ASSIGNMENT NO. 1:- STA301 SPRING 2022 DUE DATE: JUNE 25, 2022

STA301

SPRING 2022

DUE DATE: JUNE 25, 2022

QUESTION NO. 1:-

From the given data, calculate regression equation X on Y taking deviation from mean.

X	2	5	3	7	6
Y	8	3	1	4	2

SOLUTION:-

We know that:

$\begin{array}{l} X\limits^ \wedge = a + bY\\ \left( {X - X\limits^ - } \right) = {b_{xy}}\left( {Y - Y\limits^ - } \right) \end{array}$

Formula:-

${b_{xy}} = \frac{{\sum \left( {X - X\limits^ - } \right)\left( {Y - Y\limits^ - } \right)}}{{\sum {{\left( {Y = Y\limits^ - } \right)}^2}}}$

$\,\,\, = \frac{{\sum \left( {xy} \right)}}{{\sum {y^2}}}$

$\left( {X - X\limits^ - } \right) = {b_{xy}}\left( {Y - Y\limits^ - } \right)$

X	Y	x = $\left( {X - X\limits^ - } \right)$	y = $\left( {Y - Y\limits^ - } \right)$	xy	y²
2	8	-2.6	4.4	-11.44	19.36
5	3	0.4	-0.6	-0.24	0.36
3	1	-1.6	-2.6	4.16	6.76
7	4	2.4	0.4	0.96	0.16
6	2	1.6	-1.6	-2.24	2.56
$\sum$ x = 23	$\sum$ y = 18	$\sum$ $\left( {X - X\limits^ - } \right)$ = 0	$\sum$ $\left( {Y - Y\limits^ - } \right)$ = 0	$\sum$ xy = -8.8	$\sum$ y²= 29.2

$X\limits^ - = \frac{{\sum X}}{n}\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\, Y\limits^ - = \frac{{\sum Y}}{n}\,\,\,\,$

$= \frac{{23}}{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{18}}{5}\,\,$

$= 4.6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 3.6$

$\left( {X - X\limits^ - } \right) = {b_{xy}}\left( {Y - Y\limits^ - } \right)$

$\left( {X - 4.6} \right) = \frac{{ - 8.8}}{{29.2}}\left( {Y = 3.6} \right)$

QUESTION NO. 2

From the given data, find standard error of estimate for regression equation Y on X.

X	5	7	2	9	3	8
Y	4	6	3	1	5	2

SOLUTION:-

We know that:-

Standard error Formula:

${S_{yx}} = \sqrt {\frac{{\sum {Y^2} - a\sum Y - b\sum XY}}{{n - 2}}}$

X	X²	Y	Y²	XY
5	25	4	16	20
7	49	6	36	42
2	4	3	9	6
9	81	1	1	9
3	9	5	25	15
8	64	2	4	16
$\sum$ X = 34	$\sum$ X² = 232	$\sum$ Y = 21	$\sum$ Y²= 91	$\sum$ XY = 108

Formula:-

$\sum Y = na + b\sum X\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sum XY = a\sum X + b\sum {X^2}$

$21 = 6a + 34b\,\,\,\, \to \left( 1 \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,108 = 34a + 232b\,\,\,\,\, \to \left( 2 \right)$