Question

28 40 25 40 41 15 3. A farmer would like to investigate the relationship between the obtained yield of apple trees and the amount of weeds found in their roots. For this reason, nine apple trees of the same type were randomly selected and the amount of weeds in their roots (7 grams) was recorded together with their yield (y kilograms). Year #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Weeds in roots (2) 30 32 25 25 24 22 24 35 Yield (y) 30 27 42 50 30 25 The summary statistics for these data are: Sum of x data: 285 Sum of the squares of x data: 8419 Sum of y data: 355 Sum of the squares of y data: 13349 Sum of the products of u and y data: 9718 (a) i. Draw a scatter diagram of these data. Label the diagram carefully. ii. Calculate the sample correlation coefficient. Interpret your findings. iii. Calculate the least squares line of y on x and draw the line on the scatter diagram. iv. Based on the regression equation in part (iii.), what will be the predicted yield of apple for a tree with 37 grams of weeds in its roots? Will you trust this value? Justify your answer. v. Briefly comment on the suitability of the simple linear regression model for the data of this question.

Answer 1

(a). Scatterplot and fitted least square line.
X SCATTERPLOT y = -1.3474x + 73.901 R2 = 0.7211 30 25 28 60 30 27 32 50 25 40 42 25 40 Yield 30 24 22 24 41 50 45 20 35 30 10 40 25 o 0 5 10 15 20 25 30 35 40 45 Weeds in roots

(11) formula for Sample Correlation coefficient is given og 94 - n (Exy) - (en) (sy) V (n Ex?- (8x) =) ( nsy" (sy)) 10(9718) (285)(355) (10C 8419) - (285)° ) (10013349) -(355)?) 971 80 - 101175 (84190 - 81225) (133490 126025) 91 = -3995 -3995 2965) (7465) 22133725 94 S 3995 4704.65 HE 0.8498 Since correlation 1 and yields Coefficient between weeds in root comes out to be this an Inveuse relation negative between means there is $ and of Strong Magnitude ()

we have to find Regression Equation to confch is given by y* - Bo + Boxe where Bo 8 Intercept Bi : slope we have to find po and Bi conich are given by ßi = n sny (Encey n (Ena) -({x)

and B.Ex Bo -हि - u

putting values Bi 10/9718) - (285)(355) 1018419) - (285) 2 Bi = 97180- 10175 84190 81225 Bi - 3995 2965 . B = 0 -1.347

and, Во 2 3SS (-1-347) 28s) 10 Во 11 35S + 383. 24S Bo +39. ООЯ To В. +3.9 с. - Tree b же regression Equation becomes. yo 13.9 - 1.34

(1) we x= 37 are that given puting value in cussion Equation. reg 73.9 1.347 (37) go = 73.9 - 49.839 = ya 94.061 weeds in it rools is with 37 There fore predicted yield foor apple, for a tree gooms of Cappoox) kilogram ( u) deleomination) of Since our R-squared (Coefficient) 1² (0.8492)² 0.72K. is value on is R-squared value that how much vasiahon gnifies Explained by Indepen depat Variable dependent variable how close dala fitted to line . ૦૪ the is x- There fore, regression 72.110% variation is explained consider it as a suitable fit. by So