Question

The excel file gives the city and highway gas mileage for 21 two-seater cars, including the Leaf hybrid car:

Type	City	Hwy
T	17	24
T	20	28
T	20	28
T	17	25
T	18	25
T	12	20
T	11	16
T	10	16
T	17	23
T	60	66
T	9	15
T	9	13
T	15	22
T	12	17
T	22	28
T	16	23
T	13	19
T	20	26
T	20	29
T	15	23
T	26	32
M	12	19
M	21	29
M	19	27
M	19	28
M	16	23
M	18	26
M	16	23
M	18	23
M	25	32
M	23	31
M	20	29
M	18	26
M	14	22

Make a scatterplot of highway mileage y against city mileage x for all 21 cars. There is a strong positive linear association. The Leaf lies far from the other points. Does the Leaf extend the linear pattern of the other cars, or is it far from the line they form?
(b) Find the correlation between city and the highway mileage both without and with the Leaf. Based on your answer to (a), explain why r changes in this direction when you add the Leaf.

Please, give a detailed answer. Thank you!

Answer 1

a)

Yes the Leaf extend the linear pattern of the other cars.

b)

Without Leaf

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	379	518	2256.952381	2324.7	2275.33
mean	18.05	24.67	SSxx	SSyy	SSxy

sample size ,   n =   21          
here, x̅ = Σx / n=   18.05   ,     ȳ = Σy/n =   24.67  
                  
SSxx =    Σ(x-x̅)² =    2256.9524          
SSxy=   Σ(x-x̅)(y-ȳ) =   2275.3          
                  
estimated slope , ß1 = SSxy/SSxx =   2275.3   /   2256.952   =   1.0081
                  
intercept,   ß0 = y̅-ß1* x̄ =   6.4721          
                  
so, regression line is   Ŷ =   6.4721   +   1.0081   *x
                  
SSE=   (SSxx * SSyy - SS²xy)/SSxx =    30.803          
                  
std error ,Se =    √(SSE/(n-2)) =    1.273          
                  
correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.9934         

With Leaf:

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	618	856	2404.941176	2514.9	2433.94
mean	18.18	25.18	SSxx	SSyy	SSxy

sample size ,   n =   34          
here, x̅ = Σx / n=   18.18   ,     ȳ = Σy/n =   25.18  
                  
SSxx =    Σ(x-x̅)² =    2404.9412          
SSxy=   Σ(x-x̅)(y-ȳ) =   2433.9          
                  
estimated slope , ß1 = SSxy/SSxx =   2433.9   /   2404.941   =   1.0121
                  
intercept,   ß0 = y̅-ß1* x̄ =   6.7808          
                  
so, regression line is   Ŷ =   6.7808   +   1.0121   *x
                  
SSE=   (SSxx * SSyy - SS²xy)/SSxx =    51.650          
                  
std error ,Se =    √(SSE/(n-2)) =    1.270          
                  
correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.9897          

R changes a bit in negative direction(less) when we added the Leaf because few of the hybrid x can not be explained by the model.

Please revert in case of any doubt.

Please upvote. Thanks in advance