Question

Consider a set of 10 data points:
x: 1 2 3 4 4 5 5 6 6 7
Y: 7 8 9 8 9 11 10 13 14 13
Derive the simple linear regression model

Answer 1

Answer on Question#71918 – Math – Statistics and Probability

Question. Consider a set of 10 data points:

Derive the simple linear regression model.

Solution. We must calculate the coefficients $a$ and $b$ in the equation $y = a + bx$. Now we calculate the next values:

$$\bar{x} = \frac{1 + 2 + 3 + 4 + 4 + 5 + 5 + 6 + 6 + 7}{10} = 4.3;$$ $$\bar{y} = \frac{7 + 8 + 9 + 8 + 9 + 11 + 10 + 13 + 14 + 13}{10} = 10.2;$$ $$\overline{xy} = \frac{1 \cdot 7 + 2 \cdot 8 + 3 \cdot 9 + 4 \cdot 8 + 4 \cdot 9 + 5 \cdot 11 + 5 \cdot 10 + 6 \cdot 13 + 6 \cdot 14 + 7 \cdot 13}{10} = 47.6;$$ $$\sigma_x^2 = \frac{(1 - 4.3)^2 + (2 - 4.3)^2 + (3 - 4.3)^2 + (4 - 4.3)^2 + (5 - 4.3)^2 + (6 - 4.3)^2 + (7 - 4.3)^2}{10} = 3.21.$$

Now we can obtain the coefficients using the following formulas:

(see https://www.easycalculation.com/statistics/learn-regression.php)

$$b = \frac{\overline{xy} - \bar{x} \cdot \bar{y}}{\sigma_x^2} = 1.165; \quad a = \bar{y} - b\bar{x} = 5.19.$$

The regression equation for the given data is $y = 5.19 + 1.165 \cdot x$.

Answer. $y = 5.19 + 1.165 \cdot x$.

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