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Chapter 3: Linear Regression
Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 3 Linear RegressionExercise 3.1 [Pages 41 - 42]
The HRD manager of a 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.
Years of service (X) | 11 | 7 | 9 | 5 | 8 | 6 | 10 |
Monthly Income (₹ 1000's)(Y) | 10 | 8 | 9 | 5 | 9 | 7 | 11 |
- Find the regression equation of income on years of service.
- 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?
Calculate the regression equations of X on Y and Y on X from the following data:
X | 10 | 12 | 13 | 17 | 18 |
Y | 5 | 6 | 7 | 9 | 13 |
For a certain bivariate data on 5 pairs of observations given
∑ x = 20, ∑ y = 20, ∑ x^{2} = 90, ∑ y^{2} = 90, ∑ xy = 76
Calculate:
- cov (X, Y)
- b_{YX} and b_{XY}
- r
From the following data estimate y when x =125.
X | 120 | 115 | 120 | 125 | 126 | 123 |
Y | 13 | 15 | 14 | 13 | 12 | 14 |
The following table gives the aptitude test scores and productivity indices of 10 workers selected at random.
Aptitude score (X) | 60 | 62 | 65 | 70 | 72 | 48 | 53 | 73 | 65 | 82 |
Productivity Index (Y) | 68 | 60 | 62 | 80 | 85 | 40 | 52 | 62 | 60 | 81 |
Obtain the two regression equations and estimate the productivity index of a worker whose test score is 95.
The following table gives the aptitude test scores and productivity indices of 10 workers selected at random.
Aptitude score (X) | 60 | 62 | 65 | 70 | 72 | 48 | 53 | 73 | 65 | 82 |
Productivity Index (Y) | 68 | 60 | 62 | 80 | 85 | 40 | 52 | 62 | 60 | 81 |
Obtain the two regression equations and estimate the test score when the productivity index is 75.
Compute the appropriate regression equation for the following data:
X [Independent Variable] |
2 | 4 | 5 | 6 | 8 | 11 |
Y [dependent Variable] | 18 | 12 | 10 | 8 | 7 | 5 |
The following are the marks obtained by the students in Economics (X) and Mathematics (Y)
X | 59 | 60 | 61 | 62 | 63 |
Y | 78 | 82 | 82 | 79 | 81 |
Find the regression equation of Y on X.
For the following bivariate data obtain the equations of two regression lines:
X | 1 | 2 | 3 | 4 | 5 |
Y | 5 | 7 | 9 | 11 | 13 |
From the following data obtain the equation of two regression lines:
X | 6 | 2 | 10 | 4 | 8 |
Y | 9 | 11 | 5 | 8 | 7 |
For the following data, find the regression line of Y on X
X | 1 | 2 | 3 |
Y | 2 | 1 | 6 |
Hence find the most likely value of y when x = 4.
From the following data, find the regression equation of Y on X and estimate Y when X = 10.
X | 1 | 2 | 3 | 4 | 5 | 6 |
Y | 2 | 4 | 7 | 6 | 5 | 6 |
The following sample gives the number of hours of study (X) per day for an examination and marks (Y) obtained by 12 students.
X | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 7 | 8 |
Y | 45 | 60 | 55 | 60 | 75 | 70 | 80 | 75 | 90 | 80 | 75 | 85 |
Obtain the line of regression of marks on hours of study.
Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 3 Linear RegressionExercise 3.2 [Pages 47 - 48]
For bivariate data. `bar x = 53, bar y = 28, "b"_"YX" = - 1.2, "b"_"XY" = - 0.3` Find Correlation coefficient between X and Y.
For bivariate data. `bar x = 53, bar y = 28, "b"_"YX" = - 1.2, "b"_"XY" = - 0.3` Find estimate of Y for X = 50.
For bivariate data. `bar x = 53, bar y = 28, "b"_"YX" = - 1.2, "b"_"XY" = - 0.3` Find estimate of X for Y = 25.
From the data of 20 pairs of observations on X and Y, following results are obtained.
`barx = 199, bary = 94,`
`sum(x_i - barx)^2 = 1200, sum(y_i - bary)^2 = 300,`
`sum(x_i - bar x)(y_i - bar y) = - 250`
Find:
- The line of regression of Y on X.
- The line of regression of X on Y.
- Correlation coefficient between X and Y.
From the data of 7 pairs of observations on X and Y, following results are obtained.
∑(x_{i} - 70) = - 35, ∑(y_{i} - 60) = - 7,
∑(x_{i} - 70)^{2} = 2989, ∑(y_{i} - 60)^{2} = 476,
∑(x_{i} - 70)(y_{i} - 60) = 1064
[Given: `sqrt0.7884` = 0.8879]
Obtain
- The line of regression of Y on X.
- The line regression of X on Y.
- The correlation coefficient between X and Y.
You are given the following information about advertising expenditure and sales.
Advertisement expenditure (₹ in lakh) (X) |
Sales (₹ in lakh) (Y) | |
Arithmetic Mean | 10 | 90 |
Standard Mean | 3 | 12 |
Correlation coefficient between X and Y is 0.8
- Obtain the two regression equations.
- What is the likely sales when the advertising budget is ₹ 15 lakh?
- What should be the advertising budget if the company wants to attain sales target of ₹ 120 lakh?
Bring out the inconsistency in the following:
b_{YX} + b_{XY} = 1.30 and r = 0.75
Bring out the inconsistency in the following:
b_{YX} = b_{XY} = 1.50 and r = - 0.9
Bring out the inconsistency in the following:
b_{YX} = 1.9 and b_{XY} = - 0.25
Bring out the inconsistency in the following:
b_{YX} = 2.6 and b_{XY} = `1/2.6`
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 respective means is 136 and 150. The sum of the product of deviations from respective means is 123. Obtain the equation of the line of regression of X on Y.
For a certain bivariate data
X | Y | |
Mean | 25 | 20 |
S.D. | 4 | 3 |
And r = 0.5. Estimate y when x = 10 and estimate x when y = 16
Given the following information about the production and demand of a commodity obtain the two regression lines:
X | Y | |
Mean | 85 | 90 |
S.D. | 5 | 6 |
The coefficient of correlation between X and Y is 0.6. Also estimate the production when demand is 100.
Given the following data, obtain a linear regression estimate of X for Y = 10, `bar x = 7.6, bar y = 14.8, sigma_x = 3.2, sigma_y = 16` and r = 0.7
An inquiry of 50 families to study the relationship between expenditure on accommodation (₹ x) and expenditure on food and entertainment (₹ y) gave the following results:
∑ x = 8500, ∑ y = 9600, σ_{X} = 60, σ_{Y} = 20, r = 0.6
Estimate the expenditure on food and entertainment when expenditure on accommodation is Rs 200.
The following data about the sales and advertisement expenditure of a firms is given below (in ₹ Crores)
Sales | Adv. Exp. | |
Mean | 40 | 6 |
S.D. | 10 | 1.5 |
Coefficient of correlation between sales and advertisement expenditure is 0.9.
Estimate the likely sales for a proposed advertisement expenditure of ₹ 10 crores.
The following data about the sales and advertisement expenditure of a firms is given below (in ₹ Crores)
Sales | Adv. Exp. | |
Mean | 40 | 6 |
S.D. | 10 | 1.5 |
Coefficient of correlation between sales and advertisement expenditure is 0.9.
What should be the advertisement expenditure if the firm proposes a sales target ₹ 60 crores?
For certain bivariate data the following information is available.
X | Y | |
Mean | 13 | 17 |
S.D. | 3 | 2 |
Correlation coefficient between x and y is 0.6. estimate x when y = 15 and estimate y when x = 10.
Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 3 Linear RegressionExercise 3.3 [Pages 49 - 50]
From the two regression equations, find r, `bar x and bar y`. 4y = 9x + 15 and 25x = 4y + 17
In a partially destroyed laboratory record of an analysis of regression data, the following data are legible:
Variance of X = 9
Regression equations:
8x − 10y + 66 = 0
and 40x − 18y = 214.
Find on the basis of above information
- The mean values of X and Y.
- Correlation coefficient between X and Y.
- Standard deviation of Y.
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.
For bivariate data, the regression coefficient of Y on X is 0.4 and the regression coefficient of X on Y is 0.9. Find the value of the variance of Y if the variance of X is 9.
The equations of two regression lines are
2x + 3y − 6 = 0
and 2x + 2y − 12 = 0 Find
- Correlation coefficient
- `sigma_"X"/sigma_"Y"`
For a bivariate data: `bar x = 53, bar y = 28,` b_{YX} = - 1.5 and b_{XY} = - 0.2. Estimate Y when X = 50.
The equations of two regression lines are x − 4y = 5 and 16y − x = 64. Find means of X and Y. Also, find correlation coefficient between X and Y.
In a partially destroyed record, the following data are available: variance of X = 25, Regression equation of Y on X is 5y − x = 22 and regression equation of X on Y is 64x − 45y = 22 Find
- Mean values of X and Y
- Standard deviation of Y
- Coefficient of correlation between X and Y.
If the two regression lines for a bivariate data are 2x = y + 15 (x on y) and 4y = 3x + 25 (y on x), find
- `bar x`,
- `bar y`,
- b_{YX}
- b_{XY}
- r [Given `sqrt0.375` = 0.61]
The two regression equations are 5x − 6y + 90 = 0 and 15x − 8y − 130 = 0. Find `bar x, bar y`, r.
Two lines of regression are 10x + 3y − 62 = 0 and 6x + 5y − 50 = 0. Identify the regression of x on y. Hence find `bar x, bar y` and r.
For certain X and Y series, which are correlated the two lines of regression are 10y = 3x + 170 and 5x + 70 = 6y. Find the correlation coefficient between them. Find the mean values of X and Y.
Regression equations of two series are 2x − y − 15 = 0 and 3x − 4y + 25 = 0. Find `bar x, bar y` and regression coefficients. Also find coefficients of correlation. [Given `sqrt0.375` = 0.61]
The two regression lines between height (X) in inches and weight (Y) in kgs of girls are,
4y − 15x + 500 = 0
and 20x − 3y − 900 = 0
Find the mean height and weight of the group. Also, estimate the weight of a girl whose height is 70 inches.
Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 3 Linear RegressionMiscellaneous Exercise 3 [Pages 51 - 54]
Choose the correct alternative:
Regression analysis is the theory of
Estimation
Prediction
Estimation and Prediction
Calculation
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
Choose the correct alternative:
There are ______ types of regression equations
4
2
3
1
Choose the correct alternative.
In the regression equation of Y on X
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.
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
Choose the correct alternative.
b_{XY} is _____
Regression coefficient of Y on X
Regression coefficient of X on Y
Correlation coefficient between X and Y
Covariance between X and Y
Choose the correct alternative.
b_{YX} is __________.
Regression coefficient of Y on X
Regression coefficient of X on Y
Correlation coefficient between X and Y
Covariance between X and Y
Choose the correct alternative.
‘r’ is __________.
Regression coefficient of Y on X
Regression coefficient of X on Y
Correlation coefficient between X and Y
Covariance between X and Y
Choose the correct alternative.
b_{XY} .b_{YX} is _________.
v(x)
`sigma_"x"`
r^{2}
`(sigma_"y")^2`
Choose the correct alternative.
b_{YX} > 1 then b_{XY} is _______
> 1
< 1
> 0
< 0
Choose the correct alternative.
|b_{xy} + b_{yx} | ≥ _______
|r|
2 |r|
r
2r
Choose the correct alternative.
b_{xy} and b_{yx} 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
Choose the correct alternative.
If u = `("x - a")/"c" and "v" = ("y - b")/"d" "then" "b"_"yx"` = _________
`"d"/"c" "b"_"vu"`
`"c"/"d" "b"_"vu"`
`"a"/"b" "b"_"vu"`
`"b"/"a" "b"_"vu"`
Choose the correct alternative.
If u = `("x - a")/"c" and "v" = ("y - b")/"d" "then" "b"_"xy"` = _________
`"d"/"c" "b"_"uv"`
`"c"/"d" "b"_"uv"`
`"a"/"b" "b"_"uv"`
`"b"/"a" "b"_"uv"`
Choose the correct alternative.
Corr (x, x) = _____
0
1
- 1
can't be found
Choose the correct alternative.
Corr (x, y) = _____
corr (x,x)
corr (y,y)
corr (y,x)
cov (y,x)
Choose the correct alternative.
Corr `("x - a"/"c", "y - b"/"d")` = - corr (x, y) if,
c and d are opposite in sign
c and d are same in sign
a and b are opposite in sign
a and b are same in sign
Choose the correct alternative.
Regression equation of X on Y is ____
`"y" - bar "y" = "b"_"yx" ("x" - bar "x")`
`"x" - bar "x" = "b"_"xy" ("y" - bar "y")`
`"y" - bar "y" = "b"_"xy" ("x" - bar "x")`
`"x" - bar "x" = "b"_"yx" ("y" - bar "y")`
Choose the correct alternative.
Regression equation of Y on X is ____
`("y" - bar "y") = "b"_"yx" ("x" - bar "x")`
`("x" - bar "x") = "b"_"xy" ("y" - bar "y")`
`("y" - bar "y") = "b"_"xy" ("x" - bar "x")`
`("x" - bar "x") = "b"_"yx" ("y" - bar "y")`
Choose the correct alternative.
b_{yx} = ______
`"r" sigma_"x"/sigma_"y"`
`"r" sigma_"y"/sigma_"x"`
`1/"r" sigma_"y"/sigma_"x"`
`1/"r" sigma_"x"/sigma_"y"`
Choose the correct alternative.
b_{xy} = ______
`"r" sigma_"x"/sigma_"y"`
`"r" sigma_"y"/sigma_"x"`
`1/"r" sigma_"y"/sigma_"x"`
`1/"r" sigma_"x"/sigma_"y"`
Choose the correct alternative.
Cov (x, y) = __________
`sum (x - bar x)(y - bar y)`
`(sum (x - bar x)(y - bar y))/"n"`
`(sum xy)/"n" - bar x bar y`
both `(sum (x - bar x)(y - bar y))/"n"` and `(sum xy)/"n" - bar x bar y`
Choose the correct alternative.
If bxy < 0 and byx < 0 then 'r' is __________
> 0
< 0
> 1
not found
Choose the correct alternative.
If equations of regression lines are 3x + 2y − 26 = 0 and 6x + y − 31 = 0 then means of x and y are __________
(7, 4)
(4, 7)
(2, 9)
(-4, 7)
Fill in the blank:
If b_{xy} < 0 and b_{yx} < 0 then ‘r’ is __________
Fill in the blank:
Regression equation of Y on X is_________
Fill in the blank:
Regression equation of X on Y is_________
Fill in the blank:
There are __________ types of regression equations.
Fill in the blank:
Corr (x, −x) = __________
Fill in the blank:
If u = `"x - a"/"c" and "v" = "y - b"/"d"` then b_{xy} = _______
Fill in the blank:
If u = `"x - a"/"c" and "v" = "y - b"/"d"` then b_{yx} = _______
Fill in the blank:
|b_{xy} + b_{yx}| ≥ ______
Fill in the blank:
If b_{yx} > 1 then b_{xy} is _______
Fill in the blank:
b_{xy} . b_{yx} = _______
State whether the following statement is True or False.
Corr (x, x) = 1
True
False
State whether the following statement is True or False.
Regression equation of X on Y is `("y" - bar "y") = "b"_"yx" ("x" - bar "x")`
True
False
State whether the following statement is True or False.
Regression equation of Y on X is `("y" - bar "y") = "b"_"yx" ("x" - bar "x")`
True
False
State whether the following statement is True or False.
Corr (x, y) = Corr (y, x)
True
False
State whether the following statement is True or False.
b_{xy} and b_{yx} are independent of change of origin and scale.
True
False
State whether the following statement is True or False.
‘r’ is regression coefficient of Y on X
True
False
State whether the following statement is True or False.
b_{yx} is correlation coefficient between X and Y
True
False
State whether the following statement is True or False.
If u = x - a and v = y - b then b_{xy} = b_{uv}
True
False
State whether the following statement is True or False.
If u = x - a and v = y - b then r_{xy} = r_{uv}
True
False
State whether the following statement is True or False.
In the regression equation of Y on X, b_{yx} represents slope of the line.
True
False
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.
X | 1 | 2 | 3 | 4 | 1 | 3 | 1 | 2 | 3 | 4 | 2 | 4 |
Y | 10 | 10 | 18 | 20 | 11 | 15 | 12 | 15 | 17 | 19 | 13 | 16 |
Find the equation of the regression line to predict the percentage increase in sales if the campaign has been in progress for 1.5 weeks.
The regression equation of y on x is given by 3x + 2y − 26 = 0. Find b_{yx}.
If for bivariate data `bar x = 10, bar y = 12,` v(x) = 9, σ_{y} = 4 and r = 0.6 estimate y, when x = 5.
The equation of the line of regression of y on x is y = `2/9` x and x on y is x = `"y"/2 + 7/6`.
Find (i) r, (ii) `sigma_"y"^2 if sigma_"x"^2 = 4`
Identify the regression equations of x on y and y on x from the following equations, 2x + 3y = 6 and 5x + 7y − 12 = 0
If for a bivariate data b_{yx} = – 1.2 and b_{xy} = – 0.3 then find r.
From the two regression equations y = 4x – 5 and 3x = 2y + 5, find `bar x and bar y`.
The equations of the two lines of regression are 3x + 2y − 26 = 0 and 6x + y − 31 = 0 Find
- Means of X and Y
- Correlation coefficient between X and Y
- Estimate of Y for X = 2
- var (X) if var (Y) = 36
Find the line of regression of X on Y for the following data:
n = 8, `sum(x_i - bar x)^2 = 36, sum(y_i - bar y)^2 = 44, sum(x_i - bar x)(y_i - bar y) = 24`
Find the equation of line of regression of Y on X for the following data:
n = 8, `sum(x_i - barx)(y_i - bary) = 120, bar x = 20, bar y = 36, sigma_x = 2, sigma_y = 3`
The following results were obtained from records of age (X) and systolic blood pressure (Y) of a group of 10 men.
X | Y | |
Mean | 50 | 140 |
Variance | 150 | 165 |
and `sum (x_i - bar x)(y_i - bar y) = 1120`
Find the prediction of blood pressure of a man of age 40 years.
The equations of two regression lines are 10x − 4y = 80 and 10y − 9x = − 40 Find:
- `bar x and bar y`
- `"b"_"YX" and "b"_"XY"`
- If var (Y) = 36, obtain var (X)
- r
If b_{YX} = − 0.6 and b_{XY} = − 0.216, then find correlation coefficient between X and Y. Comment on it.
Chapter 3: Linear Regression
Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 3 - Linear Regression
Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 3 (Linear Regression) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.
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Concepts covered in Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 3 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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