Balbharati Maharashtra State Board12th Commerce Maths Digest PdfChapter 3 Linear Regression Ex 3.2 Questions and Answers.
Maharashtra State Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.2
Question 1
Maharashtra Board Solution
For bivariate data. = 53, = 28, byx = -1.2, bxy = -0.3 Find, (i) Correlation coefficient between X and Y. (ii) Estimate Y for X = 50 (iii) Estimate X for Y = 25
Solution & Step-by-Step Answer:
(i) r2 = byx. bxy r2 = (-1.2)(-0.3) r2 = 0.36 r = ±0.6 Since, byx and bxy are negative, r = -0.6
(ii) Regression equation of Y on X is
(Y – ) = byx(X – )
Y – 28 = -1.2(50 – 53)
Y – 28 = -1.2(-3)
Y – 28 = 3.6
Y = 31.6
(iii) Regression equation of X on Y is
(X – ) = bxy(Y – )
(X – 53) = -0.3(25 – 28)
X – 53 = -0.3(-3)
X – 53 = 0.9
X = 53.9
Question 2
Maharashtra Board Solution
From the data of 20 pairs of observation on X and Y, following result are obtained = 199, = 94, = 1200, = 300 = -250 Find (i) The line of regression of Y on X. (ii) The line of regression of X on Y. (iii) Correlation coefficient between X on Y.
Solution & Step-by-Step Answer:
Question 3
Maharashtra Board Solution
From the data of 7 pairs of observations on X and Y following results are obtained. Σ(xi – 70 ) = -35, Σ(yi – 60) = -7, Σ(xi – 70)2 = 2989, Σ(yi – 60)2 = 476, Σ(xi – 70) (yi – 60) = 1064 [Given √0.7884 = 0.8879] Obtain (i) The line of regression of Y on X. (ii) The line of regression of X on Y. (iii) The correlation coefficient between X and Y.
Solution & Step-by-Step Answer:
(i) Line of regression Y on X is (Y – ) = byx (X – ) (Y – 59) = 0.36(x – 65) (Y – 59) = 0.36x – 23.4 Y = 0.36x + 35.6
(ii) Line of regression X on Y is
(X – ) = bxy(Y – )
(X – 65) = 2.19(y – 59)
(X – 65) = 2.19y – 129.21
X = 2.19y – 64.21
(iii) r2= byx. bxy
r2= (0.36) (2.19)
r2= 0.7884
r = ±√0.7884 = ±0.8879
Since byxand bxyare positive.
∴ r = 0.8879
Question 4
Maharashtra Board Solution
You are given the following information about advertising expenditure and sales. Correlation coefficient between X and Y is 0.8 (i) Obtain two regression equations. (ii) What is the likely sales when the advertising budget is ₹ 15 lakh? (iii) What should be the advertising budget if the company wants to attain sales target of ₹ 120 lakh?
Solution & Step-by-Step Answer:
Given, = 10, = 90, σx = 3, σy = 12, r = 0.8 byx = = 3.2 bxy = = 0.2 (i) Regression equation of Y on X is (Y – ) = byx (X – ) (Y – 90) = 3.2(x – 10) Y – 90 = 3.2x – 32 Y = 3.2x + 58 Regression equation of X on Y is (X – ) = bxy (Y – ) (X – 10) = 0.2(y – 90) X – 10 = 0.2y + 18 X = 0.2y – 8
(ii) When x = 15,
Y = 3.2(15) + 58
= 48 + 58
= 106 lakh
(iii) When y = 120
X = 0.2(120) – 8
= 24 – 8
= 16 lakh
Question 5
Maharashtra Board Solution
Bring out inconsistency if any, in the following: (i) byx + bxy = 1.30 and r = 0.75 (ii) byx = bxy = 1.50 and r = -0.9 (iii) byx = 1.9 and bxy = -0.25 (iv) byx = 2.6 and bxy =
Solution & Step-by-Step Answer:
(i) Given, byx + bxy = 1.30 and r = 0.75 = 0.65 But for regression coefficients byx and bxy Here, 0.65 < r = 0.75 ∴ The data is inconsistent (ii) The signs of byx, bxy and r must be same (all three positive or all three negative) ∴ The data is inconsistent.
(iii) The signs of byxand bxyshould be same (either both positive or both negative)
∴ The data is consistent.
(iv) byx. bxy= 2.6 × = 1
∴ 0 ≤ r2≤ 1
∴ The data is consistent.
Question 6
Maharashtra Board Solution
Two sample from bivariate populations have 15 observation each. The sample means of X and Y are 25 and 18 respectively. The corresponding sum of square of deviations from respective means are 136 and 150. The sum of product of deviations from respective means is 123. Obtain the equation of line of regression of X on Y.
Solution & Step-by-Step Answer:
Given, n = 15, = 25, = 18, Σ(x – ) = 136, Σ(y – ) = 150, Σ(x – ) (y – ) = 123 Regression equation of X on Y is (X – ) = bxy (Y – ) (X – 25) = 0.82(y – 18) (X – 25) = 082y – 14.76 X = 0.82y + 10.24
Question 7
Maharashtra Board Solution
For a certain bivariate data And r = 0.5 estimate y when x = 10 and estimate x when y = 16
Solution & Step-by-Step Answer:
Given, = 25, = 20, σx = 4, σy = 3, r = 0.5 byx = = 0.375 Regression equation of Y on X is (Y – ) = byx (X – ) (Y – 20) = 0.375(x – 25) Y – 20 = 0.375x – 9.375 Y = 0.375x + 10.625 When, x = 10 Y = 0.375(10) + 10.625 = 3.75 + 10.625 = 14.375 bxy = = 0.67 Regression equation of X on Y is (X – ) = byx (Y – ) (X – 25) = 0.67(y – 20) (X – 25) = 0.67y – 13.4 X = 0.67y + 11.6 When, Y = 16 x = 0.67(16) + 11.6 = 10.72 + 11.6 = 22.32
Question 8
Maharashtra Board Solution
Given the following information about the production and demand of a commodity obtain the two regression lines: Coefficient of correlation between X and Y is 0.6. Also estimate the problem when demand is 100.
Solution & Step-by-Step Answer:
Given = 85, = 90, σx = 5, σy = 6 and r = 0.6 bxy = = 0.5 byx = = 0.72 Regression equation of X on Y is (X – ) = bxy (Y – ) (X – 85) = 0.5(y – 90) (X – 85 ) = 0.5y – 45 X = 0.5y + 40 When y = 100, x = 0.5 (100) + 40 = 50 + 40 = 90 Regression equation of Y on X is (Y – ) = byx (X – ) (Y – 90) = 0.72(x – 85) (Y – 90) = 0.72x – 61.2 Y = 0.72x + 28.8
Question 9
Maharashtra Board Solution
Given the following data, obtain linear regression estimate of X for Y = 10
Solution & Step-by-Step Answer:
= 7.6, = 14.8, σx = 3.2, σy = 16 and r = 0.7 bxy = = 0.14 Regression equation of X on Y is (X – ) = bxy (Y – ) (X – 7.6) = 0.14(y – 14.8) X – 7.6 = 0.14y – 2.072 X = 0.14y + 5.528 When y = 10 x = 0.14(10) + 5.528 = 1.4 + 5.528 = 6.928
Question 10
Maharashtra Board Solution
An inquiry of 50 families to study the relationship between expenditure on accommodation (₹ x) and expenditure on food and entertainment (₹ y) gave the following result: Σx = 8500, Σy = 9600, σx = 60, σy = 20, r = 0.6 Estimate the expenditure on food and entertainment when expenditure on accommodation is ₹ 200
Solution & Step-by-Step Answer:
n = 50 (given) Regression equation of Y on X is Y – = byx (X – ) (Y – 192) = 0.2(200 – 170) Y – 192 = 0.2(30) Y = 192 + 6 Y = 198
Question 11
Maharashtra Board Solution
The following data about the sales and advertisement expenditure of a firms is given below (in ₹ crores) Also correlation coefficient between X and Y is 0.9 (i) Estimate the likely sales for a proposed advertisement expenditure of ₹ 10 crores. (ii) What should be the advertisement expenditure if the firm proposes a sales target ₹ 60 crores Let the sales be X and advertisement expenditure be Y
Solution & Step-by-Step Answer:
Given, = 40, = 6, σx = 10, σy = 1.5, r = 0.9 byx = = 0.135 bxy = = 6 (i) Regression equation of X on Y is (X – ) = bxy (Y – ) (X – 40) = 6(y – 6) X – 40 = 6y – 36 X = 6y + 4 When y = 10 x = 6 (10) + 4 = 60 + 4 = 64 crores
(ii) Regression equation Y on X is
(Y – ) = byx(X – )
(Y – 6) = 0.135 (x – 40)
Y – 6 = 0.135x – 5.4
Y = 0.135x + 0.6
When x = 60
Y = 0.135 (60) + 0.6
= 8.1 + 0.6
= 8.7 crores
Question 12
Maharashtra Board Solution
For certain bivariate data the following information are available Correlation coefficient between x and y is 0.6, estimate x when y = 15 and estimate y when x = 10.
Solution & Step-by-Step Answer:
Given, = 13, = 17, σx = 3, σy = 2, r = 0.6 byx = = 0.4 bxy = = 0.9 Regression equation of Y on X (Y – ) = byx (X – ) Y – 17 = 0.4(x – 13) Y = 0.4x + 11.8 When x = 10 Y = 0.4(10) + 11.8 = 4 + 11.8 = 15.8 Regression equation of X on Y (X – ) = bxy (Y – ) (X – 13) = 0.9(y – 17) X – 13 = 0.9y – 15.3 X = 0.9y – 2.3 When y = 15 X = 0.9(15) – 2.3 = 13.5 – 2.3 = 11.2
