L Exercise , page Exercise , page 523.

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1 Homework #7* Statitic 1 L Eercie , pae Eercie , pae Eercie , pae Eercie , pae Eercie , pae Eercie12.4.3, pae Eercie , pae Eercie12.5.3, pae Eercie , pae Eercie , pae 548. Readin Ainment: Chapter of the tet. Material Covered: Introduction to linear rereion, the correlation, fitted rereion line, inference about rereion coefficient, uin the model for etimation and prediction. * Thi homework will not be collected for radin.

2 Statitic 1 Homework #7* (a) "'t >- ' ":!" (b) To compute the ample correlation we mut firt find the um of the product of the tandardized data. The table below ummarize thee intermediate reult. The ample correlation, r, i <'.-o =.44. ote: Uin tatitical oftware with the raw data, one will obtain a lihtly different anwer due to the roundin of the tandard deviation lited in the problem um 17. mean 3.4 d 2.1 (-) (y-y) (-) (y-y) X Sy Sy (c) The hypothee are H : p= HA: p:;z':o Th(1 tet tatitic i t = r = (.44) 2 =.849 Conultin Table 4 with df= 5-2 = 3, we find that t 2 =.978. Thu, P >.4, o we do not reject H (with a=.5). There i no inificant evidence to indicate that the population correlation i different from zero.

3 (a) The hypothee are H : p = (no linear relationhip between cob weiht and denity) HA: pi: (a linear relationhip between cob weiht and denity) The tet tatitic i t =r = = (-.9418) 2 Conultin Table 4 with df= 2-2 = 18, we find that t _ 5 = Thu, P <.1, o we reject H (with a=.5). There i tron evidence that the population correlation i different from zero; the evidence i that cob weiht and plant denity are linearly related. (b) Thi tudy wa an obervational tudy. The plant denitie were merely oberved, not manipulated by the reearcher. (c) A an obervational tudy with a tatitically inificant correlation between cob weiht and plant denity, we can only ay that there i an aociation between weiht and denity. We cannot make any caual connection. That i, we do not have evidence that alterin the plant denity will, in turn, alter cob weiht (a) (a) There are 12-2 = 1 deree of freedom (b)t = r = = ( ) 2 (c) We reject H (with a=.5). There i tron evidence that the population correlation i different from zero; the evidence i that acid concentration and funu rowth are linearly related (a) The lope and intercept of the rereion line are ) bl= ( = ' b = 224.l - (-.726)(128.5) = The fitted rereion line i y = X. Se ;::; SY~= ~ = 8.4 m. (b) A plant denity increae by l plant per plot, cob weiht decreae by.72 m of rain per cob, on averae. SS(reid) ~ l337.3 ~ 8. 6 m. (c) e = n _ 2 18 The approimate value of 8.4 m found in part (a) i quite cloe to the value of 8.6 m computed here. 2.

4 (d) Prediction of cob weiht baed on the rereion model tend to be off by 8.6 m, on averae. We could alo ay that the data point deviate above or below the rereion line by 8.6 m, on averae. (e) r 2 = (-.9418) 2 =.887, o 88.7% of the variation in cob weiht i eplained by the linear relationhip between cob weiht and denity (a) The lope and intercept of the rereion line are b = ) = -.712; I ( b = (-.712)(11.5) = The fitted rereion line i y = X. (b) and (d) - t"'l '-' -5 VI ~... :::: VI Zl ::I Of) ::I - µ;,.. I Se Laetiaric Acid (µ/ml) (c). SS(reid) = / = 1. 3 mm. n (a) (See Eercie for b and b 1.) (i) Etimated mean= (. 726)(1} = m. (ii) Etimated mean= (.726)(12) = m. (b) (i) (244.34)(1) = m:::: k. (ii) ( ))(12) = 27591m::::27.6 k Th (.) d c ) h SS(reid) ~ n we require that the tandard deviation remain contant over the rane of-value... e anwer to 1 an 11 are t e ame:. = = --= In our linear model,

5 . ( ) / (a) The ample lope i b 1 = = and = --= e J 8 calculate the tandard error: We need to SE = e b, r----:, 'I/ n " l332ji9 =.65. The CI i ± (2.11)(.65) (df= 18) < p 1 < (b) We are 95% confident that a plant denity increae by 1 plant per plot, averae cob weiht decreae by between.848 and.594 m of rain per cob.. 2 l (a) From Eercie ,. = = SEb I 1.3 = =.36. S ~ 1.884"'1! (b) The hypothee are Ho: /31 = HA:/31< (c) There are 12-2 = 1 deree of freedom (d) t = = The tandard error of the lope i (e) We reject H (with a=.5). There i tron evidence that the population lope i neative, i.e., that laetiaric acid inhibit rowth of the funu (a) To contruct a 95% confidence interval we conult Table 4 with df = 7; the critical value i t = The confidence interval i 7, ± (2.365)(.9531) 4.94 < P, < 9.45 (b) We are 95% confident that a nake lenth increae by 1 cm, averae weiht increae by between 4.94 and 9.45.