Question 6: If u = and v = then bxy = ______

Step-by-step solution provided.

Maharashtra State Board Class 12 (HSC Board) Commerce Mathematics & Statistics Solutions Chapter 3 Linear Regression Miscellaneous Exercise 3 - Digest Answers | Maharashtra Board Solutions

Balbharati Maharashtra State Board12th Commerce Maths Digest PdfChapter 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 Maharashtra Board Solution

Regression analysis is the theory of (a) Estimation (b) Prediction (c) Both a and b (d) Calculation

Solution & Step-by-Step Answer:

Question 2 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

(a) Correlated

Question 3 Maharashtra Board Solution

There are ________ types of regression equation (a) 4 (b) 2 (c) 3 (d) 1

Solution & Step-by-Step Answer:

(b) 2

Question 4 Maharashtra Board Solution

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.

Solution & Step-by-Step Answer:

(a) X is independent and Y is dependent

Question 5 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

(b) Y is independent and X is dependent

Question 6 Maharashtra Board Solution

bxy 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

Solution & Step-by-Step Answer:

(b) Regression coefficient of X on Y

Question 7 Maharashtra Board Solution

byx 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

Solution & Step-by-Step Answer:

(a) Regression coefficient of Y on X

Question 8 Maharashtra Board Solution

‘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

Solution & Step-by-Step Answer:

(d) Correlation coefficient between X and Y

Question 9 Maharashtra Board Solution

bxy. byx = _________ (a) v (b) yx (c) r2 (d) (yy)2

Solution & Step-by-Step Answer:

Question 10 Maharashtra Board Solution

If byx > 1 then bxy is ______ (a) > 1 (b) < 1 (c) > 0 (d) < 0

Solution & Step-by-Step Answer:

(b) < 1

Question 11 Maharashtra Board Solution

|bxy + byx| > ______ (a) |r| (b) 2|r| (c) r (d) 2r

Solution & Step-by-Step Answer:

(b) 2|r|

Question 12 Maharashtra Board Solution

bxy and byx 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

Solution & Step-by-Step Answer:

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

Question 13 Maharashtra Board Solution

If u = and v = then byx = ________ (a) (b) (c) (d)

Solution & Step-by-Step Answer:

(a)

Question 14 Maharashtra Board Solution

If u = and v = then bxy = ________ (a) (b) (c) (d)

Solution & Step-by-Step Answer:

(b)

Question 15 Maharashtra Board Solution

Corr(x, x) = ________ (a) 0 (b) 1 (c) -1 (d) can’t be found

Solution & Step-by-Step Answer:

(b) 1

Question 16 Maharashtra Board Solution

Corr (x, y) = ________ (a) corr(x, x) (b) corr(y, y) (c) corr(y, x) (d) cov(y, x)

Solution & Step-by-Step Answer:

Question 17 Maharashtra Board Solution

Corr = -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

Solution & Step-by-Step Answer:

(a) c and d are opposite in sign

Question 18 Maharashtra Board Solution

Regression equation of X and Y is (a) y – = byx (x – ) (b) x – = bxy (y – ) (c) y – = bxy (x – ) (d) x – = byx (y – )

Solution & Step-by-Step Answer:

(b) x – = bxy (y – )

Question 19 Maharashtra Board Solution

Regression equation of Y and X is (a) y – = byx (x – ) (b) x – = bxy (y – ) (c) y – = bxy (x – ) (d) x – = byx (y – )

Solution & Step-by-Step Answer:

(a) y – = byx (x – )

Question 20 Maharashtra Board Solution

byx = ________ (a) (b) (c) (d)

Solution & Step-by-Step Answer:

(b)

Question 21 Maharashtra Board Solution

bxy = ________ (a) (b) (c) (d)

Solution & Step-by-Step Answer:

(a)

Question 22 Maharashtra Board Solution

Cov (x, y) = ________ (a) Σ(x – )(y – ) (b) (c) (d) b and c both

Solution & Step-by-Step Answer:

(d) b and c both

Question 23 Maharashtra Board Solution

If bxy < 0 and byx < 0 then ‘r’ is ________ (a) > 0 (b) < 0 (c) > 1 (d) not found

Solution & Step-by-Step Answer:

(b) < 0

Question 24 Maharashtra Board Solution

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)

Solution & Step-by-Step Answer:

(b) (4, 7)

(II) Fill in the blanks:

Question 1 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

negative

Question 2 Maharashtra Board Solution

Regression equation of Y on X is ________

Solution & Step-by-Step Answer:

(y – ) = byx (x – )

Question 3 Maharashtra Board Solution

Regression equation of X on Y is ________

Solution & Step-by-Step Answer:

(x – ) = bxy (y – )

Question 4 Maharashtra Board Solution

There are ______ types of regression equations.

Solution & Step-by-Step Answer:

Question 5 Maharashtra Board Solution

Corr (x1 – x) = ______

Solution & Step-by-Step Answer:

-1

Question 6 Maharashtra Board Solution

If u = and v = then bxy = ______

Solution & Step-by-Step Answer:

Question 7 Maharashtra Board Solution

If u = and v = then byx = ______

Solution & Step-by-Step Answer:

Question 8 Maharashtra Board Solution

|bxy + byx| ≥ ______

Solution & Step-by-Step Answer:

2|r|

Question 9 Maharashtra Board Solution

If byx > 1 then bxy is ______

Solution & Step-by-Step Answer:

< 1

Question 10 Maharashtra Board Solution

bxy. byx = ______

Solution & Step-by-Step Answer:

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

Question 1 Maharashtra Board Solution

Corr (x, x) = 1.

Solution & Step-by-Step Answer:

True

Question 2 Maharashtra Board Solution

Regression equation of X on Y is y – = bxy (x – ).

Solution & Step-by-Step Answer:

False

Question 3 Maharashtra Board Solution

Regression equation of Y on X is y – = byx (x – ).

Solution & Step-by-Step Answer:

True

Question 4 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

True

Question 5 Maharashtra Board Solution

bxy and byx are independent of change of origin and scale.

Solution & Step-by-Step Answer:

False

Question 6 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

False

Question 7 Maharashtra Board Solution

byx is the correlation coefficient between X and Y.

Solution & Step-by-Step Answer:

False

Question 8 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

True

Question 9 Maharashtra Board Solution

If u = x – a and v = y – b then rxy = ruv.

Solution & Step-by-Step Answer:

True

Question 10 Maharashtra Board Solution

In the regression equation of Y on X, byx represents the slope of the line.

Solution & Step-by-Step Answer:

True

(IV) Solve the following problems.

Question 1 Maharashtra Board Solution

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 & Step-by-Step Answer:

Let u = x – 3, v = y – 15 ∴ Regression equation of Y on X is (y – ) = byx (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 Maharashtra Board Solution

The regression equation of y on x is given by 3x + 2y – 26 = 0. Find byx.

Solution & Step-by-Step Answer:

Given, regression equation of Y on X is 3x + 2y – 26 = 0 ∴ 2y = -3x + 26 ∴ y = x + 13 ∴ byx =

Question 3 Maharashtra Board Solution

If for a bivariate data = 10, = 12, v(x) = 9, σy = 4 and r = 0.6. Estimate y when x = 5.

Solution & Step-by-Step Answer:

Given, V(x) = 9 ∴ σx = 3 byx = = 0.6 × = 0.8 ∴ Regression equation of Y on X is (y – ) = byx (x – ) (y – 12) = 0.8(5 – 10) y – 12 = 0.8(-5) y – 12 = -4 y = 8

Question 4 Maharashtra Board Solution

The equation of the line of regression of y on x is v = and x on y is x = . Find (i) r (ii) .

Solution & Step-by-Step Answer:

Question 5 Maharashtra Board Solution

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 & Step-by-Step Answer:

∴ 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 Maharashtra Board Solution

(i) If for a bivariate data byx = -1.2 and bxy = -0.3 then find r. (ii) From the two regression equations y = 4x – 5 and 3x = 2y + 5, find and .

Solution & Step-by-Step Answer:

r2 = byx. bxy r2 = (-1.2) × (-0.3) r2 = 0.36 r = ±0.6 Since, byx. bxy are negative, r = -0.6 Also,(, ) 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 ∴ = 1, = -1

Question 7 Maharashtra Board Solution

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 & Step-by-Step Answer:

(i) Since (, ) 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 ∴ = 4, = 7

Question 8 Maharashtra Board Solution

Find the line of regression of X on Y for the following data: n = 8, Σ(xi – x)2 = 36, Σ(yi – y)2 = 44, Σ(xi – x)(yi – y) = 24

Solution & Step-by-Step Answer:

Question 9 Maharashtra Board Solution

Find the equation of line regression of Y on X for the following data: n = 8, Σ(xi – )(yi – ) = 120, = 20, = 36, σx = 2, σy = 3.

Solution & Step-by-Step Answer:

Regression equation of Y on X is (y – ) = byx (x – ) (y – 36) = 3.75(x – 20) (y – 36) = 3.75x – 75 y = 3.75x – 39

Question 10 Maharashtra Board Solution

The following result was obtained from records of age (X) and systolic blood pressure (Y) of a group of 10 men. and Σ(xi – )(yi – ) = 1120. Find the Prediction of blood pressure of a man of age 40 years.

Solution & Step-by-Step Answer:

Regression equation of Y on X is (y – ) = byx (x – ) (y – 140) = 0.7(40 – 50) y – 140 = 0.7(-10) y – 140 = -7 ∴ y = 133

Question 11 Maharashtra Board Solution

The equations of two regression lines are 10x – 4y = 80 and 10y – 9x = -40 Find: (i) and (ii) byx and bxy (iii) If var(Y) = 36, obtain var(X) (iv) r

Solution & Step-by-Step Answer:

(i) Since (, ) 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 ∴ = 10, = 5 (iv) r2 = byx. bxy = 0.36 r = ±0.6 Since byx and bxy are positive ∴ r = 0.6

Question 12 Maharashtra Board Solution

If byx = -0.6 and bxy = -0.216 then find correlation coefficient between X and Y comment on it.

Solution & Step-by-Step Answer:

r2 = byx. bxy r2 = -0.6 × -0.216 r2 = 0.1296 r = ±√0.1296 r = ± 0.36 Since byx and bxy are negative r = -0.36

Chapter 3 Linear Regression Miscellaneous Exercise 3 Solutions