This table shows 10 observations of a pair of variables (x,y). The variables x and y are positively correlated, with a correlation coefficient of 0.977. Find the slope, b, of the least squares regression line, y=ax+b, for these data:
X 1 2 3 4 5 6 7 8 9 11
Y 2 4 5 8 9 12 13 14 15 16

Question

This table shows 10 observations of a pair of variables (x,y).  The variables x and y are positively correlated, with a correlation coefficient of 0.977.  Find the slope, b, of the least squares regression line, y=ax+b, for these data:
X	1	2	3	4	5	6	7	8	9	11
Y	2	4	5	8	9	12	13	14	15	16

anonymous · Answer

.The coefficients "a" and "b" for least squares regression may be found as follows:

a = [n∑(xy) - ∑(x)∑(y)]/[n∑(x²) - ∑(x)²]

b = [∑(x²)∑(y) - ∑(x)∑(xy)]/[n∑(x²) - ∑(x)²]


1. For the equation y = ax + b:

a = [(10)(688) - (56)(98)]/[(10)(406) - (56)²] = 1.5065

b = [(406)(98) - (56)(688)]/[(10)(406) - (56)²] = 1.3636


2. For the equation y = ax + b:

a = [(4)(68) - (10)(21)]/[(4)(30) - (10)²] = 3.1000

b = [(30)(21) - (10)(68)]/[(4)(30) - (10)²] = -2.5000

anonymous · Answer

1. The slope, actually, is a, by the way.

y = 1.506a + 1.364

So the slope is 1.506.

2. y = 3.1x - 2.5

The slope is 3.1.