Balbharati Maharashtra State Board 12th Commerce Maths Digest Pdf Chapter 3 Linear Regression Ex 3.1 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1

Question 1.
The HRD manager of the company wants to find a measure which he can use to fix the monthly income of persons applying for the job in the production department. As an experimental project. He collected data of 7 persons from that department referring to years of service and their monthly incomes.
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q1
(i) Find the regression equation of income on years of service.
(ii) What initial start would you recommend for a person applying for the job after having served in a similar capacity in another company for 13 years?
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q1.1
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q1.2
(i) Regression equation of Y on X is (Y – \(\bar{y}\)) = byx (x – \(\bar{x}\))
(Y – 8) = 0.75(x – \(\bar{x}\))
Y = 0.75x + 2
(ii) When x = 13
Y = 0.75(13) + 2 = 11.75
Recommended income for the person is ₹ 11750.

Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1

Question 2.
Calculate the regression equations of X on Y and Y on X from the following date:
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q2
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q2.1
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q2.2
Regression equation of X on Y is (X – \(\bar{x}\)) = bxy (Y – \(\bar{y}\))
(X – 14) = 1(Y – 8)
X – 14 = Y – 8
X = Y + 6
Regression equation Y on X is (Y – \(\bar{y}\)) = byx (X – \(\bar{x}\))
(Y – 8) = 0.87(X – 14)
Y – 8 = 0.87X – 12.18
Y = 0.87X – 4.18

Question 3.
For a certain bivariate data on 5 pairs of observations given
Σx = 20, Σy = 20, Σx2 = 90, Σy2 = 90, Σxy = 76
Calculate (i) cov(x, y), (ii) byx and bxy, (iii) r
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q3
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q3.1
Sine byx and bxy are negative, r = -0.4

Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1

Question 4.
From the following data estimate y when x = 125
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q4
Solution:
Let u = x – 122, v = y – 14
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q4.1
Regression equation of Y on X is
(Y – \(\bar{y}\)) = byx (X – \(\bar{x}\))
(Y – 13.5) = -0.21(x – 121.5)
Y – 13.5 = -0.21x + 25.52
Y = -0.21x + 39.02
When x = 125
Y = -0.21(125) + 39.02
= -26.25 + 39.02
= 12.77

Question 5.
The following table gives the aptitude test scores and productivity indices of 10 works selected at workers selected randomly.
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q5
Obtain the two regression equation and estimate
(i) The productivity index of a worker whose test score is 95.
(ii) The test score when productivity index is 75.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q5.1
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q5.2
Regression equation of Y on X,
(Y – \(\bar{y}\)) = byx (X – \(\bar{y}\))
(Y – 65) = 1.16 (x – 65)
Y – 65 = 1.16x – 75.4
Y = 1.16x – 10.4
(i) When x = 95
Y = 1.16(95) – 10.4
= 110.2 – 10.4
= 99.8
Regression equation of X on Y,
(X – \(\bar{x}\)) = bxy (Y – \(\bar{y}\))
(X – 65) = 0.59(y – 65)
(X – 65) = 0.59y – 38.35
X = 0.59y + 26.65
(ii) When y = 75
x = 0.59(75) + 26.65
= 44.25 + 26.65
= 70.9

Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1

Question 6.
Compute the appropriate regression equation for the following data.
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q6
Solution:
Since x is the independent variable, and y is the dependent variable,
we need to find regression equation of y on x
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q6.1
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q6.2
Regression equation of y on x is (y – \(\bar{y}\)) = byx (x – \(\bar{x}\))
(y – 10) = -13.4(x – 6)
y – 10 = -1.34x + 8.04
y = -1.34x + 18.04

Question 7.
The following are the marks obtained by the students in Economic (X) and Mathematics (Y)
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q7
Find the regression equation of Y and X.
Solution:
Let u = x – 61, v = y – 80
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q7.1
Regression equation of Y on X is
(Y – \(\bar{y}\)) = byx (X – \(\bar{x}\))
(Y – 80.4) = 0.3(x – 61)
Y – 80.4 = 0.3x – 18.3
Y = 0.3x + 62.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1

Question 8.
For the following bivariate data obtain the equation of two regressions lines:
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q8
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q8.1
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q8.2
Regression equation of Y on X
(Y – \(\bar{y}\)) = byx (X – \(\bar{x}\))
(Y – 9) = 2(x – 3)
Y – 9 = 2x – 6
Y = 2x + 3
Regression equation of X on Y
(X – \(\bar{x}\)) = bxy (Y – \(\bar{y}\))
(X – 3) = 0.5(y – 9)
(X – 3) = 0.5y – 4.5
X = 0.5y – 1.5

Question 9.
Find the following data obtain the equation of two regression lines:
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q9
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q9.1
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q9.2
Regression of Y on X,
(Y – \(\bar{y}\)) = byx (X – \(\bar{x}\))
(Y – 8) = 0.65(x – 6)
Y – 8 = -0.65x + 3.9
Y = -0.65x + 11.9
Regression of X on Y
(X – \(\bar{x}\)) = bxy (Y – \(\bar{y}\))
(X – 6) = -1.3(y – 8)
(X – 6) = -1.3y + 10.4
X = -1.3y + 16.4

Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1

Question 10.
For the following data, find the regression line of Y on X
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q10
Hence find the most likely value of y when x = 4
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q10.1
(Y – 3) = 2(x – 2)
Y – 3 = 2x – 4
Y = 2x – 1
When x = 4
Y = 2(4) – 1
= 8 – 1
= 7

Question 11.
Find the following data, find the regression equation of Y on X, and estimate Y when X = 10.
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q11
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q11.1
Regression equation of Y on X is
(Y – \(\bar{y}\)) = byx (X – \(\bar{x}\))
(Y – 5) = (0.63)(x – 3.5)
Y – 5 = 0.63x – 2.2
Y = 0.63x + 2.8
When x = 10
Y = 0.63(10) + 2.8
= 6.3 + 2.8
= 9.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1

Question 12.
The following sample gave the number of hours of study (X) per day for an examination and marks (Y) obtained by 12 students.
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q12
Obtain the line of regression of marks on hours of study.
Solution:
Let u = x – 5, v = y – 70
Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Linear Regression Ex 3.1 Q12.1
∴ Equation of marks on hours of study is
(Y – \(\bar{y}\)) = byx (X – \(\bar{x}\))
(Y – 70.83) = 6.6(x – 4.92)
Y – 70.83 = 6.6x – 32.47
∴ Y = 6.6x + 38.36