Name: Assumed Mean Method
Uploaded: 2019-07-19
Duration: 3 min 2 s
Description: What is the assumed mean method? How do we find out the deviations taken from the assumed mean? Watch this educative video to get all the answers!

Question

The following table show final grades of $10$ students in Algebra and Physics.

Algebra ( $x$ )	$75$	$80$	$93$	$65$	$87$	$71$	$98$	$68$	$84$	$77$
Physics ( $y$ )	$82$	$78$	$86$	$72$	$91$	$80$	$95$	$72$	$89$	$74$

Find both the least squares lines.

Accepted Answer

Let assumed mean $A = 80, B = 82$

Given data can be tabulated as

$x$ $y$ $u = x - A$ $v = y - B$ $u^{2}$ $v^{2}$ $u v$

$75$ $82$ $- 5$ $95$ $72$ $89$ $74$

$80$ $78$ $0$ $- 4$ $0$ $16$ $0$

$93$ $86$ $13$ $4$ $169$ $16$ $52$

$65$ $72$ $- 15$ $65$ $225$ $100$ $150$

$84$ $77$ $7$ $9$ $82$ $78$ $86$

$71$ $80$ $- 9$ $72$ $81$ $4$ $18$

$98$ $95$ $18$ $13$ $324$ $169$ $234$

$68$ $72$ $- 12$ $- 10$ $144$ $100$ $120$

$84$ $89$ $4$ $7$ $16$ $49$ $28$

$77$ $74$ $- 3$ $- 8$ $9$ $64$ $24$

$\sum_{}^{} x = 798$ $\sum_{}^{} y = 819$ $\sum_{}^{} u = - 2$ $\sum_{}^{} v = - 1$ $\sum_{}^{} u^{2} = 1042$ $\sum_{}^{} v^{2} = 599$ $\sum_{}^{} u v = 689$

Mean of $x, x = A + \frac{\sum_{}^{} u}{n} = 80 + \frac{- 2}{10} = 79.8$

Mean of $y, y = B + \frac{\sum_{}^{} v}{n} = 82 + \frac{- 1}{10} = 81.9$

$b_{y x} = \frac{\sum_{}^{} u v - \frac{1}{n} \sum_{}^{} u \sum_{}^{} v}{\sum_{}^{} u^{2} - \frac{1}{n} {(\sum_{}^{} u)}^{2}}, b_{x y} = \frac{\sum_{}^{} u v - \frac{1}{n} \sum_{}^{} u \sum_{}^{} v}{\sum_{}^{} v^{2} - \frac{1}{n} {(\sum_{}^{} v)}^{2}}$

$b_{y x} = \frac{689 - 0.1 (- 2) (- 1)}{1042 - 0.1 (- 2) (- 2)} = \frac{688.8}{1041.6} = 0.66$

$b_{x y} = \frac{689 - 0.1 (- 2) (- 1)}{599 - 0.1 (- 1) (- 1)} = \frac{688.8}{598.9} = 1.15$

Regression line of $y$ on $x$ is $y - y = b_{y x} (x - x)$

$y - 81.9 = 0.66 (x - 79.8) y = 0.66 x + 29.23$

Regression line of $x$ on $y$ is $x - x = b_{x y} (y - y)$

$(x - 79.8) = 1.15 (y - 81.9) x = 1.15 y - 14.4$

The following table show final grades of $10$ students in Algebra and Physics.

Algebra ( $x$ ) $75$ $80$ $93$ $65$ $87$ $71$ $98$ $68$ $84$ $77$

Physics ( $y$ ) $82$ $78$ $86$ $72$ $91$ $80$ $95$ $72$ $89$ $74$

Find both the least squares lines.

Important Questions on Linear Regression

Learn and Prepare for any exam you want !

Algebra (x)	$144$	$100$	$120$	$65$	$87$	$71$	$98$	$68$	$84$	$77$
Physics (y)	$77$	$74$	$- 3$	$72$	$91$	$80$	$95$	$72$	$89$	$74$

The following table show final grades of 10 students in Algebra and Physics. Algebra (x) 75 80 93 65 87 71 98 68 84 77 Physics (y) 82 78 86 72 91 80 95 72 89 74 Find both the least squares lines.

Important Questions on Linear Regression

Learn and Prepare for any exam you want !

The following table show final grades of $10$ students in Algebra and Physics.

Algebra ( $x$ ) $75$ $80$ $93$ $65$ $87$ $71$ $98$ $68$ $84$ $77$

Physics ( $y$ ) $82$ $78$ $86$ $72$ $91$ $80$ $95$ $72$ $89$ $74$

Find both the least squares lines.