Balbharati Maharashtra State Board 12th Commerce Maths Digest Pdf Chapter 3 Linear Regression Miscellaneous Exercise 3 Questions and Answers.

## Maharashtra State Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Miscellaneous Exercise 3

(I) Choose the correct alternative.

Question 1.

Regression analysis is the theory of

(a) Estimation

(b) Prediction

(c) Both a and b

(d) Calculation

Answer:

(c) Both a and b

Question 2.

We can estimate the value of one variable with the help of other known variable only if they are

(a) Correlated

(b) Positively correlated

(c) Negatively correlated

(d) Uncorrelated

Answer:

(a) Correlated

Question 3.

There are ________ types of regression equation

(a) 4

(b) 2

(c) 3

(d) 1

Answer:

(b) 2

Question 4.

In the regression equation of Y on X

(a) X is independent and Y is dependent

(b) Y is independent and X is dependent

(c) Both X and Y are independent

(d) Both X and Y are dependent.

Answer:

(a) X is independent and Y is dependent

Question 5.

In the regression equation of X on Y

(a) X is independent and Y is dependent

(b) Y is independent and X is dependent

(c) Both X and Y are independent

(d) Both X and Y are dependent

Answer:

(b) Y is independent and X is dependent

Question 6.

b_{xy} is ________

(a) Regression coefficient of Y on X

(b) Regression coefficient of X on Y

(c) Correlation coefficient between X and Y

(d) Covariance between X and Y

Answer:

(b) Regression coefficient of X on Y

Question 7.

b_{yx} is ________

(a) Regression coefficient of Y on X

(b) Regression coefficient of X on Y

(c) Correlation coefficient between X and Y

(d) Covariance between X and Y

Answer:

(a) Regression coefficient of Y on X

Question 8.

‘r’ is ________

(a) Regression coefficient of Y on X

(b) Regression coefficient of X on Y

(c) Correlation coefficient between X and Y

(d) Covariance between X and Y

Answer:

(d) Correlation coefficient between X and Y

Question 9.

b_{xy} . b_{yx} = _________

(a) v

(b) y_{x}

(c) r^{2}

(d) (y_{y})^{2}

Answer:

(c) r^{2}

Question 10.

If b_{yx} > 1 then b_{xy} is ______

(a) > 1

(b) < 1

(c) > 0

(d) < 0

Answer:

(b) < 1

Question 11.

|b_{xy} + b_{yx}| > ______

(a) |r|

(b) 2|r|

(c) r

(d) 2r

Answer:

(b) 2|r|

Question 12.

b_{xy} and b_{yx} are ________

(a) Independent of change of origin and scale

(b) Independent of change of origin but not of the scale

(c) Independent of change of scale but not of origin

(d) Affected by change of origin and scale

Answer:

(b) Independent of change of origin but not of the scale

Question 13.

If u = \(\frac{x-a}{c}\) and v = \(\frac{y-b}{d}\) then b_{yx} = ________

(a) \(\frac{d}{c} b_{v u}\)

(b) \(\frac{c}{d} b_{v u}\)

(c) \(\frac{a}{b} b_{v u}\)

(d) \(\frac{b}{a} b_{v u}\)

Answer:

(a) \(\frac{d}{c} b_{v u}\)

Question 14.

If u = \(\frac{x-a}{c}\) and v = \(\frac{y-b}{d}\) then b_{xy} = ________

(a) \(\frac{d}{c} b_{u v}\)

(b) \(\frac{c}{d} b_{u v}\)

(c) \(\frac{a}{b} b_{u v}\)

(d) \(\frac{b}{a} b_{u v}\)

Answer:

(b) \(\frac{c}{d} b_{u v}\)

Question 15.

Corr(x, x) = ________

(a) 0

(b) 1

(c) -1

(d) can’t be found

Answer:

(b) 1

Question 16.

Corr (x, y) = ________

(a) corr(x, x)

(b) corr(y, y)

(c) corr(y, x)

(d) cov(y, x)

Answer:

(c) corr(y, x)

Question 17.

Corr\(\left(\frac{x-a}{c}, \frac{y-b}{d}\right)\) = -corr(x, y) if,

(a) c and d are opposite in sign

(b) c and d are same in sign

(c) a and b are opposite in sign

(d) a and b are same in sign

Answer:

(a) c and d are opposite in sign

Question 18.

Regression equation of X and Y is

(a) y – \(\bar{y}\) = b_{yx} (x – \(\bar{x}\))

(b) x – \(\bar{x}\) = b_{xy} (y – \(\bar{y}\))

(c) y – \(\bar{y}\) = b_{xy} (x – \(\bar{x}\))

(d) x – \(\bar{x}\) = b_{yx} (y – \(\bar{y}\))

Answer:

(b) x – \(\bar{x}\) = b_{xy} (y – \(\bar{y}\))

Question 19.

Regression equation of Y and X is

(a) y – \(\bar{y}\) = b_{yx} (x – \(\bar{x}\))

(b) x – \(\bar{x}\) = b_{xy} (y – \(\bar{y}\))

(c) y – \(\bar{y}\) = b_{xy} (x – \(\bar{x}\))

(d) x – \(\bar{x}\) = b_{yx} (y – \(\bar{y}\))

Solution:

(a) y – \(\bar{y}\) = b_{yx} (x – \(\bar{x}\))

Question 20.

b_{yx} = ________

(a) \(r \frac{\sigma_{x}}{\sigma_{y}}\)

(b) \(r \frac{\sigma_{y}}{\sigma_{x}}\)

(c) \(\frac{1 \sigma_{y}}{r \sigma_{x}}\)

(d) \(\frac{1 \sigma_{y}}{r \sigma_{y}}\)

Answer:

(b) \(r \frac{\sigma_{y}}{\sigma_{x}}\)

Question 21.

b_{xy} = ________

(a) \(r \frac{\sigma_{x}}{\sigma_{y}}\)

(b) \(r \frac{\sigma_{y}}{\sigma_{x}}\)

(c) \(\frac{1 \sigma_{y}}{r \sigma_{x}}\)

(d) \(\frac{1 \sigma_{y}}{r \sigma_{y}}\)

Answer:

(a) \(r \frac{\sigma_{x}}{\sigma_{y}}\)

Question 22.

Cov (x, y) = ________

(a) Σ(x – \(\bar{x}\))(y – \(\bar{y}\))

(b) \(\frac{\sum(x-\bar{x})(y-\bar{y})}{n}\)

(c) \(\frac{\sum x y}{n}-\bar{x} \bar{y}\)

(d) b and c both

Answer:

(d) b and c both

Question 23.

If b_{xy} < 0 and b_{yx} < 0 then ‘r’ is ________

(a) > 0

(b) < 0

(c) > 1

(d) not found

Answer:

(b) < 0

Question 24.

If equation of regression lines are 3x + 2y – 26 = 0 and 6x + y – 31 = 0 then means of x and y are ________

(a) (7, 4)

(b) (4, 7)

(c) (2, 9)

(d) (-4, 7)

Answer:

(b) (4, 7)

(II) Fill in the blanks:

Question 1.

If b_{xy} < 0 and b_{yx} < 0 then ‘r’ is ________

Answer:

negative

Question 2.

Regression equation of Y on X is ________

Answer:

(y – \(\bar{y}\)) = b_{yx} (x – \(\bar{x}\))

Question 3.

Regression equation of X on Y is ________

Answer:

(x – \(\bar{x}\)) = b_{xy} (y – \(\bar{y}\))

Question 4.

There are ______ types of regression equations.

Answer:

2

Question 5.

Corr (x_{1} – x) = ______

Answer:

-1

Question 6.

If u = \(\frac{x-a}{c}\) and v = \(\frac{y-b}{d}\) then b_{xy} = ______

Answer:

\(\frac{c}{d} b_{u v}\)

Question 7.

If u = \(\frac{x-a}{c}\) and v = \(\frac{y-b}{d}\) then b_{yx} = ______

Answer:

\(\frac{d}{c} b_{v u}\)

Question 8.

|b_{xy} + b_{yx}| ≥ ______

Answer:

2|r|

Question 9.

If b_{yx} > 1 then b_{xy} is ______

Answer:

< 1

Question 10.

b_{xy} . b_{yx} = ______

Answer:

r^{2}

(III) State whether each of the following is True or False.

Question 1.

Corr (x, x) = 1.

Answer:

True

Question 2.

Regression equation of X on Y is y – \(\bar{y}\) = b_{xy} (x – \(\bar{x}\)).

Answer:

False

Question 3.

Regression equation of Y on X is y – \(\bar{y}\) = b_{yx} (x – \(\bar{x}\)).

Answer:

True

Question 4.

Corr (x, y) = Corr (y, x).

Answer:

True

Question 5.

b_{xy} and b_{yx} are independent of change of origin and scale.

Answer:

False

Question 6.

‘r’ is the regression coefficient of Y on X.

Answer:

False

Question 7.

b_{yx} is the correlation coefficient between X and Y.

Answer:

False

Question 8.

If u = x – a and v = y – b then b_{xy} = b_{uv}.

Answer:

True

Question 9.

If u = x – a and v = y – b then r_{xy} = r_{uv}.

Answer:

True

Question 10.

In the regression equation of Y on X, b_{yx} represents the slope of the line.

Answer:

True

(IV) Solve the following problems.

Question 1.

The data obtained on X, the length of time in weeks that a promotional project has been in progress at a small business, and Y the percentage increase in weekly sales over the period just prior to the beginning of the campaign.

Find the equation of regression line to predict percentage increase in sales if the company has been in progress for 1.5 weeks.

Solution:

Let u = x – 3, v = y – 15

∴ Regression equation of Y on X is

(y – \(\bar{y}\)) = b_{yx} (x – \(\bar{x}\))

(y – 14.67) = 2.6(x – 2.5)

y – 14.67 = 2.6x – 6.5

y = 2.6x + 8.17

When x = 1.5

y = (2.6)(1.5) + 8.17

= 3.9 + 8.17

= 12.07

Question 2.

The regression equation of y on x is given by 3x + 2y – 26 = 0. Find b_{yx}.

Solution:

Given, regression equation of Y on X is

3x + 2y – 26 = 0

∴ 2y = -3x + 26

∴ y = \(\frac{-3}{2}\)x + 13

∴ b_{yx} = \(\frac{-3}{2}\)

Question 3.

If for a bivariate data \(\bar{x}\) = 10, \(\bar{y}\) = 12, v(x) = 9, σ_{y} = 4 and r = 0.6. Estimate y when x = 5.

Solution:

Given, V(x) = 9

∴ σ_{x} = 3

b_{yx} = \(\frac{r \cdot \sigma_{y}}{\sigma_{x}}\)

= 0.6 × \(\frac{4}{3}\)

= 0.8

∴ Regression equation of Y on X is

(y – \(\bar{y}\)) = b_{yx} (x – \(\bar{x}\))

(y – 12) = 0.8(5 – 10)

y – 12 = 0.8(-5)

y – 12 = -4

y = 8

Question 4.

The equation of the line of regression of y on x is v = \(\frac{2}{9} x\) and x on y is x = \(\frac{y}{2}+\frac{7}{6}\). Find (i) r (ii) \(\sigma_{y}^{2} \text { if } \sigma_{x}^{2}=4\).

Solution:

Question 5.

Identify the regression equations of x on y and y on x from the following equations.

2x + 3y = 6 and 5x + 7y – 12 = 0

Solution:

∴ Our assumption is correct

∴ Regression equation of Y on X is 2x + 3y = 6

∴ Regression equation of X on Y is 5x + 7y – 12 = 0

Question 6.

(i) If for a bivariate data b_{yx} = -1.2 and b_{xy} = -0.3 then find r.

(ii) From the two regression equations y = 4x – 5 and 3x = 2y + 5, find \(\bar{x}\) and \(\bar{y}\).

Solution:

r^{2} = b_{yx} . b_{xy}

r^{2} = (-1.2) × (-0.3)

r^{2} = 0.36

r = ±0.6

Since, b_{yx} . b_{xy} are negative, r = -0.6

Also,(\(\bar{x}\), \(\bar{y}\)) is the point of intersection of the regression lines

y = 4x – 5, 3x = 2y + 5

8x – 2y = 10

3x – 2y = 5

on subtracting,

5x = 5

x = 1

Substituting x = 1 in y = 4x – 5

y = 4(1) – 5

y = -1

∴ \(\bar{x}\) = 1, \(\bar{y}\) = -1

Question 7.

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

(i) Means of X and Y

(ii) Correlation coefficient between X on Y

(iii) Estimate of Y for X = 2

(iv) var (X) if var (Y) = 36

Solution:

(i) Since (\(\bar{x}\), \(\bar{y}\)) is the point of intersection of regression lines

3x + 2y = 26

6x + y = 31

3x + 2y = 26 …….(i)

12x + 2y = 62 ……..(ii)

on subtracting,

-9x = -36

x = 4

Substituting x = 4 in equation (i)

3(4) + 2y = 26

2y = 14

y = 7

∴ \(\bar{x}\) = 4, \(\bar{y}\) = 7

Question 8.

Find the line of regression of X on Y for the following data:

n = 8, Σ(x_{i} – x)^{2} = 36, Σ(y_{i} – y)^{2} = 44, Σ(x_{i} – x)(y_{i} – y) = 24

Solution:

Question 9.

Find the equation of line regression of Y on X for the following data:

n = 8, Σ(x_{i} – \(\bar{x}\))(y_{i} – \(\bar{y}\)) = 120, \(\bar{x}\) = 20, \(\bar{y}\) = 36, σ_{x} = 2, σ_{y} = 3.

Solution:

Regression equation of Y on X is

(y – \(\bar{y}\)) = b_{yx} (x – \(\bar{x}\))

(y – 36) = 3.75(x – 20)

(y – 36) = 3.75x – 75

y = 3.75x – 39

Question 10.

The following result was obtained from records of age (X) and systolic blood pressure (Y) of a group of 10 men.

and Σ(x_{i} – \(\bar{x}\))(y_{i} – \(\bar{x}\)) = 1120. Find the Prediction of blood pressure of a man of age 40 years.

Solution:

Regression equation of Y on X is

(y – \(\bar{y}\)) = b_{yx} (x – \(\bar{x}\))

(y – 140) = 0.7(40 – 50)

y – 140 = 0.7(-10)

y – 140 = -7

∴ y = 133

Question 11.

The equations of two regression lines are 10x – 4y = 80 and 10y – 9x = -40 Find:

(i) \(\bar{x}\) and \(\bar{y}\)

(ii) b_{yx} and b_{xy}

(iii) If var(Y) = 36, obtain var(X)

(iv) r

Solution:

(i) Since (\(\bar{x}\), \(\bar{y}\)) is the point of intersection of regression

10x – 4y = 80 ……(i)

-9x + 10y = -40 ……..(ii)

50x – 20y = 400

-18x + 20y = -80

32x = 320

x = 10

x = 10 in equation (i)

10(10) – 4y = 80

4y = 20

y = 5

∴ \(\bar{x}\) = 10, \(\bar{y}\) = 5

(iv) r^{2} = b_{yx} . b_{xy} = 0.36

r = ±0.6

Since b_{yx} and b_{xy} are positive

∴ r = 0.6

Question 12.

If b_{yx} = -0.6 and b_{xy} = -0.216 then find correlation coefficient between X and Y comment on it.

Solution:

r^{2} = b_{yx} . b_{xy}

r^{2} = -0.6 × -0.216

r^{2} = 0.1296

r = ±√0.1296

r = ± 0.36

Since b_{yx} and b_{xy} are negative

r = -0.36