From the data of 7 pairs of observations on X and Y, following results are obtained. ∑(xi - 70) = - 35, ∑(yi - 60) = - 7, ∑(xi - 70)2 = 2989, ∑(yi - 60)2 = 476, ∑(xi - 70)(yi - 60) = 1064 [Given: `sqrt0.7884` = 0.8879] Obtain The line of regression of Y on X. The line regression of X on Y. The correlation coefficient between X and Y.

From the data of 7 pairs of observations on X and Y, following results are obtained.

प्रश्न

From the data of 7 pairs of observations on X and Y, following results are obtained.

∑(x_i - 70) = - 35, ∑(y_i - 60) = - 7,

∑(x_i - 70)² = 2989, ∑(y_i - 60)² = 476,

∑(x_i - 70)(y_i - 60) = 1064

[Given: `sqrt0.7884` = 0.8879]

Obtain

The line of regression of Y on X.
The line regression of X on Y.
The correlation coefficient between X and Y.

बेरीज

उत्तर

Given: n =7, ∑(x_i - 70) = - 35, ∑(y_i - 60) = - 7,

∑(x_i - 70)² = 2989, ∑(y_i - 60)² = 476,

∑(x_i - 70)(y_i - 60) = 1064

Let u_i = x_i - 70 and v_i = y_i - 60

∴ ∑ u_i = - 35, ∑ v_i = - 7

`sum "u"_"i"^2 = 2989, sum "v"_"i"^2 = 479`

∑ u_i v_i = 1064

∴ `bar "u" = (sum "u"_"i")/"n" = (-35)/7 = - 5`

∴ `bar "v" = (sum "v"_"i")/"n" = (-7)/7 = - 1`

Now, `sigma_"u"^2 = (sum "u"_"i"^2)/"n" - (bar"u")^2`

`= 2989/7 - (- 5)^2` = 427 - 25 = 402

and `sigma_"v"^2 = (sum "v"_"i"^2)/"n" - (bar"v")^2`

`= 476/7 - (- 1)^2 = 68 - 1 = 67`

cov(u, v) = `(sum "u"_"i" "v"_"i")/"n" - bar"uv"`

`= 1064/7 - (- 5)(- 1)` = 152 - 5 = 147

Since the regression coefficients are independent of change of origin,

b_YX = b_VU and b_XY = b_UV

∴ b_YX = b_VU = `("cov" ("u", "v"))/sigma_"U"^2 = 147/402 = 0.36`

and b_XY = b_UV= `("cov" ("u", "v"))/sigma_"V"^2 = 147/67 = 2.19`

Also, `bar x = bar u` + 70 = - 5 + 70 = 65

and `bar y = bar v` + 60 = - 1 + 60 = 59

(i) The line of regression of Y on X is

`("Y" - bar y) = "b"_"YX" ("X" - bar x)`

∴ (Y - 59) = (0.36)(X - 65)

∴ Y - 59 = 0.36X - 23.4

∴ Y = 0.36X + 59 - 23.4

∴ Y = 0.36X + 35.6

(ii) The line of regression of X on Y is

`("X" - bar x) = "b"_"XY" ("Y" - bar y)`

∴ (X - 65) = (2.19)(Y - 59)

∴ X - 65 = 2.19Y - 129.21

∴ X = 2.19Y + 65 - 129.21

∴ X = 2.19Y - 64.21

(iii) r = `+-sqrt("b"_"YX" * "b"_"XY")`

`= +- sqrt((0.36)(2.19))`

`= +- sqrt0.7884 = +- 0.8879`

Since b_YX and b_XY both are positive,

r is also positive.

∴ r = 0.8879

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Properties of Regression Coefficients

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Q 3 Q 2 Q 4

पाठ 3: Linear Regression - Exercise 3.2 [पृष्ठ ४७]

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बालभारती Mathematics and Statistics 2 (Commerce) [English] Standard 12 Maharashtra State Board

पाठ 3 Linear Regression
Exercise 3.2 | Q 3 | पृष्ठ ४७

संबंधित प्रश्‍न

For bivariate data. `bar x = 53, bar y = 28, "b"_"YX" = - 1.2, "b"_"XY" = - 0.3` Find estimate of Y for X = 50.

Bring out the inconsistency in the following:

b_YX = b_XY = 1.50 and r = - 0.9

Bring out the inconsistency in the following:

b_YX = 1.9 and b_XY = - 0.25

Bring out the inconsistency in the following:

b_YX = 2.6 and b_XY = `1/2.6`

Given the following information about the production and demand of a commodity obtain the two regression lines:

	X	Y
Mean	85	90
S.D.	5	6

The coefficient of correlation between X and Y is 0.6. Also estimate the production when demand is 100.

Two samples from bivariate populations have 15 observations each. The sample means of X and Y are 25 and 18 respectively. The corresponding sum of squares of deviations from respective means is 136 and 150. The sum of the product of deviations from respective means is 123. Obtain the equation of the line of regression of X on Y.

The following data about the sales and advertisement expenditure of a firms is given below (in ₹ Crores)

	Sales	Adv. Exp.
Mean	40	6
S.D.	10	1.5

Coefficient of correlation between sales and advertisement expenditure is 0.9.

Estimate the likely sales for a proposed advertisement expenditure of ₹ 10 crores.

For certain bivariate data the following information is available.

	X	Y
Mean	13	17
S.D.	3	2

Correlation coefficient between x and y is 0.6. estimate x when y = 15 and estimate y when x = 10.

The equations of two regression lines are
2x + 3y − 6 = 0
and 3x + 2y − 12 = 0 Find

Correlation coefficient
`sigma_"X"/sigma_"Y"`

If the two regression lines for a bivariate data are 2x = y + 15 (x on y) and 4y = 3x + 25 (y on x), find

`bar x`,
`bar y`,
b_YX
b_XY
r [Given `sqrt0.375` = 0.61]

The two regression equations are 5x − 6y + 90 = 0 and 15x − 8y − 130 = 0. Find `bar x, bar y`, r.

Regression equations of two series are 2x − y − 15 = 0 and 3x − 4y + 25 = 0. Find `bar x, bar y` and regression coefficients. Also find coefficients of correlation. [Given `sqrt0.375` = 0.61]

Choose the correct alternative:

If for a bivariate data, b_YX = – 1.2 and b_XY = – 0.3, then r = ______

Choose the correct alternative:

|b_yx + b_xy| ≥ ______

Choose the correct alternative:

If r = 0.5, σ_x = 3, σ_y² = 16, then b_xy = ______

State whether the following statement is True or False:

If b_yx = 1.5 and b_xy = `1/3` then r = `1/2`, the given data is consistent

State whether the following statement is True or False:

Corr(x, x) = 0

State whether the following statement is True or False:

Cov(x, x) = Variance of x

If the sign of the correlation coefficient is negative, then the sign of the slope of the respective regression line is ______

The geometric mean of negative regression coefficients is ______

b_yx is the ______ of regression line of y on x

The equations of two lines of regression are 3x + 2y – 26 = 0 and 6x + y – 31 = 0. Find variance of x if variance of y is 36

The equations of the two lines of regression are 2x + 3y − 6 = 0 and 5x + 7y − 12 = 0. Find the value of the correlation coefficient `("Given" sqrt(0.933) = 0.9667)`

Given the following information about the production and demand of a commodity.

Obtain the two regression lines:

	Production (X)	Demand (Y)
Mean	85	90
Variance	25	36

Coefficient of correlation between X and Y is 0.6. Also estimate the demand when the production is 100 units.

The equations of the two lines of regression are 6x + y − 31 = 0 and 3x + 2y – 26 = 0. Find the value of the correlation coefficient

If n = 5, Σx = Σy = 20, Σx² = Σy² = 90 , Σxy = 76 Find Covariance (x,y)

x	y	`x - barx`	`y - bary`	`(x - barx)(y - bary)`	`(x - barx)^2`	`(y - bary)^2`
1	5	– 2	– 4	8	4	16
2	7	– 1	– 2	`square`	1	4
3	9	0	0	0	0	0
4	11	1	2	2	4	4
5	13	2	4	8	1	16
Total = 15	Total = 45	Total = 0	Total = 0	Total = `square`	Total = 10	Total = 40

Mean of x = `barx = square`

Mean of y = `bary = square`

b_xy = `square/square`

b_yx = `square/square`

Regression equation of x on y is `(x - barx) = "b"_(xy) (y - bary)`

∴ Regression equation x on y is `square`

Regression equation of y on x is `(y - bary) = "b"_(yx) (x - barx)`

∴ Regression equation of y on x is `square`

Mean of x = 25

Mean of y = 20

`sigma_x` = 4

`sigma_y` = 3

r = 0.5

b_yx = `square`

b_xy = `square`

when x = 10,

`y - square = square (10 - square)`

∴ y = `square`

x	y	xy	x²	y²
6	9	54	36	81
2	11	22	4	121
10	5	50	100	25
4	8	32	16	64
8	7	`square`	64	49
Total = 30	Total = 40	Total = `square`	Total = 220	Total = `square`

b_xy = `square/square`

b_yx = `square/square`

∴ Regression equation of x on y is `square`

∴ Regression equation of y on x is `square`

b_XY . b_YX = ______.

From the data of 7 pairs of observations on X and Y, following results are obtained.

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From the data of 7 pairs of observations on X and Y, following results are obtained.

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