Write the regression equation of Y on X and its normal equations.

Regression equation of Y on X is Y = a + b X . We know that in the method of least squares, ∑ i = 1 n y i - y ^ i = 0 and E ( a , b ) = ∑ i = 1 n y i - y ^ i 2 . Consider, E ( a , b ) = ∑ i = 1 n y i - y ^ i 2 ⇒ E ( a , b ) = ∑ i = 1 n y i - a - b x i 2 Differentiation of E ( a , b ) with respect to a and b and equating them to zero. ⇒ ∂ E ( a , b ) ∂ a = - 2 ∑ i = 1 n y i - a - b x i = 0 → ( 1 ) ⇒ ∂ E ( a , b ) ∂ b = - 2 ∑ i = 1 n x i y i - a - b x i = 0 → ( 2 ) From equation 1 and 2 we get ⇒ n a + b ∑ i = 1 n x i = ∑ i = 1 n y i ⇒ a ∑ i = 1 n x i + b ∑ i = 1 n x i 2 = ∑ i = 1 n x i y i These equations are popularly known as normal equations.

Write the regression equation of X on Y and its normal equations.

Regression equation of X on Y is X = a + b Y We know that in the method of least squares, ∑ i = 1 n x i - x ^ i = 0 and E ( a , b ) = ∑ i = 1 n x i - x ^ i 2 . Consider, E ( a , b ) = ∑ i = 1 n x i - x ^ i 2 ⇒ E ( a , b ) = ∑ i = 1 n x i - a - b y i 2 Differentiation of E ( a , b ) with respect to a and b and equating them to zero ⇒ ∂ E ( a , b ) ∂ a = - 2 ∑ i = 1 n x i - a - b y i = 0 → ( 1 ) ⇒ ∂ E ( a , b ) ∂ b = - 2 ∑ i = 1 n y i x i - a - b y i = 0 → ( 2 ) From equation 1 and 2 we get ⇒ n a + b ∑ i = 1 n y i = ∑ i = 1 n x i ⇒ a ∑ i = 1 n y i + b ∑ i = 1 n y i 2 = ∑ i = 1 n x i y i These equations are popularly known as normal equations.

The following table shows the age ( X ) and systolic blood pressure ( Y ) of 8 persons. Age ( X ) 56 42 60 50 54 49 39 45 Blood pressure ( Y ) 160 130 125 135 145 115 140 120 Fit a simple linear regression model, Y on X and estimate the blood pressure of a person of 60 years.

Given that, the data of age ( X ) and blood pressure ( Y ) . Age ( X ) Blood pressure ( Y ) X 2 Y 2 X . Y 56 160 3136 25600 8960 42 130 1764 16900 5460 60 125 3600 15625 7500 50 135 2500 18225 6750 54 145 2916 21025 7830 49 115 2401 13225 5635 39 140 1521 19600 5460 45 120 2025 14400 5400 Σ X = 395 Σ Y = 1070 Σ X 2 = 19863 Σ Y 2 = 144600 Σ X Y = 52995 We know that, regression equation of Y on X is Y = a + b Y X X , where b Y X = n ∑ i = 1 n x i y i - ∑ i = 1 n x i × ∑ i = 1 n y i n ∑ i = 1 n x i 2 - ∑ i = 1 n x i 2 and a ^ = 1 n ∑ i = 1 n y i - b ^ 1 n ∑ i = 1 n x i ⇒ b Y X = 8 52995 - 395 × 1070 8 19863 - 395 2 = 0 . 455 ⇒ a = 1070 8 - 0 . 455 8 395 = 111 . 284 ∴ Regression equation of Y on X is Y = 111 . 284 + 0 . 455 X .

Find the regression equation of X on Y given that n = 5 , ∑ x = 30 , ∑ y = 40 , ∑ x y = 214 , ∑ x 2 = 220 , ∑ y 2 = 340 .

Given that, n = 5 , ∑ x = 30 , ∑ y = 40 , ∑ x y = 214 , ∑ x 2 = 220 , ∑ y 2 = 340 . We know that, regression equation of X on Y is X = c + b X Y Y ,where b X Y = n ∑ i = 1 n x i y i - ∑ i = 1 n x i ∑ i = 1 n y i n ∑ i = 1 n y i 2 - ∑ i = 1 n y i 2 and c = 1 n ∑ i = 1 n x i - b ^ 1 n ∑ i = 1 n y i ⇒ b X Y = 5 ( 214 ) - 30 × 40 5 ( 340 ) - 1600 = - 1 . 3 ⇒ c = 30 5 + 1 . 3 5 ( 40 ) = 16 . 4 ∴ Regression equation of X on Y is 16 . 4 - 1 . 3 Y .

Write the regression equation of y on x and its normal equations.

Regression equation of y on x is y = a + b x . We know that,the method of least square is given by, ∑ i = 1 n y i - y ^ i = 0 and E ( a , b ) = ∑ i = 1 n y i - y ^ i 2 . Consider, E ( a , b ) = ∑ i = 1 n y i - y ^ i 2 ⇒ E ( a , b ) = ∑ i = 1 n y i - a - b x i 2 Differentiation of E ( a , b ) with respect to a and b and equating them to zero we get, ⇒ ∂ E ( a , b ) ∂ a = - 2 ∑ i = 1 n y i - a - b x i = 0 → ( 1 ) ⇒ ∂ E ( a , b ) ∂ b = - 2 ∑ i = 1 n x i y i - a - b x i = 0 → ( 2 ) From equation 1 and 2 we get ⇒ n a + b ∑ i = 1 n x i = ∑ i = 1 n y i ⇒ a ∑ i = 1 n x i + b ∑ i = 1 n x i 2 = ∑ i = 1 n x i y i These equations are popularly known as normal equations.

Find the regression equation of X on Y given that n = 5 , ∑ x = 40 , ∑ y = 50 , ∑ x y = 300 , ∑ x 2 = 350 , ∑ y 2 = 400 .

Given that, n = 5 , ∑ x = 40 , ∑ y = 50 , ∑ x y = 300 , ∑ x 2 = 350 , ∑ y 2 = 400 . We know that, regression equation of X on Y is X = c + b X Y Y ,where b X Y = n ∑ i = 1 n x i y i - ∑ i = 1 n x i ∑ i = 1 n y i n ∑ i = 1 n y i 2 - ∑ i = 1 n y i 2 and c = 1 n ∑ i = 1 n x i - b ^ 1 n ∑ i = 1 n y i ⇒ b X Y = 5 ( 300 ) - 40 × 50 5 ( 400 ) - 2500 = 1 ⇒ c = 40 5 + 1 5 ( 50 ) = 18 ∴ Regression equation of X on Y is X = 18 + Y .

Write the regression equation of Y on X and its normal equations.

Regression equation of Y on X is Y = a + b X . We know that in the method of least squares, ∑ i = 1 n y i - y ^ i = 0 and E ( a , b ) = ∑ i = 1 n y i - y ^ i 2 . Consider, E ( a , b ) = ∑ i = 1 n y i - y ^ i 2 ⇒ E ( a , b ) = ∑ i = 1 n y i - a - b x i 2 Differentiation of E ( a , b ) with respect to a and b and equating them to zero. ⇒ ∂ E ( a , b ) ∂ a = - 2 ∑ i = 1 n y i - a - b x i = 0 → ( 1 ) ⇒ ∂ E ( a , b ) ∂ b = - 2 ∑ i = 1 n x i y i - a - b x i = 0 → ( 2 ) From equation 1 and 2 we get ⇒ n a + b ∑ i = 1 n x i = ∑ i = 1 n y i ⇒ a ∑ i = 1 n x i + b ∑ i = 1 n x i 2 = ∑ i = 1 n x i y i These equations are popularly known as normal equations.

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Using the method of least square, calculate the regression equation of $X$ on $Y$ and $Y$ on $X$ from the following data and estimate $X$ where $Y$ is $16$ .

$X$ $10$ $12$ $13$ $17$ $18$

$Y$ $5$ $6$ $7$ $9$ $13$

Also determine the value of correlation coefficient

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Important Questions on Linear Regression

EASY

Mathematics>Statistics>Linear Regression>The Method of Least Squares

Find the regression equation of

X

Y

given that

n = 5

\sum x = 30

\sum y = 40

\sum x y = 214

\sum x^{2} = 220

\sum y^{2} = 340

EASY

Mathematics>Statistics>Linear Regression>The Method of Least Squares

The method applied for deriving the regression equations is known as

MEDIUM

Mathematics>Statistics>Linear Regression>The Method of Least Squares

Using the method of least square, calculate the regression equation of $X$ on $Y$ from the following data.

$X$	$10$	$12$	$14$	$16$	$18$
$Y$	$6$	$7$	$8$	$9$	$10$

EASY

Mathematics>Statistics>Linear Regression>The Method of Least Squares

The regression equation of $y$ on $x$ for the following data:

$\begin{array}{l} x & 41 & 82 & 62 & 37 & 58 & 96 & 127 & 74 & 123 & 100 \\ y & 28 & 56 & 35 & 17 & 42 & 85 & 105 & 61 & 98 & 73 \end{array}$

Is given by

EASY

Mathematics>Statistics>Linear Regression>The Method of Least Squares

Write the regression equation of

Y

X

and its normal equations.

EASY

Mathematics>Statistics>Linear Regression>The Method of Least Squares

Write the regression equation of

X

Y

and its normal equations.

EASY

Mathematics>Statistics>Linear Regression>The Method of Least Squares

Find the linear regression equation of percentage worms $(Y)$ on size of the crop $(X)$ based on the following seven observations.

Size of the crop $(X)$	$16$	$15$	$11$	$27$	$39$	$22$	$20$
Percentage worms $(Y)$	$24$	$25$	$34$	$40$	$35$	$20$	$23$

EASY

Mathematics>Statistics>Linear Regression>The Method of Least Squares

The following table shows the age $(X)$ and systolic blood pressure $(Y)$ of $8$ persons.

Age $(X)$	$56$	$42$	$60$	$50$	$54$	$49$	$39$	$45$
Blood pressure $(Y)$	$160$	$130$	$125$	$135$	$145$	$115$	$140$	$120$

Fit a simple linear regression model, $Y$ on $X$ and estimate the blood pressure of a person of $60$ years.

EASY

Mathematics>Statistics>Linear Regression>The Method of Least Squares

Find the regression equation of

X

Y

given that

n = 5

\sum x = 30

\sum y = 40

\sum x y = 214

\sum x^{2} = 220

\sum y^{2} = 340

EASY

Mathematics>Statistics>Linear Regression>The Method of Least Squares

Using the method of least square, calculate the regression equation of $X$ on $Y$ and $Y$ on $X$ from the following data and estimate $X$ where $Y$ is $16$ .

$X$	$10$	$12$	$13$	$17$	$18$
$Y$	$5$	$6$	$7$	$9$	$13$

Also determine the value of correlation coefficient

MEDIUM

Mathematics>Statistics>Linear Regression>The Method of Least Squares

Write the regression equation of

y

x

and its normal equations.

MEDIUM

Mathematics>Statistics>Linear Regression>The Method of Least Squares

Using the method of least square, calculate the regression equation of $X$ on $Y$ from the following data.

$X$	$10$	$12$	$14$	$16$	$18$
$Y$	$6$	$7$	$8$	$9$	$10$

EASY

Mathematics>Statistics>Linear Regression>The Method of Least Squares

Find the regression equation of

X

Y

given that

n = 5

\sum x = 40

\sum y = 50

\sum x y = 300

\sum x^{2} = 350

\sum y^{2} = 400

MEDIUM

Mathematics>Statistics>Linear Regression>The Method of Least Squares

Find the linear regression equation of percentage worms $(Y)$ on size of the crop $(X)$ based on the following seven observations.

Size of the crop $(X)$	$16$	$15$	$10$	$25$	$35$	$20$	$28$
Percentage worms $(Y)$	$24$	$25$	$30$	$40$	$32$	$20$	$22$

MEDIUM

Mathematics>Statistics>Linear Regression>The Method of Least Squares

Write the regression equation of

Y

X

and its normal equations.

Using the method of least square, calculate the regression equation of X on Y and Y on X from the following data and estimate X where Y is 16. X 10 12 13 17 18 Y 5 6 7 9 13 Also determine the value of correlation coefficient

Important Questions on Linear Regression

Learn and Prepare for any exam you want !

Using the method of least square, calculate the regression equation of $X$ on $Y$ and $Y$ on $X$ from the following data and estimate $X$ where $Y$ is $16$ .

$X$ $10$ $12$ $13$ $17$ $18$

$Y$ $5$ $6$ $7$ $9$ $13$

Also determine the value of correlation coefficient