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

Earn 100

Give an explanation for the existence of two regression lines and write the general form of simple linear regression equation.

Important Questions on Regression Analysis

MEDIUM

Mathematics>Statistics>Regression Analysis>Why Are There Two Regression Lines?

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>Regression Analysis>Why Are There Two Regression Lines?

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>Regression Analysis>Why Are There Two Regression Lines?

\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>Regression Analysis>Why Are There Two Regression Lines?

Y = 9 X

and

X = 0.01 Y

, then

r

is equal to:

EASY

Mathematics>Statistics>Regression Analysis>Why Are There Two Regression Lines?

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>Regression Analysis>Why Are There Two Regression Lines?

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>Regression Analysis>Why Are There Two Regression Lines?

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>Regression Analysis>Why Are There Two Regression Lines?

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>Regression Analysis>Why Are There Two Regression Lines?

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>Regression Analysis>Why Are There Two Regression Lines?

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>Regression Analysis>Why Are There Two Regression Lines?

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______.

MEDIUM

Mathematics>Statistics>Regression Analysis>Why Are There Two Regression Lines?

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>Regression Analysis>Why Are There Two Regression Lines?

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>Regression Analysis>Why Are There Two Regression Lines?

b_{y x} = - 1.6

and

b_{x y} = - 0.4

, then

r_{x y}

will be_____.

HARD

Mathematics>Statistics>Regression Analysis>Why Are There Two Regression Lines?

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>Regression Analysis>Why Are There Two Regression Lines?

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

HARD

Mathematics>Statistics>Regression Analysis>Why Are There Two Regression Lines?

The following observations are given :

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

Estimate the values of $y$ when the value of $x$ is $10$ and also estimate the value of $x$ when the value of $y$ is $5 .$

EASY

Mathematics>Statistics>Regression Analysis>Why Are There Two Regression Lines?

If the regression equations are

\hat{X} = 64 - 0.95 y

\hat{Y} = 7.25 - 0.95 x

then find the correlation coefficient.

EASY

Mathematics>Statistics>Regression Analysis>Why Are There Two Regression Lines?

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>Regression Analysis>Why Are There Two Regression Lines?

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

Give an explanation for the existence of two regression lines and write the general form of simple linear regression equation.

Important Questions on Regression Analysis

Learn and Prepare for any exam you want !

Give an explanation for the existence of two regression lines and write the general form of simple linear regression equation.