STAB27-Winter 2006 Term test February 8,2006 Last Name: First Name: Student #: Tutorial Section / Room: Dayffime (Tutorial): INSTRUCTIONS Duration: hour, 45 minutes Statistical table(s) attached at the end Please do not detach them There are 4 pages including this page Please check to see you have all the pages You will get a mark out of 60 for this test Marks are shown in brackets Show your work and answer in the space provided, in ink Pencil may be used, but then no remarking will be allowed Use back of pages for rough work This test may be long, so be sure to proportion your time carefully among the questions and limit your time spent on any single question Good luck!!
) The marketing department a firm is concerned about the effect of advertising on sales of the firm's major product To investigate the relationship between advertising and sales, the data on the two variables were gathered from a random sample of 8 sales districts In the data the sales are expressed in thousands of dollars and advertising expenditures are expressed in hundreds of dollars Some useful MINIT AB outputs for analyzing these data are given below: 800 Scatterplot of salesl vs ADV 700 600 jsoo 400 300 : 2oo~ 50 200 250 300 ArN 350 400 450 Descriptive Statistics: ADV, sales Variable ADV sales N N* 8 0 8 0 Mean 2953 57 SE Mean 92 386 StDev 8 5 636 Minimum 600 2000 Q 200 3650 Median 325 5625 Q3 3688 6525 Maximum 450 7400 Pearson correlation of ADV and sales = 0988 Regression Analysis: sales versus ADV Predictor Constant ADV Coef A 98285 SE Coef 2387 007809 T B 2539 P 0007 0000 S = 26236 R-Sq = C R-Sq(adj) = D Analysis of Variance Regression Residual Error Total DF SS E 443784 F G 7 MS H I F P J 0000 - E :: I F= J9--) s-}~ ) i f- -, ~ ~~ li 2
Predicted Values for New Observations New Obs Fit 5209 SE Fit 69 95% CI (50795, 53422) 95% PI (46394, 57823) Values of Predictors for New Observations New Obs ADV 300 Normal Probability Plot of the Residuals (respoose is salesl) 9S so 8) 70 t! '" so t 3D '" : 0,, so -25 0 25 50 Residual ResIduals Versus the Fitted Values (respoose is salesl) 25-i so 3 o -25 -soj,,,,,, 200 300 400 500 600 700 800 IItted Va i) [0] Some entries of the above MINITAB output have been replaced by the letters A-J Fill in the deleted entries (Note that there are 0 deleted entries) p
ii) [2] What do you learn from the above residual plots 'f - las~ riot d~ +0 ~ ~~ i Lv ' ~/ Y~l~~ ~ ta-!pye>f< hcfyw, - 'YM\'~ v') -bi'h y~ J 0) ~ ~ ~~ ~~p~ r'~~f':~q") iii) [3] Test the hypothesis Ho : PI = versus Ho : PI * at the 5% level of significance ~t- "t Y (PI denotes the sl~e of the regression line) ~;: tj':: (,' ~ru"r - I, <;'E~/ :: lowd, (~\~C) / /7- \I~( < 0 ((}(JO~- 'f-l- /\ f6lj' H-o iv) [3] Give a 95%confidenceintervalfor the slopeof the regressionline v) [2] 95% prediction interval for sales for a district with advertising expenditures of $30 000 is given in the above output If we calculate a 95% prediction interval sales for a district with advertising expenditures of $25 000, will it be wider or narrower than the 95% prediction interval at $30 OOO?Why? \ :5o-D L" ~ \DV ( :2-"LS:5), 4
2) As cheddar cheese matures, a variety of chemical processes take place The taste of matured cheese is related to the concentration of several chemicals in the fmal product In a study of cheddar cheese, samples of cheese were analyzed for their chemical composition and taste Three of the chemicals whose concentrations were measured were acetic acid, hydrogen sulfide (denoted h2s) and lactic acid Some MINITAB outputs obtained from this study are given below: GI ow III 8 ow Scatterplot of taste vs acetic, h2s, lactic acetic h2s lactic 45 50 55 60 652 4 6 8 0 60 45 30-5- 0, I I 00 25 50 75 200 60 45 30 5 o Correlations: taste, acetic, h2s, lactic taste acetic h2s acetic 0550 0002 h2s 0756 068 0000 0000 lactic 0704 0604 0645 0000 0000 0000 Cell Contents: Pearson correlation P-Value Regression Analysis: taste versus lactic, h2s, acetic The regression equation is taste = - 289 + 97 lactic + 39 h2s + 033 acetic Predictor Coef SE Coef T P Constant -2888 974-46 055 lactic 967 8629 228 003 h2s 392 248 33 0004 acetic 0328 4460 007 0942 S = 0307 R-Sq = 652% R-Sq(adj) = 62% 5
Analysis of Variance Regression Residual Error Total DF 3 26 29 SS 49945 26684 76629 MS 6648 026 F 622 P 0000 lactic h2s acetic DF Seq SS 38004 935 06 Unusual Observations Obs 5 lactic 52 taste 5490 Fit 2945 SE Fit 304 Residual 2545 St Resid 263R R denotes an observation with a large standardized residual Predicted Values for New Observations New Obs Fit 9224 SE Fit 4482 95% CI (02, 8437) 95% PI (-22, 8670) Values of Predictors for New Observations New Obs lactic 540 h2s 370 acetic 30 6
Residual plots for taste versus lactic, h2s, acetic Residual Plots for taste Normal Probabirlty Plot of the Residuals 99 90 0-20 -0 o 0 Residual 20 ResidualsVersus the 20 0~ 'a i 0 I -0-20 o 5 30 Fitted Value Fitted Values 45 60 60 Histogram of the Residuals Residuals Versus the Order of the Data ~ 45 t 30 5 0 'a i -0 00-0 o 0 Residual 20-20 2 4 6 8 0 2 4 6 8 20 22 24 26 28 30 Observation Order Regression Analysis: taste versus h2s The regression equation is taste = - 979 + 578 h2s Predictor Constant h2s Coef -9787 5776 SE Coef 5958 09458 T -64 6 02 0000 P S = 08334 R-Sq = 57% R-Sq(adj) = 556% Analysis of Variance DF SS MS F P Regression 43767 43767 3729 0000 Residual Error 28 3286 74 Total 29 76629 i) [2] What do you learn from the scatter plots and the correlations between taste and explanatory variables? I - '\ g I ( I,- 5 ~ P i':) ] P b<t] OVI I'/S) ~t; wa-<-k~ 6Y g k;~ (~ ""- b- ~~) ~, R-\~~dv: (~\ \ ~ Y-h-) ~ ~ ~ e //Lq \;~ ' (\,~ -h, 0 '),~RA ~~7 ~'rca-~) ~--k~ ~ k--h~3l;r~ ~-
ii) [3] Test whether there is a regression relation Use IX= 005 State the null and the alternative hypotheses -!-to' pic f 2- I J, + ff- ~ g :::- ~J - D I /7( tj- ~ / 0 (0--v~l (~ AtVovl}) :: CJ t C)()-i)' < () D0 - I " t 6J,"",:e, ~ ~ 0-, y "'~J"(~' CM 're/ <>--76'-) iii) [2] Givea 95% confidenceintervalfor the coefficientofh2s in the regressionmodel with all threepredictors S \ 0, / - :t? 0 s-t f< t ~ 2-\f-8 iv) [3] Give a 90% confidence interval for the mean taste of cheddar cheese with lactic = 540, h2s = 370 and acetic = 30 v) [3] Give the 90% prediction interval for a sample of cheddar cheese with lactic = 540, h2s = 370 and acetic = 30 - ~ " - 9 ~ 'f! 2-, 6S -b J 'f \f ~- t r 67-' -b vi) [3] Test whether acetic can be dropped from the model, given that the other two variables are retained Use IX= 005 State the null and the alternative hypotheses ~ lrfo '\ t::>s-o ') H, " r::>t V-voJ ::- OICf~L- D > ' O~-- fa,~cy-f; C ~ k,ryr5f~ ~ fu ~ ~O V0ey,'~~ ~~i~~~'
vii) [3] Test whether both acetic and h2s can be dropped from the model given that lactic isrem;~~st ::~:re ;:null7 ~~~tc::;: crvj i f,<- cry(!)0 L - F ( (I q 3,s- -t o ::,')/'- ~ ~ I ~ '2-- ~ r- - I - ~, 0 l < b~v~' L <:>0 f U (I)-vCAI CO ODS-;, viii) [3] Test whetherboth acetic and lacticcan be droppedfrom the modelgiventhat h2s fj 0 ~ (0t-,fis retaineduse a = 005 State the null and the alternativehypotheses 'f}r0jj 60-"!-Io',ft >-f:s'"0 V5 +(I---I- tl"'-hr>" i> f'/ c (?> 'f~ F- (5~~b'l ~ 2-b6~V)/~' ~< I cnd ~ - ~ --- - ~, 6 r r-v~ A c\~} - / \/ b- ~ ~ ~I ( ( () 2- b ~I', I~J I 3) Runners are concerned about their form racing One measure of the form is the stride -Lt l-l2s I~ 0 rate, the number of steps taken per second As running speed increases, the stride rate should also increase In a study, the researchers measured the stride rate for different speeds The MINITAB outputs below are obtained from the regression analysis of stride rate on speed In these outputs speedl\2 = speed2 and speedl\3= speed3 Scatterplot 0stride vs speed 35 34 :j! 33 I 32~ l 3,] 30 ~ u v ~ w ~ D n speed Regression Analysis: stride versus speed The regression equation is stride = 77 + 00803 speed Predictor Constant speed Coef 76608 0080284 SE Coef T P 003068 5757 0000 00067 4966 0000 S = 00090684 R-Sq = 998% R-Sq(adj) = 998% 9
Analysis of Variance Regression Residual Error Total OF 5 6 SS 020279 00004 020320 MS 020279 000008 F 246593 P 0000 Residuals Versus the Fitted Values (response is stride) 000 0005! :II 0000 i -0005-000 30 3 32 33 litted V 34 35 36 i) [2] What do you learn from the above residual plot? lfa; 6vLyv(/~ l~ R\CC~ ~ k0~ 0-~ ~ Regression Analysis: stride versus speed, speedj\2 The regression equation is stride = 248 + 00039 speed + 00020 speeda2 Predictor Constant speed speeda2 Coef 24824 000395 00020083 SE Coef 067 00240 00003258 T 228 032 66 P 0000 0766 0004 S = 00032866 R-Sq = 000% R-Sq(adj) = 000% Analysis of Variance Regression Residual Error Total OF 2 4 6 SS 02036 000004 020320 MS 0058 00000 F 037749 P 0000 speed speeda2 OF Seq SS 020279 000037 0
Residuals Versus the Fitted Val- (respoose is stride) 0004 0003 0002 000 :II i 0000-000 -0002-0003 3:0 3 32 33 fittedvalue 34 35 36 ii) [2] What do you learn from the residual plots for the quadratic model? - M~ YO\/~ C ~ pq4e-r~ ~~~ &-< ~ -Ir +&t (; ia-l"'-y~ ~ iii) [2] Does the second order term significantly improve the fit of the model? Why? (ie does the second order term make a significant contribution to the model?) (Use a = 005 ) )~ 'I f> 2" ;;() \IS --/', fil' F"2- c:r () r~v"j:;; 0'0 0 y- <- 0 '05' jx<j \ ~ ~GsvJ;J ~ ~ ~ CL- lj' ~;, ~~ Regression Analysis: stride versus speed, speeda2, speeda3 The regression equation is stride = 263-0020 speed + 00033 speeda2-0000023 speeda3 Predictor Coef SE Coef T P Constant 2635 42 87 059 speed -00205 02259-009 0934 speeda2 000330 0098 028 080 speeda3-00000228 0000204-0 092 S = 000360563 R-Sq = 000% R-Sq(adj) = 000%
Analysis of Variance Regression Residual Error Total DF SS 3 02036 3 0000039 6 0203200 MS 0067720 000003 F P 520902 0000 speed speed"2 speed"3 DF Seq SS 0202789 0000372 0000000 i 000 3 i 0000 Residuals Versus the FilfEd Values (responseis stride) 0004 0003 0002-000 -0002-0003 3:0 3 32 33 34 35 36 fitted Value iv) [2] Does the third order term significantly improve the fit of the model? Why? (ie does the third order term make a significant contribution to the model?) (Use a ~ 005) Uo ' f3 ~ ::::- D Vs ~ " p~ to 3 r--v~\:= ooz2-/» ~ I Q:S- V0 )~ 3,W ~ Wt-v-- ~ fae v) [3] Test whether both quadratic and cubic terms can be dropped from the model (Use a = 005) State the null and the alternative hypotheses \ li) CA-+ ~3 \ ()DO5~~ --td-~ )/- C>60bO)3 ~ p-v",-\ L 0,> N?J) ~6\V/+ ~ UiL-, Itt,S/ 2