Question

1. We try to study the relationship between a personality scale that measures willingness to take orders (on a scale from 0 to 20, where 20 represents eagerness to take orders) and the number of days served in county prison for misdemeanors.  Here are the data from a sample of six prisoners:

Prisoner	Willing-ness	Days Served	(x-xbar)	(y-ybar)	(x-xbar)^2	(y-ybar)^2	(x-xbar)* (y-ybar)
Jake	20	4	9	-20	81	16	-180
Jason	3	65	-8	41	64	4225	-328
Sarah	10	20	-1	-4	1	400	4
D	11	9	0	-15	0	81	0
Dean	18	7	7	-17	49	49	-119
Joan	4	39	-7	15	49	1521	105
Sum	66	144	0	0	244	6292	-728
Mean	11	24	Variance==>		40.6666
Median	10.5	14.5	Standard Deviation=>		6.3770

Use the above table and briefly answer the following questions:

1. What is the Coefficient of Determination (R²) and what is this telling us?

1. What is the sample coefficient of correlation (r) and what does this tell us?

Answer 1

a.

X Values
∑ = 66
Mean = 11
∑(X - M_x)² = SS_x = 244

Y Values
∑ = 144
Mean = 24
∑(Y - M_y)² = SS_y = 2836

X and Y Combined
N = 6
∑(X - M_x)(Y - M_y) = -728

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = -728 / √((244)(2836)) = -0.8752

So r^2=-0.8752^2=0.7660

So 76.60% of variation in y is explained by x

b. X Values
∑ = 66
Mean = 11
∑(X - M_x)² = SS_x = 244

Y Values
∑ = 144
Mean = 24
∑(Y - M_y)² = SS_y = 2836

X and Y Combined
N = 6
∑(X - M_x)(Y - M_y) = -728

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = -728 / √((244)(2836)) = -0.8752

Hence we see that there is strong negative correlation between x and y