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Chapters
Chapter 1.2: Matrices
Chapter 1.3: Differentiation
Chapter 1.4: Applications of Derivatives
Chapter 1.5: Integration
Chapter 1.6: Definite Integration
Chapter 1.7: Application of Definite Integration
Chapter 1.8: Differential Equation and Applications
Chapter 2.1: Commission, Brokerage and Discount
Chapter 2.2: Insurance and Annuity
Chapter 2.3: Linear Regression
Chapter 2.4: Time Series
Chapter 2.5: Index Numbers
Chapter 2.6: Linear Programming
Chapter 2.7: Assignment Problem and Sequencing
Chapter 2.8: Probability Distributions
Chapter 2: Linear Regression
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2 Linear Regression Q.1
MCQ [1 Mark]
Choose the correct alternative:
If for a bivariate data, b_{YX} = – 1.2 and b_{XY} = – 0.3, then r = ______
– 0.06
0.06
0.6
– 0.6
Choose the correct alternative:
If the regression equation X on Y is 3x + 2y = 26, then b_{xy} equal to
`3/2`
`2/3`
` 3/2`
`  2/3`
Choose the correct alternative:
If b_{yx} < 0 and b_{xy} < 0, then r is ______
< 0
> 0
c = 0
> 1
Choose the correct alternative:
b_{yx} + b_{xy} ≥ ______
r
2r
r
– r
Choose the correct alternative:
Find the value of the covariance between X and Y, if the regression coefficient of Y on X is 3.75 and σ_{x} = 2, σ_{y} = 8
7
30
15
1.875
Choose the correct alternative:
The slope of the line of regression of y on x is called the ______
regression coefficient of x on y
correlation coefficient between y and x
covariance between y and x
regression coefficient of y on x
Choose the correct alternative:
Regression analysis is the theory of
Estimation
Prediction
Estimation and Prediction
Calculation
Choose the correct alternative:
We can estimate the value of one variable with the help of other known variable only if they are
Correlated
Positively correlated
Negatively correlated
Uncorrelated
Choose the correct alternative:
There are ______ types of regression equations
4
2
3
1
Choose the correct alternative:
In the regression equation of X on Y
X is independent and Y is dependent
Y is independent and X is dependent
Both X and Y are independent
Both X and Y are dependent
Choose the correct alternative:
b_{xy} and b_{yx} are ______
Independent of change of origin and scale
Independent of change of origin but not of scale
Independent of change of scale but not of origin
Affected by change of origin and scale
Choose the correct alternative:
If the lines of regression of Y on X is y = `x/4` and X on Y is x = `y/9 + 1` then the value of r is
`1/6`
0
` 1/4`
` 1/6`
Choose the correct alternative:
If r = 0.5, σ_{x} = 3, `σ_"y"^2` = 16, then b_{yx} = ______
0.375
0.667
2.667
0.093
Choose the correct alternative:
The regression line is obtained by
Minimizing the sum of squares of deviations of the predicted values from the observed values
Minimizing the sum of deviations of the predicted values from the observed values
Maximizing the sum of squares deviations of the predicted values from the observed values
Maximizing the sum of deviations of the predicted values from the observed values
Choose the correct alternative:
u = `(x  20)/5` and v = `(y  30)/4`, then b_{xy} =
`4/5` b_{vu}
`4/5` b_{uv}
`5/4` b_{uv}
`5/4` b_{uv}
Choose the correct alternative:
y = 5 – 2.8x and x = 3 – 0.5 y be the regression lines, then the value of b_{yx} is
– 0.5
– 2.8
0.5
– 2
Choose the correct alternative:
If r = 0.5, σ_{x} = 3, σ_{y}^{2} = 16, then b_{xy} = ______
0.375
0.667
2.667
0.093
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2 Linear Regression Q.2
[1 Mark]
State whether the following statement is True or False:
The equations of two regression lines are 10x – 4y = 80 and 10y – 9x = 40. Then b_{xy} = 0.9
True
False
State whether the following statement is True or False:
y = 5 + 2.8x and x = 3 + 0.5y be the regression lines of y on x and x on y respectively, then b_{yx} = – 0.5
True
False
Choose the correct alternative:
Both the regression coefficients cannot exceed 1
True
False
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
True
False
State whether the following statement is True or False:
Correlation analysis is the theory of games
True
False
State whether the following statement is True or False:
If b_{xy} < 0 and b_{yx} < 0 then ‘r’ is > 0
True
False
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
True
False
State whether the following statement is True or False:
If u = x – a and v = y – b then b_{xy} = b_{uv}
True
False
State whether the following statement is True or False:
If equation of regression lines are 3x + 2y – 26 = 0 and 6x + y – 31= 0, then mean of X is 7
True
False
State whether the following statement is True or False:
b_{xy} is the slope of regression line of y on x
True
False
State whether the following statement is True or False:
Corr(x, x) = 0
True
False
State whether the following statement is True or False:
Corr(x, x) = 1
True
False
State whether the following statement is True or False:
Cov(x, x) = Variance of x
True
False
State whether the following statement is True or False:
Regression analysis is used for measuring the degree of the relationship between the variables
True
False
State whether the following statement is True or False:
Regression coefficient of x on y is the slope of regression line of x on y
True
False
State whether the following statement is True or False:
The variable used for predicting the response is called the independent variable
True
False
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2 Linear Regression Q.3
Fill in the following blanks: [1 Mark]
If n = 5, ∑xy = 76, ∑x^{2} = ∑y^{2} = 90, ∑x = 20 = ∑y, the covariance = ______
b_{xy} + b_{yx} ≥ ______
Among the given regression lines 6x + y – 31 = 0 and 3x + 2y – 26 = 0, the regression line of x on y is ______
If u = `(x  "a")/"c"` and v = `(y  "b")/"d"`, then b_{xy} = ______
If the regression equations are 8x – 10y + 66 = 0 and 40x – 18y = 214, the mean value of y is ______
If the sign of the correlation coefficient is negative, then the sign of the slope of the respective regression line is ______
The value of product moment correlation coefficient between x and x is ______
Arithmetic mean of positive values of regression coefficients is greater than or equal to ______
If u = `(x  20)/5` and v = `(y  30)/4`, then b_{yx} = ______
The geometric mean of negative regression coefficients is ______
Dependent variables are also known as ______
b_{yx} is the ______ of regression line of y on x
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2 Linear Regression Q.4
Answer the following: [4 Marks]
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
Given the following information about the production and demand of a commodity.
Obtain the two regression lines:
ADVERTISEMENT (x) (₹ in lakhs) 
DEMAND (y) (₹ in lakhs) 

Mean  10  90 
Variance  9  144 
Coefficient of correlation between x and y is 0.8.
What should be the advertising budget if the company wants to attain the sales target of ₹ 150 lakhs?
The equations of the two lines of regression are 2x + 3y − 6 = 0 and 5x + 7y − 12 = 0. Identify the regression lines
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)`
The age in years of 7 young couples is given below. Calculate husband’s age when wife’s age is 38 years.
Husband (x)  21  25  26  24  22  30  20 
Wife (y)  19  20  24  20  22  24  18 
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. Identify the regression lines
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
The equations of the two lines of regression are 6x + y − 31 = 0 and 3x + 2y – 26 = 0. Calculate the mean values of x and y
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 means are 136 and 148 respectively. The sum of product of deviations from respective means is 122. Obtain the regression equation of x on y
For 50 students of a class, the regression equation of marks in statistics (X) on the marks in Accountancy (Y) is 3y − 5x + 180 = 0.
The mean marks in accountancy is 44 and the variance of marks in statistics is `(9/16)^"th"` of the variance of marks in accountancy. Find the mean marks in statistics and the correlation coefficient between marks in two subjects.
If n = 5, Σx = Σy = 20, Σx^{2} = Σy^{2} = 90 , Σxy = 76 Find Covariance (x,y)
If n = 5, Σx = Σy = 20, Σx^{2} = Σy^{2} = 90, Σxy = 76 Find the regression equation of x on y
If n = 6, Σx = 36, Σy = 60, Σxy = –67, Σx^{2} = 50, Σy^{2} =106, Estimate y when x is 13
Compute the appropriate regression equation for the following data:
x (Dependent Variable)  10  12  13  17  18 
y (Independent Variable)  5  6  7  9  13 
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
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2 Linear Regression Q.5
Activity questions: [4 Marks]
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 = 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`
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`
The regression equation of y on x is 2x – 5y + 60 = 0
Mean of x = 18
`2 square  5 bary + 60` = 0
∴ `bary = square`
`sigma_x : sigma_y` = 3 : 2
∴ b_{yx} = `square/square`
∴ b_{yx} = `square/square`
∴ r = `square`
The regression equation of x on y is 40x – 18y = 214 ......(i)
The regression equation of y on x is 8x – 10y + 66 = 0 ......(ii)
Solving equations (i) and (ii),
`barx = square`
`bary = square`
∴ b_{yx} = `square/square`
∴ b_{xy} = `square/square`
∴ r = `square`
Given variance of x = 9
∴ b_{yx} = `square/square`
∴ `sigma_y = 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`
Chapter 2: Linear Regression
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 2  Linear Regression
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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 2 Linear Regression are Regression, Types of Linear Regression, Fitting Simple Linear Regression, The Method of Least Squares, Lines of Regression of X on Y and Y on X Or Equation of Line of Regression, Properties of Regression Coefficients.
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