Given that the points three, negative nine and two negative, four lie on a regression line 𝑦 on 𝑥, which of the following points does not lie on the same line?
The first thing we can do is use these two points to calculate a function for this line. To do that, we’ll need the slope and the 𝑦-intercept. The formula for calculating the slope given two points is 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. We’ll label our points 𝑥 one, 𝑦 one; 𝑥 two, 𝑦 two. And we get the slope equals four minus negative nine over two minus three. Four minus negative nine equals five. Two minus three equals negative one. And our slope here equals negative five.
Our formula now says 𝑦 equals negative five 𝑥 plus 𝑏. where 𝑏 is a 𝑦-intercept. To find 𝑏 we plug in a point that we already know falls on the line — here; three negative, nine — and then solve for 𝑏. By adding 15 to both sides, I see that the 𝑏 value equals six. And that makes the function for this line of regression 𝑦 equals negative five 𝑥 plus six. From there, we’ll consider each case, a through d. Is negative 69 equal to negative five times 16 plus six? Does 56 equal negative five times negative 10 plus six, negative 54 equal to negative five times 12 plus six, and negative 94 equal to negative five times 20 plus six?
For the first option, if we multiply negative five times 16 plus six, we do not get negative 69. It’s equal to negative 74. If we go back and look at the instructions, we’re looking for the point that does not fall on the same line. Because negative five times 16 plus six doesn’t equal negative 69, 16, negative 69 is not a point on this line of regression. Looking at the other three options, negative five times negative 10 equals 50 plus six equals 56. Negative five times 12 equals negative 60 plus six equals negative 54. And negative five times 20 equals negative 100 plus six equals negative 94.
This verifies that the other three options do fall on the line of regression. The one that does not fall on the same line is 16, negative 69.