Advertisements
Advertisements
Question
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.
Advertisements
Solution
Given, `sigma_"X"^2 = 9`
∴ σX = 3
(i) The two regression equations are
8x - 10y + 66 = 0
i.e., 8x - 10y = - 66 ...(i)
and 40x - 18y = 214 ....(ii)
By 5 × (i) - (ii), we get
40x - 50y = - 330
40x - 18y = 214
(-) (+) (-)
- 32y = - 544
∴ y = `544/32 = 17`
Substituting y = 17 in (i), we get
8x - 10 × 17 = - 66
∴ 8x - 170 = - 66
∴ 8x = - 66 + 170
∴ 8x = 104
∴ x = `104/8 = 13`
Since the point of intersection of two regression lines is `(bar x, bar y)`,
`bar x` = mean value of X = 13, and
`bar y` = mean value of X = 17.
(ii) Let 8x - 10y + 66 = 0 be the regression equation of Y on X.
∴ The equation becomes 10Y = 8X + 66
i.e., Y = `8/10 "X" + 66/10`
i.e., Y = `4/5 "X" + 33/5`
Comparing it with Y = bYX X + a, we get
`"b"_"YX" = 4/5`
Now, the other equation, i.e., 40x - 18y = 214 is the regression equation of X on Y.
∴ The equation becomes X = `18/40 "Y" + 214/40`
i.e., X = `9/20 "Y" + 107/20`
Comparing it with X = bXY Y + a', we get
`"b"_"XY" = 9/20`
r = `+- sqrt("b"_"XY" * "b"_"YX")`
∴ r = `+- sqrt(9/20 xx 4/5) = +- sqrt(9/25) = +- 3/5 = +- 0.6`
Since bYX and bXY both are positive,
r is also positive.
∴ r = 0.6
(iii) `"b"_"YX" = "r" sigma_"Y"/sigma_"X"`
∴ `4/5 = 0.6 xx sigma_"Y"/3`
∴ `4/5 = sigma_"Y"/5`
∴ `sigma_"Y" = 4`
APPEARS IN
RELATED QUESTIONS
For bivariate data. `bar x = 53, bar y = 28, "b"_"YX" = - 1.2, "b"_"XY" = - 0.3` Find estimate of X for Y = 25.
Bring out the inconsistency in the following:
bYX = bXY = 1.50 and r = - 0.9
Bring out the inconsistency in the following:
bYX = 1.9 and bXY = - 0.25
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
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.
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, σX = 60, σY = 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?
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`,
- bYX
- bXY
- r [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.
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.
Choose the correct alternative:
If for a bivariate data, bYX = – 1.2 and bXY = – 0.3, then r = ______
Choose the correct alternative:
|byx + bxy| ≥ ______
Choose the correct alternative:
Find the value of the covariance between X and Y, if the regression coefficient of Y on X is 3.75 and σx = 2, σy = 8
Choose the correct alternative:
Both the regression coefficients cannot exceed 1
Arithmetic mean of positive values of regression coefficients is greater than or equal to ______
The geometric mean of negative regression coefficients is ______
Given the following information about the production and demand of a commodity.
Obtain the two regression lines:
| Production (X) |
Demand (Y) |
|
| Mean | 85 | 90 |
| Variance | 25 | 36 |
Coefficient of correlation between X and Y is 0.6. Also estimate the demand when the production is 100 units.
The equations of the two lines of regression are 6x + y − 31 = 0 and 3x + 2y – 26 = 0. Find the value of the correlation coefficient
Mean of x = 53
Mean of y = 28
Regression coefficient of y on x = – 1.2
Regression coefficient of x on y = – 0.3
a. r = `square`
b. When x = 50,
`y - square = square (50 - square)`
∴ y = `square`
c. When y = 25,
`x - square = square (25 - square)`
∴ x = `square`
Mean of x = 25
Mean of y = 20
`sigma_x` = 4
`sigma_y` = 3
r = 0.5
byx = `square`
bxy = `square`
when x = 10,
`y - square = square (10 - square)`
∴ y = `square`
For a bivariate data:
`sum(x - overlinex)^2` = 1200, `sum(y - overliney)^2` = 300, `sum(x - overlinex)(y - overliney)` = – 250
Find:
- byx
- bxy
- Correlation coefficient between x and y.
