Question

obtain the two regression equations from the following data and estimate the value of X when Y = 50 and the value of Y when X = 45. x= 40 50 38 60 65 48 30. y=38 60 55 70 60 48 30 ​

Answer 1

Answer:

Regression equation of Y on X ... Calculate the two regression equations of X on Y and Y on X from the data given below, taking deviations ...

Step-by-step explanation:

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Answer 2

Answer:

The regression equation of X on Y: $X = 0.73Y + 9.7$.

And the value of X when Y is $50$ = $46.2$.

The regression equation of Y on X: $Y = 0.91X + 8.6$.

And the value of Y when X is $45$ = $49.5$.

Step-by-step explanation:

Given data,

X                     Y

$40$                    $38$

$50$                    $60$

$38$                    $55$

$60$                    $70$

$65$                    $60$

$48$                    $48$

$30$                    $30$

$\sum X = 331$    $\sum Y = 361$

The two regression coefficient equations of X on Y and Y on X =?

The value of X when Y is $50$ =?

The value of Y when X is $45$ =?

Firstly, we have to find out the mean of X and Y.

Number of terms in both X and Y = $7$

• $\bar{X}$ = $\frac{\sum X}{N }$ = $\frac{331}{7}$ = $47.2$
• $\bar{Y}$ = $\frac{\sum Y}{N }$ = $\frac{361}{7}$ = $51.5$

Now, the regression coefficient of X on Y:

• $b_{xy}$ = $\frac{N\sum XY - (\sum X )( \sum Y )}{N\sum Y^{2}- (\sum Y )^{2} }$

And the regression coefficient of Y on X:

• $b_{yx}$ = $\frac{N\sum XY - (\sum X )( \sum Y )}{N\sum X^{2}- (\sum X )^{2} }$

For this, we have to make a table to find the above mentioned terms in the formula:

X                     Y                        X²                         Y²                      XY

$40$                    $38$                      $1600$                       $1444$                  $1520$

$50$                    $60$                      $2500$                       $3600$                  $3000$

$38$                    $55$                      $1444$                       $3025$                  $2090$

$60$                    $70$                      $3600$                       $4900$                  $4200$

$65$                    $60$                      $4225$                       $3600$                  $3900$

$48$                    $48$                      $2304$                       $2304$                  $2304$

$30$                    $30$                      $900$                         $900$                    $900$

$\sum X = 331$   $\sum Y = 361$   $\sum X^{2}$ = $16573$     $\sum Y^{2} =$ $19773$     $\sum XY =$ $17914$

Now, the coefficient of regression of X on Y:

• $b_{xy}$ = $\frac{ 7( 17914) - (331 )( 361 )}{7(19773)- (361 )^{2} }$ = $\frac{125398 - 119491}{138411- 130321 }$  = $\frac{5907}{8090 }$ = $0.73$

And the coefficient of regression of Y on X:

• $b_{yx}$ = $\frac{7(17914) - (331 )( 361 )}{7(16573)- (331 )^{2} }$ = $\frac{125398 - 119491}{116011- 109561 }$ = $\frac{5907}{6450 }$ = $0.91$

Now, the regression equation of X on Y: ( Y = $50$ )

• $X - \bar{ X} = b_{xy}( Y - \bar{ Y} )$
• $X - 47.2 = 0.73( Y - 51.5 )$
• $X - 47.2 = 0.73Y - 37.5$
• $X = 0.73Y + 9.7$

As given Y = $50$

•  $X = 0.73(50) + 9.7$ = $36.5 + 9.7$ = $46.2$  

Now, the equation of regression of Y on X: ( X = $45$ )

• $Y - \bar{ Y} = b_{yx}( X - \bar{ X} )$
• $Y - 51.5 = 0.91( X -47.2 )$
• $Y - 51.5 = 0.91X -42.9$
• $Y = 0.91X + 8.6$

As given X = $45$

• $Y = 0.91(45) + 8.6$   = $40.9 + 8.6$ = $49.5$

Hence,

• The value of X when Y is $50$ = $46.2$.

• The value of Y when X is $45$ = $49.5$.