x j = jth observaton at th level (n th sample), µ = overall factor mean, L = mean effect of th level of factor relatve to µ, where L = 0, ε j = nherent random varaton ~ N( 0, σ ). Note that as a result, H 0 : µ = µ (all ) H 0 : L = 0 (all ) Estmates of µ, L and ε j can be calculated from observed measurements by T n, T T and n n x j T, respectvely. n Thus for the example on 60W electrc lght bulbs for whch the observed measurements (x j ) were Brand wth n = 15, n 1 = n = n 3 = 5, T = 315, T 1 = 80, T = 100 and T 3 = 135. Hence, estmates of µ, L 1, L and L 3 are 1, 5, 1 and +6, respectvely. Estmates of the ε j are best tabulated as shown. Brand (estmates of ε j )

12 Notce that, relatve to the orgnal measurements, these values representng nherent random varaton are qute small. What s the sum of these values? What s sum of squares of these values and how was t found earler? Actvty 7 Calculate estmates of µ, L and ε j for the data you collected n ether of Actvtes 1 and. Exercse 7A 1. Four treatments for fever blsters, ncludng a placebo (A), were randomly assgned to 0 patents. The data below show, for each treatment, the numbers of days from ntal appearance of the blsters untl healng s complete. Treatment Number of days A B C D Test the hypothess, at the 5% sgnfcance level, that there s no dfference between the four treatments wth respect to mean tme of healng.. The followng data gve the lfetmes, n hours, of three types of battery. I Type II III Analyse these data for a dfference between mean lfetmes. (Use a 5% sgnfcance level.) 3. Three dfferent brands of magnetron tubes (the key component n mcrowave ovens) were subjected to stress testng. The number of hours each operated before needng repar was recorded. A Brand B Although these tmes may not represent lfetmes, they do ndcate how well the tubes can wthstand stress. Use a one way analyss of varance procedure to test the hypothess that the mean lfetme under stress s the same for the three brands. What assumptons are necessary for the valdty of ths test? Is there reason to doubt these assumptons for the gven data? 4. Three specal ovens n a metal workng shop are used to heat metal specmens. All the ovens are supposed to operate at the same temperature. It s known that the temperature of an oven vares, and t s suspected that there are sgnfcant mean temperature dfferences between ovens. The table below shows the temperatures, n degrees centgrade, of each of the three ovens on a random sample of heatngs. Oven Temperature ( o C) Statng any necessary assumptons, test for a dfference between mean oven temperatures. Estmate the values of µ (1 value), L (3 values) and ε j (15 values) for the model (temperature) j = x j = µ + L +ε j. Comment on what they reveal. C

13 5. Eastsde Health Authorty has a polcy whereby any patent admtted to a hosptal wth a suspected coronary heart attack s automatcally placed n the ntensve care unt. The table below gves the number of hours spent n ntensve care by such patents at fve hosptals n the area. Hosptal A B C D E Use a one factor analyss of varance to test, at the 1% level of sgnfcance, for dfferences between hosptals. (AEB) 6. An experment was conducted to study the effects of varous dets on pgs. A total of 4 smlar pgs were selected and randomly allocated to one of the fve groups such that the control group, whch was fed a normal det, had 8 pgs and each of the other groups, to whch the new dets were gven, had 4 pgs each. After a fxed tme the gans n mass, n klograms, of the pgs were measured. Unfortunately by ths tme two pgs had ded, one whch was on det A and one whch was on det C. The gans n mass of the remanng pgs are recorded below. Dets Gan n mass (kg) Normal A B C D Use a one factor analyss of varance to test, at the 5% sgnfcance level, for a dfference between dets. What further nformaton would you requre about the dead pgs and how mght ths affect the conclusons of your analyss? (AEB) 7.4 Two way (factor) anova Ths s an extenson of the one factor stuaton to take account of a second factor. The levels of ths second factor are often determned by groupngs of subjects or unts used n the nvestgaton. As such t s often called a blockng factor because t places subjects or unts nto homogeneous groups called blocks. The desgn tself s then called a randomsed block desgn. Example A computer manufacturer wshes to compare the speed of four of the frm's complers. The manufacturer can use one of two expermental desgns. (a) (b) Use 0 smlar programs, randomly allocatng 5 programs to each compler. Use 4 copes of any 5 programs, allocatng 1 copy of each program to each compler. Whch of (a) and (b) would you recommend, and why? 135

14 Soluton In (a), although the 0 programs are smlar, any dfferences between them may affect the complaton tmes and hence perhaps any conclusons. Thus n the 'worst scenaro', the 5 programs allocated to what s really the fastest compler could be the 5 requrng the longest complaton tmes, resultng n the compler appearng to be the slowest! If used, the results would requre a one factor analyss of varance; the factor beng compler at 4 levels. In (b), snce all 5 programs are run on each compler, dfferences between programs should not affect the results. Indeed t may be advantageous to use 5 programs that dffer markedly so that comparsons of complaton tmes are more general. For ths desgn, there are two factors; compler (4 levels) and program (5 levels). The factor of prncpal nterest s compler whereas the other factor, program, may be consdered as a blockng factor as t creates 5 blocks each contanng 4 copes of the same program. Thus (b) s the better desgned nvestgaton. The actual complaton tmes, n mllseconds, for ths two factor (randomsed block) desgn are shown n the followng table. Compler Program A Program B Program C Program D Program E Assumptons and nteracton The three assumptons for a two factor analyss of varance when there s only one observed measurement at each combnaton of levels of the two factors are as follows. 1. The populaton at each factor level combnaton s (approxmately) normally dstrbuted.. These normal populatons have a common varance, σ. 3. The effect of one factor s the same at all levels of the other factor. Hence from assumptons 1 and, when one factor s at level and the other at level j, the populaton has a dstrbuton whch s N ( µ j, σ ). 136

15 Assumpton 3 s equvalent to statng that there s no nteracton between the two factors. Now nteracton exsts when the effect of one factor depends upon the level of the other factor. For example consder the effects of the two factors: sugar (levels none and teaspoons), and strrng (levels none and 1 mnute), on the sweetness of a cup of tea. Strrng has no effect on sweetness f sugar s not added but certanly does have an effect f sugar s added. Smlarly, addng sugar has lttle effect on sweetness unless the tea s strred. Hence factors sugar and strrng are sad to nteract. Interacton can only be assessed f more than one measurement s taken at each combnaton of the factor levels. Snce such stuatons are beyond the scope of ths text, t wll always be assumed that nteracton between the two factors does not exst. Thus, for example, snce t would be most unusual to fnd one compler partcularly suted to one program, the assumpton of no nteracton between complers and programs appears reasonable. Is t lkely that the assumpton of no nteracton s vald for the data you collected n each of Actvtes 3 and 4? Notaton and computatonal formulae As llustrated earler, the data for a two factor anova can be dsplayed n a two-way table. It s thus convenent, n general, to label the factors as a row factor and a column factor. Notaton, smlar to that for the one factor case, s then as follows. Number of levels of row factor = r Number of levels of column factor = c Total number of observatons = rc Observaton n ( j) th cell of table = x j (th level of row factor and = 1,, K, r jth level of column factor) j = 1,, K, c 137

16 Sum of c observatons n th row = T R = x j j Sum of r observatons n jth column = T Cj = Sum of all rc observatons = T = x j = T R = T Cj These lead to the followng computatonal formulae whch agan are smlar to those for one factor anova except that there s an addtonal sum of squares, etc for the second factor. Total sum of squares, SS T = x j T rc Between rows sum of squares, SS R = j x j j T R c T rc j Between columns sum of squares, SS C = j T Cj r T rc Error (resdual) sum of squares, SS E = SS T SS R SS C What are the degrees of freedom for SS T, SS R and SS C when there are 0 observatons n a table of 5 rows and 4 columns? What s then the degrees of freedom of SS E? Anova table and hypothess tests For a two factor analyss of varance ths takes the followng form. Source of Sum of Degrees of Mean F rato varaton squares freedom square Between rows SS R r 1 MS R MS R MS E Between columns SS C c 1 MS C MS C MS E Error (resdual) SS ( r 1) ( c 1) MS E E Total SS T rc 1 138

17 Notes: 1. The three sums of squares, SS R, SS C and SS E are ndependently dstrbuted.. For the degrees of freedom: ( r 1)+ ( c 1)+ ( r 1) ( c 1)= ( rc 1). Usng the F ratos, tests for sgnfcant row effects and for sgnfcant column effects can be undertaken. H 0 : no effect due to row factor H 0 : no effect due to column factor H 1 : an effect due to row factor H 1 : an effect due to column factor Crtcal regon, Crtcal regon, ( α) F > F [( r 1), ( r 1) ( c 1) ] Test statstc, F R = MS R MS E ( α) F > F [( c 1), ( r 1) ( c 1) ] Test statstc, F C = MS C MS E Example Returnng to the complaton tmes, n mllseconds, for each of fve programs, run on four complers. Test, at the 1% sgnfcance level, the hypothess that there s no dfference between the performance of the four complers. Has the use of programs as a blockng factor proved worthwhle? Explan. The data, gven earler, are reproduced below. Compler Program A Program B Program C Program D Program E

18 Soluton To ease computatons, these data have been transformed (coded) by x = 100 ( tme 5) to gve the followng table of values and totals. Compler Row totals T R Program A Program B Program C Program D Program E ( ) Column totals ( T Cj ) = T x j = The sums of squares are now calculated as follows. (Rows = Programs, Columns = Complers) SS T = = SS R = 1 ( ) = SS C = 1 ( ) SS E = = 9354 = Anova table Source of Sum of Degrees of Mean F rato varaton squares freedom square Between programs Between complers Error (resdual) Total

19 H 0 : no effect on complaton tmes due to complers H 1 : an effect on complaton tmes due to complers Sgnfcance level, α = 0.01 Crtcal regon Degrees of freedom, v 1 = c 1 = 3 H 0 reject H 0 1% Crtcal regon s F > and v = ( r 1) ( c 1)= 4 3=1 0 Accept H F Test statstc F C = 4.55 Ths value does not le n the crtcal regon. Thus there s no evdence, at the 1% sgnfcance level, to suggest a dfference n complaton tmes between the four complers. The use of programs as a blockng factor has been very worthwhle. From the anova table (a) SS R accounts for 100 = 97.65% of the total varaton n the observatons, much of whch would have been ncluded n SS E had not programs been used as a blockng varable, (b) F R = whch ndcates sgnfcance at any level! Actvty 8 Carry out a two factor analyss of varance for the data you collected n ether or both of Actvtes 3 and 4. In each case dentfy the blockng factor, and explan whether or not t has made a sgnfcant contrbuton to the analyss. Model Wth x j denotng the one observaton n the th row and jth column, (j)th cell, of the table, then x j ~ N ( µ j,σ ) for = 1,, K, r and j = 1,, K, c or x j µ j = ε j ~ N( 0, σ ) However, t s assumed that the two factors do not nteract but smply have an addtve effect, so that 141

20 µ j = µ + R +C wth R = C j j µ = overall mean = 0, where R = mean effect of th level of row factor relatve to µ C j = mean effect of jth level of column factor relatve to µ ε j = nherent random varaton As a result, when testng for an effect due to rows, the hypotheses may be wrtten as H 0 : R = 0 (all ) H 1 : R 0 (some ) What are the correspondng hypotheses when testng for an effect due to columns? If requred, estmates of µ, R, C j and ε j can be calculated from observed measurements by T rc, T R c T rc, T Cj r T, rc x j T R c T Cj r + T, rc respectvely. What are the estmates of µ, R 1, C and ε 1 n the prevous example, based upon the transformed data? Actvty 9 Calculate estmates of some of µ, R, C j and ε j for the data you collected n ether of Actvtes 3 and 4. Exercse 7B 1. Pror to submttng a quotaton for a constructon project, companes prepare a detaled analyss of the estmated labour and materals costs requred to complete the project. A company whch employs three project cost assessors, wshed to compare the mean values of these assessors' cost estmates. Ths was done by requrng each assessor to estmate ndependently the costs of the same four constructon projects. These costs, n 0000s, are shown n the next column. Assessor A B C Project Project Project Project Perform a two factor analyss of varance on these data to test, at the 5% sgnfcance level, that there s no dfference between the assessors' mean cost estmates. 14

21 . In an experment to nvestgate the warpng of copper plates, the two factors studed were the temperature and the copper content of the plates. The response varable was a measure of the amount of warpng. The resultant data are as follows. Copper content (%) Temp ( o C) Statng all necessary assumptons, analyse for sgnfcant effects. 3. In a study to compare the body szes of slkworms, three genotypes were of nterest: heterozygous (HET), homozygous (HOM) and wld (WLD). The length, n mllmetres, of a separately reared cocoon of each genotype was measured at each of fve randomly chosen stes wth the followng results. Ste Slkworm HOM HET WLD Identfy the blockng factor. Has t proved useful? Explan. Test, at the 1% sgnfcance level, for a dfference n mean lengths between genotypes. 4. Four dfferent washng solutons were beng compared to study ther effectveness n retardng bactera growth n mlk contaners. The study was undertaken n a laboratory, and only four trals could be run on any one day. As days could represent a potental source of varablty, the expermenter decded to use days as a blockng varable. Observatons were made for fve days wth the followng (coded) results. Day Soluton A The Marathon of the South West took place n Brstol n Aprl 198. The table below gves the tmes taken, n hours, by twelve compettors to complete the course, together wth ther type of occupaton and tranng method used. Types of occupaton Tranng Offce Manual Professonal methods worker worker sportsperson A B C D Carry out an analyss of varance and test, at the 5% level of sgnfcance, for dfferences between types of occupaton and between tranng methods. The age and sex of each of the above compettors are subsequently made avalable to you. Is ths nformaton lkely to affect your conclusons and why? (AEB) 6. Informaton about the current state of a complex ndustral process s dsplayed on a control panel whch s montored by a techncan. In order to fnd the best dsplay for the nstruments on the control panel, three dfferent arrangements were tested by smulatng an emergency and observng the reacton tmes of fve dfferent techncans. The results, n seconds, are gven below. Techncan Arrangement P Q R S T A B C Carry out an analyss of varance and test for dfferences between techncans and between arrangements at the 5% sgnfcance level. Currently arrangement C s used and t s suggested that ths be replaced by arrangement A. Comment, brefly, on ths suggeston and on what further nformaton you would fnd useful before comng to a defnte decson. (AEB) B C D Statng any necessary assumptons, analyse for sgnfcant dfferences between solutons. Was the expermenter wse to use days as a blockng factor? Justfy your answer. 143

22 7.5 Mscellaneous Exercses 1. After completng a sx month typng course wth the Speedyfngers Insttute, four people, A, B, C and D, had ther typng speed measured, n words per mnute, on each of fve knds of work. The results are gven n the table below. 144 Legal Busness Numerc Prose I Prose II A B C D Carry out an analyss of varance and test, at the 5% level of sgnfcance, for dfferences between the people and between knds of work. Subsequently t transpred that A and C used electrc typewrters, whlst B and D used manual typewrters. Does ths nformaton affect your conclusons and why? (AEB). A batch of brcks was randomly dvded nto three parts and each part was stored by a dfferent method. After one week the percentage water content of a number of brcks stored by each method was measured. Method of storage % water content Makng any necessary assumptons, use a one factor analyss of varance to test, at the 5% sgnfcance level, for dfferences between methods of storage. If low water content s desrable, state whch method of storage you would recommend, and calculate a 95% confdence nterval for ts mean percentage water content after one week. [You may assume that the estmated varance of a sample mean s gven by (Wthn samples mean square) (sample sze).] (AEB) 3. A textle factory produces a slcone proofed nylon cloth for makng nto ranwear. The chef chemst thought that a slcone soluton of about 1% strength would yeld a cloth wth a maxmum waterproofng ndex. It was also suspected that there mght be some batch to batch varaton because of slght dfferences n the cloth. To test ths, fve dfferent strengths of soluton were tested on each of three dfferent batches of cloth. The followng values of the waterproofng ndex were obtaned. Strength of slcone soluton (%) Cloth A B C [You may assume that the total sum of squares of the observatons x ( ) = ] Carry out an analyss of varance to test, at the 5% sgnfcance level, for dfferences between strengths of slcone soluton and between cloths. Comment on the chef chemst's orgnal belefs n the lght of these results and suggest what actons the chef chemst mght take. (AEB) 4. (a) A caterng frm wshes to buy a meat tenderser, but was concerned wth the effect on the weght loss of meat durng cookng. The followng results were obtaned for the weght loss of steaks of the same pre-cooked weght when three dfferent tendersers were used. Weght loss n grams A Tenderser B C Carry out a one factor analyss of varance and test at the 5% sgnfcance level whether there s a dfference n weght loss between tendersers. (b) Tme and temperature are mportant factors n the weght loss durng cookng. As these had not been taken account of durng the frst tral, a further set of results was obtaned where all the steaks were cooked at the same temperature and cookng tmes of 0, 5 and 30 mnutes were used. An analyss of these data led to the followng table. Source of Sum of Degrees of varaton squares freedom Between tendersers 31 Between tmes 697 Error 85 4 Total Test at the 5% sgnfcance level for dfferences between tendersers and between tmes. (c) Contrast the results obtaned n (a) and (b) and comment on why the two sets of data can lead to dfferent conclusons. (AEB)

23 5. A commuter n a large cty can travel to work by car, bcycle or bus. She tmes four journeys by each method wth the followng results, n mnutes. Car Bcycle Bus (a) Carry out an analyss of varance and test at the 5% sgnfcance level whether there are dfferences n the mean journey tmes for the three methods of transport. (b) The tme of day at whch she travels to work vares. Bearng n mnd that ths s lkely to affect the tme taken for the journey, suggest a better desgn for her experment and explan brefly why you beleve t to be better. (c) Suggest a factor other than leavng tme whch s lkely to affect the journey tme and two factors other than journey tme whch mght be consdered when choosng a method of transport. (AEB) 6. (a)as part of a project to mprove the steerablty of trucks, a manufacturer took three trucks of the same model and ftted them wth soft, standard and hard front sprngs, respectvely. The turnng radus (the radus of the crcle n whch the truck could turn full crcle) was measured for each truck usng a varety of drvers, speeds and surface condtons. Use the followng nformaton to test for a dfference between sprngs at the 5% sgnfcance level. Source Sum of Degrees of squares freedom Between sprngs 37.9 Wthn sprngs Total (b) A statstcan suggested that the experment would be mproved f the same truck was used all the tme wth the front sprngs changed as necessary and f the speed of the truck was controlled. The followng results for turnng crcle, n metres, were obtaned. Sprngs Speed Soft Standard Hard 15 km/h km/h Carry out a two factor analyss of varance and test at the 5% sgnfcance level for dfferences between sprngs and between speeds. [You may assume that the total sum of squares about the mean (SS T ) s 9.] (c) Compare the two experments and suggest a further mprovement to the desgn. (AEB) 7. A drug s produced by a fermentaton process. An experment was run to compare three smlar chemcal salts, X, Y and Z, n the producton of the drug. Snce there were only three of each of four types of fermenter A, B, C and D avalable for use n the producton, three fermentatons were started n each type of fermenter, one contanng salt X, another salt Y and the thrd salt Z. After several days, samples were taken from each fermenter and analysed. The results, n coded form, were as follows. Fermenter type A B C D X 67 Y 73 X 7 Z 70 Z 68 Z 65 Y 80 X 68 Y 78 X 69 Z 73 Y 69 State the type of expermental desgn used. Test, at the 5% level of sgnfcance, the hypothess that the type of salt does not affect the fermentaton. Comment on what assumpton you have made about the nteracton between type of fermenter and type of salt. (AEB) 8. A factory s to ntroduce a new product whch wll be assembled from a number of components. Three dfferent desgns are consdered and four employees are asked to compare ther speed of assembly. The tral s carred out one mornng and each of the four employees assembled desgn A from 8.30 am to 9.30 am, desgn B from am to am and desgn C from am to 1.30 pm. The number of products completed by each of the employees s shown n the followng table. Employee Desgn A B C (a) Carry out a two factor analyss of varance and test at the 5% sgnfcance level for dfferences between desgns and between employees. [You may assume that the total sum of squares about the mean (SS T ) s ] 145

24 (b) Comment on the fact that all employees assembled the desgns n the same order. Suggest a better way of carryng out the experment. (c) The two factor analyss assumes that the effects of desgn and employee may be added. Comment on the sutablty of ths model for these data and suggest a possble mprovement. (AEB) 9. In a hot, thrd world country, mlk s brought to the captal cty from surroundng farms n churns carred on open lorres. The keepng qualty of the mlk s causng concern. The lorres could be covered to provde shade for the churns relatvely cheaply or refrgerated lorres could be used but these are very expensve. The dfferent methods were tred and the keepng qualty measured. (The keepng qualty s measured by testng the ph at frequent ntervals and recordng the tme taken for the ph to fall by 0.5. A hgh value of ths tme s desrable.) Transport method Keepng qualty (hours) Open Covered Refrgerated (a) Carry out a one factor analyss of varance and test, at the 5% level, for dfferences between methods of transport. (b) Examne the method means and comment on ther mplcatons. (c) Dfferent farms have dfferent breeds of cattle and dfferent journey tmes to the captal, both of whch could have affected the results. How could the data have been collected and analysed to allow for these dfferences? (AEB) 10. A hosptal doctor wshed to compare the effectveness of 4 brands of pankller A, B, C and D. She arranged that when patents on a surgcal ward requested pankllers they would be asked f ther pan was mld, severe or very severe. The frst patent who sad mld would be gven brand A, the second who sad mld would be gven brand B, the thrd brand C and the fourth brand D. Pankllers would be allocated n the same way to the frst four patents who sad ther pan was severe and to the frst four patents who sad ther pan was very severe. The patents were then asked to record the tme, n mnutes, for whch the pankllers were effectve. The followng data were collected. Brand A B C D mld severe very severe (a) Carry out a two factor analyss of varance and test at the 5% sgnfcance level for dfferences between brands and between symptoms. You may assume that the total sum of squares ( SS T ) = (b) Crtcse the experment and suggest mprovements. (AEB) 146