Question

What is the correlation between the number of calories consumed and BMI?

What is the slope coefficient for the linear relationship between the number of calories consumed and BMI, assuming that caloric intake is the independent variable? Round answer to 3 decimal places.

What is the intercept for the linear relationship between the number of calories consumed and BMI, assuming caloric intake is the independent variable?

Which of the following can you conclude based on your analyses?a) consumption of more calories leads to greater BMI b) consuming more calories leads to lower BMI c) calorie consumption has no relationship

calories consumed	BMI
3000	31.0
2200	28.2
2540	26.3
1750	22.1
2005	25.3

Answer 1

a)

The provided data are shown in the table below

X	Y
3000	31.0
2200	28.2
2540	26.3
1750	22.1
2005	25.3

Also, the following calculations are needed to compute the correlation coefficient:

	X	Y	X*Y	X2	Y2
	3000	31.0	93000	9000000	961
	2200	28.2	62040	4840000	795.24
	2540	26.3	66802	6451600	691.69
	1750	22.1	38675	3062500	488.41
	2005	25.3	50726.5	4020025	640.09
Sum =	11495	132.9	311243.5	27374125	3576.43

The correlation coefficient r is computed using the following expression:

r = ​​SS_XY / (​ SS_X * SS_Y )​

where

In this case, based on the data provided, we get that

SS_{XY} = 311243.5−​(11495×132.9)5 = 5706.4

SS_XX = 27374125−​(11495)2/5 = 947120

SS_YY =  3576.43−​(132.9)2/5 = 43.948

Therefore, based on this information, the sample correlation coefficient is computed as follows

r = 5706.4/sqrt(947120* 43.948 ) = 0.884

r = 0.884

which completes the calculation.

b) Slope and c) intercept

Xˉ = ∑Xi​ / n = 11495/5 ​= 2299

Yˉ = ∑Yi​ / n = 132.9/5 = 26.58

Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows:

m = SS_XY/SS_XX = 5706.4/ 947120 = 0.006

n =Yˉ − Xˉ*m = 26.58 − 2299×0.006 = 12.7285

Therefore, we find that the regression equation is:

BMI = 12.7285 + 0.006*Calorie Intake

Since the correlation and slope are positive that mean more the calorie intake greater the BMI