We have an answer to "Find the regression line using the given points."

usagirl007A

Answered question

2021-03-06

Find the regression line using the given points.

Answer & Explanation

Jayden-James Duffy

Skilled2021-03-07Added 91 answers

Given:
The given bivariate data is(1,6),(3,2), and (2,4).
Calculation:
First determine $\sum x_{i} y_{i}, \sum x_{i} and \sum y_{i}$ .
$x_{i} \to$ first coordinates of the ordered pairs.
$y_{i} \to$ second coordinates of the ordered pairs.
$\sum x_{i} y_{i} = 1 \cdot 6 + 3 \cdot 2 + 2 \cdot 4 = 20$
$\sum x_{i} = 1 + 3 + 2 = 6$
$\sum x_{i}^{2} = 1^{2} + 3^{2} + 2^{2} = 14$
$\sum y_{i} = 6 + 2 + 4 = 12$
$\sum y_{i}^{2} = 6^{2} + 2^{2} + 4^{2} = 56$
Find the sample variance $s^{2}$ using the formula $s^{2} = \frac{\sum x_{i}^{2} - \frac{\sum x_{i}^{2}}{n}}{n - 1}$
$s_{x}^{2} = \frac{14 - \frac{6^{2}}{3}}{3 - 1} = \frac{14 - 12}{2} = 1$
$s_{y}^{2} = \frac{56 - \frac{12^{2}}{3}}{3 - 1} = \frac{56 - 48}{2} = 4$
Find sample standard deviation.
$s_{x} = \sqrt{1} = 1$
$s_{y} = \sqrt{4} = 2$
For covariance $s_{x y}$ using formula $s_{x y} = \frac{\sum x_{i} y_{i} - \frac{\sum x_{i} - \sum y_{i}}{n}}{n - 1}$ .
Where $n =$ number of ordered pairs.
$n = 3$
$s_{x y} = \frac{20 - \frac{6 \cdot 12}{3}}{3 - 1}$
$= \frac{20 - \frac{72}{3}}{3 - 1}$
$= \frac{20 - 24}{2}$
$= - \frac{4}{2} = - 2$
Find the correlation coefficient r using the formula $r = \frac{s_{x y}}{s_{x} s_{y}}$ .
$r = - \frac{2}{1 \cdot 2} = - 1$
Find the slope b using the formula $b = r \cdot \frac{s_{y}}{s_{x}}$
$b = - 1 \cdot \frac{2}{1} = - 2$
Find the value of y-intercept.

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