Question

The equation of a regression line, unlike the correlation, depends on the units we use to measure the explanatory and response variables. Here is the data on percent body fat and preferred amount of salt. Preferred amount of salt x 0.2 0.3 0.4 0.5 0.6 0.8 1.1 Percent body fat y 19 31 22 29 39 24 31 In calculating the preferred amount of salt, the weight of the salt was in milligrams. (a) Find the equation of the regression line for predicting percent body fat from preferred amount of salt when weight is in milligrams. (Round your answers to one decimal place.) ŷ = + x (b) A mad scientist decides to measure weight in tenths of milligrams. The same data in these units are as follows. Preferred amount of salt x 2 3 4 5 6 8 11 Percent body fat y 19 31 22 29 39 24 31 Find the equation of the regression line for predicting percent body fat from preferred amount of salt when weight is in tenths of milligrams. (Round your intercept to one decimal place and your slope to two decimal places.) ŷ = + x (c) Use both lines to predict the percent body fat from preferred amount of salt for a child with preferred amount of salt 0.9 when weight is measured in milligrams, which is the same as 9 when weight is in tenths of milligrams. (Round your answers to one decimal place.) in milligrams % body fat in tenths of milligrams % body fat Are the two predictions the same (up to any roundoff error)? Yes No

Answer 1

a)

X	Y	XY	X²	Y²
0.2	19	3.8	0.04	361
0.3	31	9.3	0.09	961
0.4	22	8.8	0.16	484
0.5	29	14.5	0.25	841
0.6	39	23.4	0.36	1521
0.8	24	19.2	0.64	576
1.1	31	34.1	1.21	961
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
3.9	195	113.1	2.75	5705

Sample size, n =	7
x̅ = Ʃx/n = 3.9/7 =	0.55714286
y̅ = Ʃy/n = 195/7 =	27.8571429
SSxx = Ʃx² - (Ʃx)²/n = 2.75 - (3.9)²/7 =	0.57714286
SSyy = Ʃy² - (Ʃy)²/n = 5705 - (195)²/7 =	272.857143
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 113.1 - (3.9)(195)/7 =	4.45714286

Slope, b = SSxy/SSxx = 4.45714/0.57714 = 7.7227723

y-intercept, a = y̅ -b* x̅ = 27.85714 - (7.72277)*0.55714 = 23.554455

Regression equation :

ŷ = 23.6 + (7.7) x

b) Slope, b = SSxy/SSxx = 44.57143/57.71429 = 0.772277228

y-intercept, a = y̅ -b* x̅ = 27.85714 - (0.77228)*5.57143 = 23.55445545

Regression equation :   

ŷ = 23.6 + (0.77) x  

-------

c) Predicted values:

In milligrams :

ŷ = 23.6 + (7.7) * 0.9 = 30.5 % body fat.

In tenth of milligrams:

ŷ = 23.6 + (0.77) * 9 = 30.5% body fat

Yes, the two predictions are the same.