Question

Do heavier cars really use more gasoline? Suppose a car is chosen at random. Let x be the weight of the car (in hundreds of pounds), and let y be the miles per gallon (mpg).

x	29	47	29	47	23	40	34	52
y	29	20	25	13	29	17	21	14

given Σx = 301, Σy = 168, Σx² = 12,089, Σy² = 3802, Σxy = 5906, and r ≈ −0.907.

Find x, and y. Then find the equation of the least-squares line  = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)

=

+ ???? x

Find the value of the coefficient of determination r². What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r² to three decimal places. Round your answers for the percentages to one decimal place.)

r2 =
explained	? %
unexplained	?%

(f) Suppose a car weighs x = 31 (hundred pounds). What does the least-squares line forecast for y = miles per gallon? (Round your answer to two decimal places.)
? mpg

Answer 1

X	Y	XY	X²	Y²
29	29	841	841	841
47	20	940	2209	400
29	25	725	841	625
47	13	611	2209	169
23	29	667	529	841
40	17	680	1600	289
34	21	714	1156	441
52	14	728	2704	196

	X	Y	XY	X²	Y²
total sum	301.000	168.000	5906.00	12089.000	3802
mean	37.6250	21.0000

sample size ,   n =   8          
here, x̅ =Σx/n =   37.62 ,   ȳ = Σy/n =   21  
                  
SSxx =    Σx² - (Σx)²/n =   763.875          
SSxy=   Σxy - (Σx*Σy)/n =   -415.000          
SSyy =    Σy²-(Σy)²/n =   274.000          
estimated slope , ß1 = SSxy/SSxx =   -415.000   /   763.875   =   -0.5433
                  
intercept,   ß0 = y̅-ß1* x̄ =   41.4410          
                  
so, regression line is   Ŷ =   41.441   +   -0.543   *x

...........

R² =    (Sxy)²/(Sx.Sy) =    0.823
explained = 82.3%

unexplained = 17.7%

..............

Predicted Y at X=   31   is                  
Ŷ =   41.441   +   -0.543   *   31   =   24.60

.................

THANKS

revert back for doubt

please upvote