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 .

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

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 .

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

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

From the given data Variable x y Mean 6 8 Standard Deviation 4 6 and coefficient correlation 2 3 . Find regression coefficients b x y and b y x .

Variable x y Mean x = 6 y = 8 Standard deviation σ x = 4 σ y = 6 Given, x = 6 , y = 8 , σ x = 4 , σ y = 6 and r = 2 3 . Let coefficient of regression of y on x be b y x . b y x = r · σ y σ x = 2 3 × 6 4 = 1 Let coefficient of regression of x on y be b x y . b x y = r · σ x σ y = 2 3 × 4 6 = 4 9 Hence, b y x = 1 , b x y = 4 9

The following table shows the ages x and systolic blood pressure y of 12 women. Determine the least squares regression line of y on x . Age ( x ) 56 42 72 36 63 47 55 49 38 42 68 60 Blood Pressure ( y ) 147 125 160 118 149 128 150 145 115 140 152 155

Given data can be tabulated as x y x - x y - y ( x - x ) 2 ( y - y ) 2 ( x - x ) ( y - y ) 56 147 3 . 67 6 . 67 13 . 5 44.5 24.5 42 125 - 10 . 33 - 15 . 33 106 . 7 235 158 . 36 72 160 19 . 67 19 . 67 386 . 9 386 . 9 386 . 9 36 118 - 16 . 33 - 22 . 33 266 . 7 498 . 6 364 . 64 63 149 10 . 67 8 . 67 113 . 85 75 . 2 92 . 5 47 128 - 5 . 33 - 12 . 33 28 . 4 152 65 . 7 55 150 2 . 67 9 . 67 7 . 1 93 . 5 25 . 82 49 145 - 3 . 33 4 . 67 11 21 . 8 - 15 . 55 38 115 - 14 . 33 - 25 . 33 205 . 35 641 . 6 363 42 140 - 10 . 33 - 0 . 33 106 . 7 0 . 1 3 . 4 68 152 15 . 67 11 . 67 245 . 5 136 . 2 183 60 155 7 . 67 14 . 67 58 . 8 215 . 2 112 . 5 ∑ x = 628 ∑ y = 1684 ∑ = 0 ∑ = 0 ∑ = 1551 ∑ = 2500 . 6 ∑ = 1765 Mean of x , x = ∑ x n = 628 12 = 52 . 33 Mean of y , y = ∑ y n = 1684 12 = 140 . 33 b y x = ∑ ( x - x ) ( y - y ) ∑ ( x - x ) 2 b y x = 1765 1551 = 1 . 14 Regression line of y on x is y - y = b y x ( x - x ) y - 140 . 33 = 1 . 14 ( x - 52 . 33 ) y = 1 . 14 x + 80 . 67

EASY

Earn 100

State the differences between correlation and regression from their definitions.

Important Questions on Linear Regression

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

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 $y$ .

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

Y = 9 X

and

X = 0.01 Y

, then

r

is equal to:

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

b_{y x} = - 1.6

and

b_{x y} = - 0.4

, then

r_{x y}

will be_____.

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

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>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 $x = 6$ .

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 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

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

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

\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

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

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 $b_{x y}$

EASY

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

From the given data

Variable	$x$	$y$
Mean	$6$	$8$
Standard Deviation	$4$	$6$

and coefficient correlation $\frac{2}{3}$ . Find regression coefficients $b_{x y}$ and $b_{y x}$ .

EASY

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

The regression lines intersect at

EASY

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

The following table shows the ages $x$ and systolic blood pressure $y$ of $12$ women.

Determine the least squares regression line of $y$ on $x$ .

Age ( $x$ )	$56$	$42$	$72$	$36$	$63$	$47$	$55$	$49$	$38$	$42$	$68$	$60$
Blood Pressure ( $y$ )	$147$	$125$	$160$	$118$	$149$	$128$	$150$	$145$	$115$	$140$	$152$	$155$

HARD

Mathematics>Statistics>Linear Regression>Properties of Regression Coefficients

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 $5 .$

State the differences between correlation and regression from their definitions.

Important Questions on Linear Regression

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