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
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.
From the two regression equations, find r, `bar x and bar y`. 4y = 9x + 15 and 25x = 4y + 17
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 a bivariate data, `bar x = 53`, `bar y = 28`, byx = −1.5 and bxy = −0.2. Estimate y when x = 50.
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]
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 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 the regression equation X on Y is 3x + 2y = 26, then bxy equal to
Choose the correct alternative:
bxy and byx are ______
Choose the correct alternative:
If r = 0.5, σx = 3, σy2 = 16, then bxy = ______
Choose the correct alternative:
Both the regression coefficients cannot exceed 1
State whether the following statement is True or False:
If byx = 1.5 and bxy = `1/3` then r = `1/2`, the given data is consistent
State whether the following statement is True or False:
The following data is not consistent: byx + bxy =1.3 and r = 0.75
State whether the following statement is True or False:
Regression coefficient of x on y is the slope of regression line of x on y
If u = `(x - "a")/"c"` and v = `(y - "b")/"d"`, then bxy = ______
Arithmetic mean of positive values of regression coefficients is greater than or equal to ______
The geometric mean of negative regression coefficients is ______
byx is the ______ of regression line of y on x
Given the following information about the production and demand of a commodity.
Obtain the two regression lines:
| ADVERTISEMENT (x) (₹ in lakhs) |
DEMAND (y) (₹ in lakhs) |
|
| Mean | 10 | 90 |
| Variance | 9 | 144 |
Coefficient of correlation between x and y is 0.8.
What should be the advertising budget if the company wants to attain the sales target of ₹ 150 lakhs?
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`
The regression equation of y on x is 2x – 5y + 60 = 0
Mean of x = 18
`2 square - 5 bary + 60` = 0
∴ `bary = square`
`sigma_x : sigma_y` = 3 : 2
∴ byx = `square/square`
∴ byx = `square/square`
∴ r = `square`
bXY . bYX = ______.
