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प्रश्न
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.
उत्तर
Let 10y = 3x + 170 be the regression equation of Y on X.
∴ The equation becomes 10y = 3x + 170
i.e., Y = `3/10 "X" + 170/10`
Comparing it with Y = bYX X + a, we get
`"b"_"YX" = 3/10`
Now other equation 5x + 70 = 6y be the regression equation of X on Y.
∴ The equation becomes 5x = 6y – 70
i.e., X = `6/5 "Y" - 70/5`
Comparing it with X = bXY Y + a', we get
`"b"_"XY" = 6/5`
∴ r = `+-sqrt("b"_"XY" * "b"_"YX") = +-sqrt(6/5 xx 3/10) = +- sqrt(9/25) +- 3/5`
Since bYX and bXY both are positive,
r is positive.
∴ r = `3/5` = 0.6
Now, two correlated lines of regression are
10y = 3x + 170
i.e., - 3x + 10y = 170 …(i)
and 5x + 70 = 6y
i.e., 5x - 6y = –70 …(ii)
By (i) × 5 + (ii) × 3, we get
- 15x + 50y = 850
+ 15x - 18y = - 210
32y = 640
∴ y = 20
Substituting y = 20 in equation (i), we get
- 3x +10(20) = 170
∴ - 3x + 200 = 170
∴ 3x = 200 - 170
∴ x = 10
Since the point of intersection of two regression lines is `(bar x, bar y)`,
`bar x` = mean value of X = 10, and
`bar y` = mean value of Y = 20.
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संबंधित प्रश्न
From the data of 7 pairs of observations on X and Y, following results are obtained.
∑(xi - 70) = - 35, ∑(yi - 60) = - 7,
∑(xi - 70)2 = 2989, ∑(yi - 60)2 = 476,
∑(xi - 70)(yi - 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.
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
From the two regression equations, find r, `bar x and bar y`. 4y = 9x + 15 and 25x = 4y + 17
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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:
|byx + bxy| ≥ ______
Choose the correct alternative:
If r = 0.5, σx = 3, `σ_"y"^2` = 16, then byx = ______
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:
Corr(x, x) = 0
The value of product moment correlation coefficient between x and x is ______
If u = `(x - 20)/5` and v = `(y - 30)/4`, then byx = ______
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?
For a certain bivariate data of a group of 10 students, the following information gives the internal marks obtained in English (X) and Hindi (Y):
| X | Y | |
| Mean | 13 | 17 |
| Standard Deviation | 3 | 2 |
If r = 0.6, Estimate x when y = 16 and y when x = 10
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`
| x | y | xy | x2 | y2 |
| 6 | 9 | 54 | 36 | 81 |
| 2 | 11 | 22 | 4 | 121 |
| 10 | 5 | 50 | 100 | 25 |
| 4 | 8 | 32 | 16 | 64 |
| 8 | 7 | `square` | 64 | 49 |
| Total = 30 | Total = 40 | Total = `square` | Total = 220 | Total = `square` |
bxy = `square/square`
byx = `square/square`
∴ Regression equation of x on y is `square`
∴ Regression equation of y on x is `square`
The following results were obtained from records of age (x) and systolic blood pressure (y) of a group of 10 women.
| x | y | |
| Mean | 53 | 142 |
| Variance | 130 | 165 |
`sum(x_i - barx)(y_i - bary)` = 1170
