Question

Compute the appropriate regression equation for the following data:
X Y
2 18
4 12
5 10
6 8
8 7
11 5

Where X is the independent variable and Y is the dependent variable.
Also find the correlation coefficient between X and Y and infer about the
relationship between X and Y.

Answer 1

Question #65454, Math / Statistics and Probability

Compute the appropriate regression equation for the following data:

Where $X$ is the independent variable and $Y$ is the dependent variable.

Also find the correlation coefficient between $X$ and $Y$ and infer about the relationship between $X$ and $Y$.

**Answer.**

$$\sum X = 36, \quad \sum Y = 60, \quad \sum X^2 = 266, \quad \sum Y^2 = 706, \quad \sum XY = 293.$$

Regression equation $y = a + bx$.

Where $a = \frac{\sum Y\sum X^2 - \sum X\sum XY}{n\sum X^2 - (\sum X)^2} = \frac{60*266 - 36*293}{6*266 - 36^2} \approx 18.04,$

$$b = \frac{n\sum XY - \sum X\sum Y}{n\sum X^2 - (\sum X)^2} = \frac{6*293 - 36*60}{6*266 - 36^2} \approx -1.34.$$

So $y = 18.04 - 1.34x$.

**Correlation coefficient**

$$r = \frac{n\sum XY - \sum X\sum Y}{\sqrt{n\sum X^2 - (\sum X)^2} \sqrt{n\sum Y^2 - (\sum Y)^2}} = \frac{6*293 - 36*60}{\sqrt{6*266 - 36^2} \sqrt{6*706 - 60^2}} \approx -0.9203.$$

There is a strong negative linear correlation between $X$ and $Y$.

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