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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`
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Solution
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  2  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 = 20  Total = 10  Total = 40 
Mean of x = `barx = (sumx)/"n"` = 3
Mean of y = `bar = (sumy)/"n"` = 9
b_{xy} = `(sum(x  barx)(y  bary))/(sum(y  bary)^2) = 20/40 = 1/2`
b_{yx} = `(sum(x  barx)(y  bary))/(sum(x  barx)^2) = 20/10` = 2
Regression equation of x on y is `(x  barx) = "b"_(xy) (y  bary)`
i.e., `("X"  3) = 1/2 ("Y"  9)`
∴ Regression equation x on y is 2X – Y + 3 = 0
Regression equation of y on x is `(y  bary) = "b"_(yx) (x  barx)`
i.e., (Y – 9) = 2(X – 3)
∴ Regression equation of y on x is 2X – Y + 3 = 0
RELATED QUESTIONS
You are given the following information about advertising expenditure and sales.
Advertisement expenditure (₹ in lakh) (X) 
Sales (₹ in lakh) (Y)  
Arithmetic Mean  10  90 
Standard Mean  3  12 
Correlation coefficient between X and Y is 0.8
 Obtain the two regression equations.
 What is the likely sales when the advertising budget is ₹ 15 lakh?
 What should be the advertising budget if the company wants to attain sales target of ₹ 120 lakh?
Bring out the inconsistency in the following:
b_{YX} + b_{XY} = 1.30 and r = 0.75
Bring out the inconsistency in the following:
b_{YX} = 2.6 and b_{XY} = `1/2.6`
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.
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.
What should be the advertisement expenditure if the firm proposes a sales target ₹ 60 crores?
For bivariate data, the regression coefficient of Y on X is 0.4 and the regression coefficient of X on Y is 0.9. Find the value of the variance of Y if the variance of X is 9.
The equations of two regression lines are
2x + 3y − 6 = 0
and 3x + 2y − 12 = 0 Find
 Correlation coefficient
 `sigma_"X"/sigma_"Y"`
In a partially destroyed record, the following data are available: variance of X = 25, Regression equation of Y on X is 5y − x = 22 and regression equation of X on Y is 64x − 45y = 22 Find
 Mean values of X and Y
 Standard deviation of Y
 Coefficient of correlation between X and Y.
The two regression equations are 5x − 6y + 90 = 0 and 15x − 8y − 130 = 0. Find `bar x, bar y`, r.
Choose the correct alternative:
If r = 0.5, σ_{x} = 3, `σ_"y"^2` = 16, then b_{yx} = ______
Choose the correct alternative:
If r = 0.5, σ_{x} = 3, σ_{y}^{2} = 16, then b_{xy} = ______
State whether the following statement is True or False:
The following data is not consistent: b_{yx} + b_{xy} =1.3 and r = 0.75
State whether the following statement is True or False:
If u = x – a and v = y – b then b_{xy} = b_{uv}
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 ______
If u = `(x  20)/5` and v = `(y  30)/4`, then b_{yx} = ______
The geometric mean of negative regression coefficients is ______
b_{yx} is the ______ of regression line of y on x
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.
If n = 5, Σx = Σy = 20, Σx^{2} = Σy^{2} = 90 , Σxy = 76 Find Covariance (x,y)
For a certain bivariate data of a group of 10 students, the following information gives the internal marks obtained in English (X) and Hindi (Y):
X  Y  
Mean  13  17 
Standard Deviation  3  2 
If r = 0.6, Estimate x when y = 16 and y when x = 10
Mean of x = 53
Mean of y = 28
Regression coefficient of y on x = – 1.2
Regression coefficient of x on y = – 0.3
a. r = `square`
b. When x = 50,
`y  square = square (50  square)`
∴ y = `square`
c. When y = 25,
`x  square = square (25  square)`
∴ x = `square`
x  y  xy  x^{2}  y^{2} 
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} = ______.
b_{xy} + b_{yz} ≥ ______.