3. What percent of the variability in sodium is explained by calories? The Excel

Question

3. What percent of the variability in sodium is explained by calories?

The Excel analysis below shows a plot of calories (x) and sodium (y) for meat hot dogs. Also shown are the means, standard deviations, and correlation of the variables. Use this information to answer questions 1 through 4 Calories Sodium 375 450 500 325 400 525 450 300 450 550 500 120 155 160 140 150 175 180 135 140 160 175 600 500 6 400 300 200 100 120 140 60 180 200 Calories mean 153.636 438.636 st. dev 18.854 80.904 corr 0.710

Explanation / Answer

Line of Regression Y on X i.e Y = bo + b1 X

calculation procedure for regression

mean of X = X / n = 153.6364

mean of Y = Y / n = 438.6364

(Xi - Mean)^2 = 3554.541

(Yi - Mean)^2 = 65454.54

(Xi-Mean)*(Yi-Mean) = 10829.541

b1 = (Xi-Mean)*(Yi-Mean) / (Xi - Mean)^2

= 10829.541 / 3554.541

= 3.047

bo = Y / n - b1 * X / n

bo = 438.6364 - 3.047*153.6364 = -29.444

value of regression equation is, Y = bo + b1 X

Y'=-29.444+3.047* X

calculation procedure for correlation
sum of (x) = x = 1690
sum of (y) = y = 4825
sum of (x^2)= x^2 = 263200
sum of (y^2)= y^2 = 2181875
sum of (x*y)= x*y = 752125
to caluclate value of r( x,y) = covariance ( x,y ) / sd (x) * sd (y)
covariance ( x,y ) = [ x*y - N *(x/N) * (y/N) ]/n-1
= 752125 - [ 11 * (1690/11) * (4825/11) ]/11- 1
= 984.504
and now to calculate r( x,y) = 984.504/ (SQRT(1/11*752125-(1/11*1690)^2) ) * ( SQRT(1/11*752125-(1/11*4825)^2)
=984.504 / (17.976*77.139)
=0.71
value of correlation is =0.71
coeffcient of determination = r^2 = 0.504
50.4% percent of the variability in sodium is explained by calories

X Y (Xi - Mean)^2 (Yi - Mean)^2 (Xi-Mean)*(Yi-Mean) 120 375 1131.4074 4049.5914 2140.4994 155 450 1.8594 129.1314 15.4954 160 500 40.4954 3765.4914 390.4934 140 325 185.9514 12913.231 1549.5914 150 400 13.2234 1492.7714 140.4974 175 525 456.4034 7458.6714 1845.0374 180 450 695.0394 129.1314 299.5854 135 300 347.3154 19220.051 2583.6834 140 450 185.9514 129.1314 -154.9586 160 550 40.4954 12401.851 708.6734 175 500 456.4034 3765.4914 1310.9474

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