# SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 2 - Linear Regression [Latest edition]

#### Chapters ## Chapter 2: Linear Regression

Q.1Q.2Q.3Q.4Q.5
Q.1

### 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]

Q.1 | Q 1

Choose the correct alternative:

If for a bivariate data, bYX = – 1.2 and bXY = – 0.3, then r = ______

• – 0.06

• 0.06

• 0.6

• – 0.6

Q.1 | Q 2

Choose the correct alternative:

If the regression equation X on Y is 3x + 2y = 26, then bxy equal to

• 3/2

• 2/3

• - 3/2

•  - 2/3

Q.1 | Q 3

Choose the correct alternative:

If byx < 0 and bxy < 0, then r is ______

• < 0

• > 0

• c = 0

• > 1

Q.1 | Q 4

Choose the correct alternative:

|byx + bxy| ≥ ______

• |r|

• 2|r|

• r

• – r

Q.1 | Q 5

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 σx = 2, σy = 8

• 7

• 30

• 15

• 1.875

Q.1 | Q 6

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

Q.1 | Q 7

Choose the correct alternative:

Regression analysis is the theory of

• Estimation

• Prediction

• Estimation and Prediction

• Calculation

Q.1 | Q 8

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

Q.1 | Q 9

Choose the correct alternative:

There are ______ types of regression equations

• 4

• 2

• 3

• 1

Q.1 | Q 10

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

Q.1 | Q 11

Choose the correct alternative:

bxy and byx 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

Q.1 | Q 12

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

Q.1 | Q 13

Choose the correct alternative:

If r = 0.5, σx = 3, σ_"y"^2 = 16, then byx = ______

• 0.375

• 0.667

• 2.667

• 0.093

Q.1 | Q 14

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

Q.1 | Q 15

Choose the correct alternative:

u = (x - 20)/5 and v = (y - 30)/4, then bxy

• 4/5 bvu

• 4/5 buv

• 5/4 buv

• 5/4 buv

Q.1 | Q 16

Choose the correct alternative:

y = 5 – 2.8x and x = 3 – 0.5 y be the regression lines, then the value of byx is

• – 0.5

• – 2.8

• 0.5

• – 2

Q.1 | Q 17

Choose the correct alternative:

If r = 0.5, σx = 3, σy2 = 16, then bxy = ______

• 0.375

• 0.667

• 2.667

• 0.093

Q.2

### 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]

Q.2 | Q 1

State whether the following statement is True or False:

The equations of two regression lines are 10x – 4y = 80 and 10y – 9x = 40. Then bxy = 0.9

• True

• False

Q.2 | Q 2

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 byx = – 0.5

• True

• False

Q.2 | Q 3

Choose the correct alternative:

Both the regression coefficients cannot exceed 1

• True

• False

Q.2 | Q 4

State whether the following statement is True or False:

If byx = 1.5 and bxy = 1/3 then r = 1/2, the given data is consistent

• True

• False

Q.2 | Q 5

State whether the following statement is True or False:

Correlation analysis is the theory of games

• True

• False

Q.2 | Q 6

State whether the following statement is True or False:

If bxy < 0 and byx < 0 then ‘r’ is > 0

• True

• False

Q.2 | Q 7

State whether the following statement is True or False:

The following data is not consistent: byx + bxy =1.3 and r = 0.75

• True

• False

Q.2 | Q 8

State whether the following statement is True or False:

If u = x – a and v = y – b then bxy = buv

• True

• False

Q.2 | Q 9

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

Q.2 | Q 10

State whether the following statement is True or False:

bxy is the slope of regression line of y on x

• True

• False

Q.2 | Q 11

State whether the following statement is True or False:

Corr(x, x) = 0

• True

• False

Q.2 | Q 12

State whether the following statement is True or False:

Corr(x, x) = 1

• True

• False

Q.2 | Q 13

State whether the following statement is True or False:

Cov(x, x) = Variance of x

• True

• False

Q.2 | Q 14

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

Q.2 | Q 15

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

Q.2 | Q 16

State whether the following statement is True or False:

The variable used for predicting the response is called the independent variable

• True

• False

Q.3

### 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]

Q.3 | Q 1

If n = 5, ∑xy = 76, ∑x2 = ∑y2 = 90, ∑x = 20 = ∑y, the covariance = ______

Q.3 | Q 2

|bxy + byx| ≥ ______

Q.3 | Q 3

Among the given regression lines 6x + y – 31 = 0 and 3x + 2y – 26 = 0, the regression line of x on y is ______

Q.3 | Q 4

If u = (x - "a")/"c" and v = (y - "b")/"d", then bxy = ______

Q.3 | Q 5

If the regression equations are 8x – 10y + 66 = 0 and 40x – 18y = 214, the mean value of y is ______

Q.3 | Q 6

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

Q.3 | Q 7

The value of product moment correlation coefficient between x and x is ______

Q.3 | Q 8

Arithmetic mean of positive values of regression coefficients is greater than or equal to ______

Q.3 | Q 9

If u = (x - 20)/5 and v = (y - 30)/4, then byx = ______

Q.3 | Q 10

The geometric mean of negative regression coefficients is ______

Q.3 | Q 11

Dependent variables are also known as ______

Q.3 | Q 12

byx is the ______ of regression line of y on x

Q.4

### 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]

Q.4 | Q 1

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

Q.4 | Q 2

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?

Q.4 | Q 3. a.

The equations of the two lines of regression are 2x + 3y − 6 = 0 and 5x + 7y − 12 = 0. Identify the regression lines

Q.4 | Q 3. b.

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)

Q.4 | Q 4

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
Q.4 | Q 5

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.

Q.4 | Q 6. a.

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

Q.4 | Q 6. b.

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

Q.4 | Q 6. c.

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

Q.4 | Q 7

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

Q.4 | Q 8

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.

Q.4 | Q 9. a.

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

Q.4 | Q 9. b.

If n = 5, Σx = Σy = 20, Σx2 = Σy2 = 90, Σxy = 76 Find the regression equation of x on y

Q.4 | Q 10

If n = 6, Σx = 36, Σy = 60, Σxy = –67, Σx2 = 50, Σy2 =106, Estimate y when x is 13

Q.4 | Q 11

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
Q.4 | Q 12

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

Q.5

### 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]

Q.5 | Q 1
 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

bxy = square/square

byx = 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

Q.5 | Q 2

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

Q.5 | Q 3

Mean of x = 25

Mean of y = 20

sigma_x = 4

sigma_y = 3

r = 0.5

byx = square

bxy = square

when x = 10,

y - square = square (10 - square)

∴ y = square

Q.5 | Q 4

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

∴ byx = square/square

∴ byx = square/square

∴ r = square

Q.5 | Q 5

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

∴ byx = square/square

∴ bxy = square/square

∴ r = square

Given variance of x = 9

∴ byx = square/square

∴ sigma_y = square

Q.5 | Q 6
 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

bxy = square/square

byx = square/square

∴ Regression equation of x on y is square

∴ Regression equation of y on x is square

## Chapter 2: Linear Regression

Q.1Q.2Q.3Q.4Q.5 ## 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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