Question

A used car dealer wants to develop a regression equation that determines mileage as a function of the age of a car in years. He collects the data shown below for the 12 cars he has on his lot. Assignment 1094 data a) What is the slope of the regression equation? Give your answer to two decimal places. | b) What is the value of the correlation coefficient? Give your answer to two decimal places. c) A 4 year old car is delivered to his lot with 160000 miles. Manually enter these values in the data table above and rerun the regression analysis. What is the value of the slope? Give your answer to two decimal places. d) Including the additional car, what is the value of the correlation coefficient? Give your answer to two decimal places. e) Did the additional car strengthen or weaken the linear relationship between age and mileage? Can not be determined. It strengthened the linear relationship. It weakened the linear relationship.

Age	Mileage
7	52654
4	49300
13	113671
7	63001
7	69850
7	68048
12	106979
8	82719
7	80894
10	109258
14	149804
8	98915

Answer 1

Solution: The following data are passed: Age Mileage 52654 7 4 49300 13 113671 7 63001 7 69850 7 68048 12 106979 8 82719 7 80894 10 109258 14 149804 8 98915 The Independent variable is Age, and the dependent variable is Mileage. In order to compute the regression coefficients, the following table needs to be used: Age Mileage Age Mil eage Age" Mileage 2 277244 7 52654 368578 49 3716 4 49300 197200 16 243049 0000 13 113671 147772 3 169 129210 96241 7 63001 441007 49 396912 6001 7 69850 488950 49 487902 2500 68048 476336 463053 0304 12 106979 128374 8 144 114445 06441 8 82719 661752 64 684243 2961 7 80894 566258 49 654383 9236 10 109258 109258 0 100 119373 10564 14 149804 209725 6 196 224412 38416 8 98915 791320 64 978417 7225 Sum - 104 104509 3 994270 8 998 100596 213605 Based on the above table, the following is calculated: 1. = 104 12 = 8.6666666666667 T2 71 Y = - 72 Y 1045093 12 = 87091.083333333 2 1 SSxxx = 998 1042/12 = 96.666666666667 n =] i=1 2 11 1 SSyy-Y 100596213605 - 10450932/12 = 9577932050.9167 72 1 SSχY - ΣX,Y, - 3xx -- (*) ) 9942708 - 104 x 1045093/12-885235.33333333

Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows: m = SSXY SSxx 885235.33333333 96.666666666667 =9157.6069 n = Y - im · m = 87091.083333333 - 8.6666666666667 x 9157.6069 = 7725.1569 Therefore, we find that the regression equation is: Mileage = 7725.1569 + 9157.6069 Age Graphically: Scatter Plot and Regression Line 19199.20 19199.20 19199.20 1919.20 191-919.20 19199.20 9199.20 9199.202 4 6 8 10 12 14 16 Age Regression equation: Y = 7725.1569 + 9157.6069 * X
a) slope of the regression equation is = 9157.6

b)

The correlation coefficient r is computed using the following expression: Υ - SSXY SSxx SSYY where 1 SSry - ΣΧΥ (Σκ) (ΣΥ) n 1=1 2 η 1 SSxx ΣΧ? (Σ» n 1=1 i=1 2 SSYy = ΣΥ? Σ' (Σ) In this case, based on the data provided, we get that 1 SSXY = 9942708 (104 Χ 1045093) 12 885235.333 1 SSxx 998 (104)? = 96.667 12 1 SSYY = 100596213605 (1045093)? - 9577932050.917 12 Therefore, based on this information, the sample correlation coefficient is computed as follows 885235.333 r SSXY SSXX SSyy = 0.92 96.667 x 9577932050.917

Solution: The following data are passed: Age Mileage 7 52654 4 49300 13 113671 7 63001 7 69850 7 68048 12 106979 8 82719 7 80894 10 109258 14 149804 8 98915 4 16000 The independent variable is Age, and the dependent variable is Mileage. In order to compute the regression coefficients, the following table needs to be used: Age Mileage Age*Mil eage Age? Mileage 2 7 52654 368578 49 277244 3716 4 49300 197200 16 243049 0000 13 113671 147772 3 169 129210 96241 7 63001 441007 49 396912 6001 7 69850 488950 49 487902 2500 7 68048 476336 49 463053 0304 12 106979 128374 8 144 114445 06441 8 82719 661752 64 684243 2961 7 80894 566258 49 49 654383 9236 10 109258 109258 0 100 119373 10564 14 149804 209725 6 196 224412 38416 8 98915 791320 64 978417 7225 4 16000 64000 16 256000 000 Sum - 108 106109 3 100067 08 1014 100852 213605 Based on the above table, the following is calculated: X- 108 13 8.3076923076923 n =1 12 1 1061093 Y 81622.538461538 92 =1 13 2 TL SSXX = > X = 1014 - 1082/13 = 116.76923076923 12

the value of the slope = 10203.6627

2 SSxx = ΣΧ? ΣΧ; = 1014 - 1082 /13 = 116.76923076923 1=1 2 1 = 100852213605 – 10610932 /13 = 14243109401.231 5ης - Σ - (Σ.) ΣΧ. - (Σκ) (Σ SSXY ΣΥ, = 10006708 - 108 x 1061093/13 = 1191473.8461538 1=1 Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows: SSXY η = 1191473.8461538 116.76923076923 10203.6627 SSxx η =Y – Χ. η • m = 81622.538461538 – 8.3076923076923 x 10203.6627 = -3146.3518 Therefore, we find that the regression equation is: Mileage =-3146.3518 + 10203.6627Age Graphically: Scatter Plot and Regression Line 19239.20 1929.20 19239.20 4 6 8 ΠΟ 12 14 15 .0760.80 Age Regression equation: Y = -3146.3518 + 10203.6627 * Χ

d)
Solution: The provided data are shown in the table below X Y 7 52654 4 49300 13 113671 7 63001 7 69850 7 68048 12 106979 8 82719 7 80894 10 109258 14 149804 8 98915 4 16000 Also, the following calculations are needed to compute the correlation coefficient: х Y X*Y X2 Y2 7 52654 368578 49 277244 3716 4 49300 197200 16 243049 0000 13 113671 147772 3 169 129210 96241 7 63001 441007 49 396912 6001 7 69850 488950 49 487902 2500 7 68048 476336 49 463053 0304 12 106979 128374 8 144 114445 06441 8 82719 661752 64 684243 2961 7 80894 566258 49 654383 9236 109258 10 109258 100 119373 10564 0 14 149804 209725 6 196 224412 38416 00 98915 791320 64 978417 7225 4 16000 64000 16 256000 000 Sum = 108 106109 3 100067 08 1014 100852 213605

r = S8xx = x 2) The correlation coefficient r is computed using the following expression: SSXY VSSxx SSyy where SSXY x;y; ΣΧ; ΣΥ; ΣΧ; n 1 = i=1 2 n n 1 n SSyy = Y i=1 In this case, based on the data provided, we get that 1 SSXY = 10006708 (108 x 1061093) 13 1191473.846 SSxx 1014 1 (108)? = 116.769 13 1 S Syy 100852213605 (1061093)2 = 14243109401.231 13 Therefore, based on this information, the sample correlation coefficient is computed as follows r = SSXY SSXX SSyy 1191473.846 V116.769 x 14243109401.231 0.924
e) It weakened the linear realtionship.