The following results were obtained with respect to two variables x and y : ∑ x = 15 , ∑ y = 25 , ∑ x y = 83 , ∑ x 2 = 55 , ∑ y 2 = 135 and n = 5 Find the regression equation of x on y .

Given, ∑ x = 15 , ∑ y = 25 , ∑ x y = 83 , ∑ x 2 = 55 , ∑ y 2 = 135 and n = 5 First we find: b x y We know that b x y = Σ x y - Σ x Σ y n Σ y 2 - ( Σ y ) 2 n = 83 - 15 25 5 135 - 25 2 5 = 83 - 75 135 - 125 = 8 10 Hence, b x y = 4 5 Now, we know that the regression equation of x on y is given by x - x ¯ = b x y ( y - y ¯ ) ⇒ x - 15 5 = 4 5 y - 25 5 ⇒ 5 x - 15 = 4 y - 5 ⇒ 5 x - 15 = 4 y - 20 ⇒ 5 x - 4 y = - 5 Hence, the required equation is given by 5 x - 4 y + 5 = 0

If x ¯ = 18 , y ¯ = 100 , σ x = 14 , σ y = 20 , and correlation coefficient r x y = 0 · 8 , find the regression equation of y on x .

Given, x ¯ = 18 , y ¯ = 100 , σ x = 14 , σ y = 20 , and r x y = 0 · 8 , We know that the slope of equation y on x is given by b y x = r x y · σ y σ x = 0 . 8 × 20 14 = 8 7 Hence, the regression equation of y on x is given by y - y ¯ = b y x ( x - x ¯ ) ⇒ y - 100 = 8 7 x - 18 ⇒ 7 y - 700 = 8 x - 144 ⇒ 8 x - 7 y + 556 = 0 , which is the required regression equation of y on x .

The following results were obtained with respect to two variables x and y : ∑ x = 15 , ∑ y = 25 , ∑ x y = 83 , ∑ x 2 = 55 , ∑ y 2 = 135 and n = 5 Find the regression coefficient b x y

Given, ∑ x = 15 , ∑ y = 25 , ∑ x y = 83 , ∑ x 2 = 55 , ∑ y 2 = 135 and n = 5 To find: b x y We know that b x y = Σ x y - Σ x Σ y n Σ y 2 - ( Σ y ) 2 n = 83 - 15 25 5 135 - 25 2 5 = 83 - 75 135 - 125 = 8 10 Hence, b x y = 4 5

The correlation coefficient between x and y is 0 . 6 . If the variance of x is 225 , the variance of y is 400 , mean of x is 10 and mean of y is 20 , find the equations of two regression lines.

Given, correlation coefficient r x y = 0 . 6 , Variance of x , σ x 2 = 225 ⇒ σ x = 15 Variance of y , σ y 2 = 400 ⇒ σ y = 20 Mean of x is x ¯ = 10 Mean of y is y ¯ = 20 Equation of regression line for y on x is, y - y ¯ = r x y σ y σ x x - x ¯ ⇒ y - 20 = 0 . 6 × 20 15 x - 10 ⇒ y - 20 = 6 10 × 4 3 x - 10 ⇒ y - 20 = 4 5 x - 10 ⇒ 5 y - 100 = 4 x - 40 ⇒ 5 y - 4 x = 60 Equation of regression line for x on y is, x - x ¯ = r x y σ x σ y y - y ¯ ⇒ x - 10 = 0 . 6 × 15 20 y - 20 ⇒ x - 10 = 6 10 × 3 4 y - 20 ⇒ x - 10 = 9 20 y - 20 ⇒ 20 x - 200 = 9 y - 180 ⇒ 20 x - 9 y = 20 Hence, equations of regression lines are 5 y - 4 x = 60 and 20 x - 9 y = 20 .

The correlation coefficient between x and y is 0 . 6 . If the variance of x is 225 , the variance of y is 400 , mean of x is 10 and mean of y is 20 , find the expected value of y when x = 2 .

Given, correlation coefficient r x y = 0 . 6 , Variance of x , σ x 2 = 225 ⇒ σ x = 15 Variance of y , σ y 2 = 400 ⇒ σ y = 20 Mean of x is x ¯ = 10 Mean of y is y ¯ = 20 Equation of regression line for y on x is, y - y ¯ = r x y σ y σ x x - x ¯ ⇒ y - 20 = 0 . 6 × 20 15 x - 10 ⇒ y - 20 = 6 10 × 4 3 x - 10 ⇒ y - 20 = 4 5 x - 10 ⇒ 5 y - 100 = 4 x - 40 ⇒ 5 y - 4 x = 60 At x = 2 , ⇒ 5 y - 4 2 = 60 ⇒ 5 y = 68 ⇒ y = 68 5 Hence, expected value of y when x = 2 is, y = 68 5 .

Find the line of regression of y on x from the following table. x 1 2 3 4 5 y 7 6 5 4 3 Hence, estimate the value of y when x = 6 .

Given that: x y x y x 2 1 7 7 1 2 6 12 4 3 5 15 9 4 4 16 16 5 3 15 25 Total ∑ x = 15 ∑ y = 25 ∑ x y = 65 ∑ x 2 = 55 Here, x ¯ = Σ x n = 15 5 = 3 and y ¯ = Σ y n = 25 5 = 5 Now, b y x = cov ⋅ ( x , y ) σ x 2 = Σ x y − Σ x Σ y n Σ x 2 − ( Σ x ) 2 n = 65 − 15 × 25 5 55 − 15 × 15 5 = 65 − 75 55 − 45 = − 10 10 = − 1 Regression of y on x is: y − y ¯ = b y x ( x − x ¯ ) ⇒ y − 5 = ( − 1 ) ( x − 3 ) ⇒ y − 5 = − x + 3 ⇒ x + y − 8 = 0 So, Value of y when x = 6 is given by 6 + y - 8 = 0 ⇒ y = 2

EASY

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Find the coefficient of correlation from the following regression lines $x - 2 y + 3 = 0$ , $4 x - 5 y + 1 = 0$ .(Answer up to four decimal places).

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Important Questions on Linear Regression

MEDIUM

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

The following results were obtained with respect to two variables $x$ and $y$ :

$\sum x = 15, \sum y = 25, \sum x y = 83, \sum x^{2} = 55, \sum y^{2} = 135 and n = 5$

Find the regression equation of $x$ on $y$ .

EASY

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

If the relationship between

x

and

y

is given by

2 x + 3 y = 10

and the range of

y

10

, then what is the range of

x

MEDIUM

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

\bar{x} = 18, \bar{y} = 100, σ_{x} = 14, σ_{y} = 20

, and correlation coefficient

r_{x y} = 0 \cdot 8,

find the regression equation of

y

x

MEDIUM

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

If the slope of the regression line is calculated to be

5.5

and the intercept

15

then the value of

Y

when

X

6

is______.

EASY

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

The following results were obtained with respect to two variables $x$ and $y$ :

$\sum x = 15, \sum y = 25, \sum x y = 83, \sum x^{2} = 55, \sum y^{2} = 135 and n = 5$

Find the regression coefficient $b_{x y}$

MEDIUM

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

The correlation coefficient between

x

and

y

0.6

. If the variance of

x

225

, the variance of

y

400

, mean of

x

10

and mean of

y

20

, find the equations of two regression lines.

MEDIUM

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

If two lines of regression are

3 x + 12 y = 19

and

9 x + 3 y = 46

, then the value of

r_{x y}

will be

MEDIUM

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

The correlation coefficient between

x

and

y

0.6

. If the variance of

x

225

, the variance of

y

400

, mean of

x

10

and mean of

y

20

, find the expected value of

y

when

x = 2

EASY

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

For the lines of regression

4 x - 2 y = 4

and

2 x - 3 y + 6 = 0

, find the mean of

' x'

and the mean of

' y'

MEDIUM

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

Find the equation of the regression line of $y$ on $x$ , if the observations $(x, y)$ are as follows:

$(1, 4), (2, 8), (3, 2), (4, 12), (5, 10), (6, 14), (7, 16), (8, 6), (9, 18)$ .

Also, find the estimated value of $y$ when $x = 14$ .

MEDIUM

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

Y = 9 X

and

X = 0.01 Y

, then

r

is equal to:

MEDIUM

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

Find the line of regression of $y$ on $x$ from the following table.

$x$	$1$	$2$	$3$	$4$	$5$
$y$	$7$	$6$	$5$	$4$	$3$

Hence, estimate the value of $y$ when $x = 6$ .

MEDIUM

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

If the equation of the two regression lines are

2 x - 3 y = 0

and

4 y - 5 x = 8

then the correlation coefficient between

x

and

y

is equal to______.

MEDIUM

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

b_{y x} = - 1.6

and

b_{x y} = - 0.4

, then

r_{x y}

will be_____.

HARD

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

Find the regression coefficients $b_{y x}, b_{x y}$ and correlation coefficient $' r'$ for the following data:

$(2, 8), (6, 8), (4, 5), (7, 6), (5, 2)$

EASY

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

If the two lines of regression are

4 x + 3 y + 7 = 0

and

3 x + 4 y + 8 = 0

, then the means of

x

and

y

are

EASY

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

A manufacturer of optical lenses has the following data on the cost per unit (in Rs.) of a certain quality of lens and the number of units made in each order:

Order Number

Number of units (

x

)

Cost per unit (

y

)

$1$

$2$

$3$

$4$

$5$

$6$

$1$

$3$

$5$

$7$

$10$

$15$

$58$

$52$

$46$

$40$

$37$

$22$

Find the correlation coefficient. Also determine the two regression equations.

EASY

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

In a correlation analysis, between production

X

and price of a commodity

Y

we get the following details. Variance of

X = 36

. The regression equations are:

12 X - 15 Y + 99 = 0

and

60 X - 27 Y = 321

. Calculate the average value of

X

and

Y

EASY

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

The lines of regression in a bivariate distribution are $x + 2 y = 5 and 2 x + 3 y - 8 = 0 .$ Find the regression coefficients.

EASY

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

The lines of regression in a bivariate distribution are $x + 2 y = 5 and 2 x + 3 y - 8 = 0 .$ Find Coefficient of correlation.

Find the coefficient of correlation from the following regression lines x-2y+3=0, 4x-5y+1=0.(Answer up to four decimal places).

Important Questions on Linear Regression

Learn and Prepare for any exam you want !

Find the coefficient of correlation from the following regression lines $x - 2 y + 3 = 0$ , $4 x - 5 y + 1 = 0$ .(Answer up to four decimal places).