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 29 22 30 38 24 29

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 29 22 30 38 24 29

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

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
0.2	19	0.1276	68.6531	2.9592
0.3	29	0.0661	2.9388	-0.4408
0.4	22	0.0247	27.9388	0.8306
0.5	30	0.0033	7.3673	-0.1551
0.6	38	0.0018	114.7959	0.4592
0.8	24	0.0590	10.7959	-0.7980
1.1	29	0.2947	2.9388	0.9306

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	3.90	191.00	0.58	235.43	3.79
mean	0.56	27.29	SSxx	SSyy	SSxy

a)

sample size ,   n =   7          
here, x̅ = Σx / n=   0.557   ,     ȳ = Σy/n =   27.286  
                  
SSxx =    Σ(x-x̅)² =    0.5771          
SSxy=   Σ(x-x̅)(y-ȳ) =   3.8          
                  
estimated slope , ß1 = SSxy/SSxx =   3.8   /   0.577   =   6.55941
                  
intercept,   ß0 = y̅-ß1* x̄ =   23.63119          
                  
so, regression line is   Ŷ =   23.6 +   6.6 *x

b)  Ŷ =   23.63 +   0.66 *x

c)

Predicted Y at X=   0.9   is          
Ŷ =   23.6312   +   6.5594   *0.9=   29.535
---------------

Predicted Y at X=   9   is          
Ŷ =   23.6312   +   0.6559   *9=   29.535

Are the two predictions the same (up to any roundoff error)?

Yes