Balbharati solutions for Mathematics and Statistics 2 (Commerce) [English] 12 Standard HSC Maharashtra State Board chapter 3 - Linear Regression [Latest edition]

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.

1. Find the regression equation of income on years of service.
2. 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:

For the certain bivariate data on 5 pairs of observations given

∑x = 20, ∑y = 20, ∑x² = 90, ∑y² = 90, ∑xy = 76

Calculate: 

1. cov(x, y)
2. b_yx and b_xy
3. r

From the following data estimate y when x = 125.

The following table gives the aptitude test scores and productivity indices of 10 workers selected at random.

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.

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:

The following are the marks obtained by the students in Economics (X) and Mathematics (Y)

Find the regression equation of Y on X.

For the following bivariate data obtain the equations of two regression lines:

From the following data obtain the equation of two regression lines:

For the following data, find the regression line of Y on X

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.

The following sample gives the number of hours of study (X) per day for an examination and marks (Y) obtained by 12 students.

Obtain the line of regression of marks on hours of study.

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:

1. The line of regression of Y on X.
2. The line of regression of X on Y.
3. 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

1. The line of regression of Y on X.
2. The line regression of X on Y.
3. The correlation coefficient between X and Y.

You are given the following information about advertising expenditure and sales.

Correlation coefficient between X and Y is 0.8

1. Obtain the two regression equations.
2. What is the likely sales when the advertising budget is ₹ 15 lakh?
3. 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

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:

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)

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)

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.

Correlation coefficient between x and y is 0.6. estimate x when y = 15 and estimate y when x = 10.

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

1. The mean values of X and Y.
2. Correlation coefficient between X and Y.
3. 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 variance of marks in statistics is `(9/16)^"th"` of the variance of marks in accountancy. Find 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 3x + 2y − 12 = 0 Find 

1. Correlation coefficient
2. `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

1. Mean values of X and Y
2. Standard deviation of Y
3. 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

1. `bar x`,
2. `bar y`,
3. b_YX
4. b_XY
5. 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.

Choose the correct alternative:

Regression analysis is the theory of

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

Choose the correct alternative:

There are ______ types of regression equations

Choose the correct alternative.

In the regression equation of Y on X

Choose the correct alternative:

In the regression equation of X on Y

Choose the correct alternative.

b_XY is _____

b_YX is ______.

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‘r’ is __________.

Choose the correct alternative.

b_XY .b_YX is _________.

Choose the correct alternative.

b_YX > 1 then b_XY is _______

|b_xy + b_yx | ≥ _______

b_xy and b_yx are _______.

Choose the correct alternative.

If u = `("x - a")/"c" and "v" = ("y - b")/"d"  "then"   "b"_"yx"` = _________

Choose the correct alternative.

If u = `("x - a")/"c" and "v" = ("y - b")/"d"  "then"   "b"_"xy"` = _________

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Corr (x, x) = _____

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Corr (x, y) = _____

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Corr `("x - a"/"c", "y - b"/"d")` = - corr (x, y) if,

Regression equation of X on Y is ______

Choose the correct alternative.

Regression equation of Y on X is ____

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b_yx = ______

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b_xy = ______

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Cov (x, y) = __________

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If bxy < 0 and byx < 0 then 'r' is __________

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 __________

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_________

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 = _______

Corr (x, x) = 1

Regression equation of X on Y is `("y" - bar "y") = "b"_"yx" ("x" - bar "x")`

State whether the following statement is True or False.

Regression equation of Y on X is `("y" - bar "y") = "b"_"yx" ("x" - bar "x")`

State whether the following statement is True or False.

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

State whether the following statement is True or False.

b_xy and b_yx are independent of change of origin and scale. 

‘r’ is regression coefficient of Y on X

State whether the following statement is True or False.

b_yx is correlation coefficient between X and Y

State whether the following statement is True or False.

If u = x - a and v = y - b then b_xy = b_uv 

State whether the following statement is True or False.

If u = x - a and v = y - b then r_xy = r_uv 

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

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 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

1. Means of X and Y
2. Correlation coefficient between X and Y
3. Estimate of Y for X = 2
4. 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 the line of regression of Y on X for the following data:

n = 8, `sum(x_i - barx).(y_i - bary) = 120, barx = 20, bary = 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.

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:

1. `bar x and bar y`
2. b_YX and b_XY
3. If var (Y) = 36, obtain var (X)
4. r

If b_YX = − 0.6 and b_XY = − 0.216, then find correlation coefficient between X and Y. Comment on it.