Question

Probability and Statistics

1. Linear Regression Given 4 data points: X Y 5 15 Use simple linear regression to estimate ßo and ß, for the best-fit line ỹ ß0 + ßqx Calculate these values: x | 7 | S | Spy | Bo | Big Sketch the regression line and the data points below

Answer 1

We are given data:

X	Y	$X^2$	$Y^2$	$XY$
1	5	1	25	5
2	7	4	49	14
3	9	9	81	27
4	15	16	225	60
		$\sum X^2~=~30$	$\sum Y^2~=~380$	$\sum XY~=~106$

Average of X = $\bar{X}~=~\frac{1+2+3+4}{4}~=~2.5$

Average of Y = $\bar{Y}~=~\frac{5+7+9+15}{4}~=~9$

Now,

$S_{xx}$:

X	$X-\bar{X}$	$\left (X-\bar{X} \right )^2$
1	-1.5	2.25
2	0.5	0.25
3	0.5	0.25
4	1.5	2.25
		$\sum\left (X-\bar{X} \right )^2~=~5$

So,

$S_{xx}~=~5$

For, $S_{xy}$:

Y	$Y-\bar{Y}$	$\left (X-\bar{X} \right )^2$
5	-4	16
7	-2	4
9	0	0
15	6	36
		$\sum\left (Y-\bar{Y} \right )^2~=~56$

We know that $\hat \beta_{0}~\&~\hat{\beta_{1}}$ are the coefficients of the linear equations:

$\hat{\beta_{1}}$ is the slope of the regression line which is nothing but the ratio of $S_{xy}$ and $S_{xx}$

So,

$\hat\beta_{1}~=~\frac{n\left ( \sum{XY} \right )-\left ( \sum{X} \right )\left ( \sum{Y} \right )}{n\left ( \sum{X^2} \right )- \left ( \sum X \right )^2}~=~\frac{424-360}{120-100}~=~\frac{64}{20}~=~3.2$

and $\hat{\beta_{0}}$ is the intercept for managing the differences:

$\hat\beta_{0}~=~\frac{\left ( \sum{Y} \right )\left ( \sum X^2 \right )-\left ( \sum{X} \right )\left ( \sum{XY} \right )}{n\left ( \sum{X}^2 \right )-\left ( \sum{X} \right )^2}~=~\frac{20}{20}~=~1$

Thus, the table obtained as:

$\bar{X}$	$\bar{Y}$	$S_{xx}$	$S_{xy}$	$\hat{\beta{0}}$	$\hat{\beta{1}}$
2.5	9	5	56	1	3.2

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