Advertisements
Advertisements
प्रश्न
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
उत्तर
(i) Given equations of regression are
3x + 2y - 26 = 0
i.e., 3x + 2y = 26 ….(i)
and 6x + y - 31 = 0
i.e., 6x + y = 31 ….(ii)
By (i) - 2 × (ii), we get
3x + 2y = 26
12x + 2y = 62
- - -
- 9x = - 36
∴ x = `(-36)/-9 = 4`
Substituting x = 4 in (ii), we get
6 × 4 + y = 31
∴ 24 + y = 31
∴ y = 31 - 24
∴ y = 7
Since the point of intersection of two regression lines is `(bar x, bar y)`,
`bar x` = mean of X = 4, and
`bar y` = mean of Y = 7
(ii) Let 3x + 2y 26 = 0 be the regression equation of Y on X.
∴ The equation becomes 2Y = 3X + 26
i.e., Y = `(-3)/2"X" + 26/2`
Comparing it with Y = bYX X + a, we get
`"b"_"YX" = (- 3)/2`
Now, the other equation 6x + y - 31 = 0 is the regression equation of X on Y.
∴ The equation becomes 6X = - Y + 31
i.e., X = `(-1)/6 "Y" + 31/6`
Comparing it with X = bXY Y+ a', we get
`"b"_"XY" = (-1)/6`
∴ r = `+-sqrt("b"_"XY" * "b"_"YX")`
`= +- sqrt((- 1)/6 xx (- 3)/2) = +- sqrt(1/4) = +- 1/2 = +- 0.5`
Since the values of bXY and bYX are negative,
r is also negative.
∴ r = - 0.5
(iii) The regression equation of Y on X is
Y = `(- 3)/2 "X" + 26/2`
For X = 2, we get
Y = `(- 3)/2 xx 2 + 26/2 = - 3 + 13 = 10`
(iv) Given, Var (Y) = 36, i.e., `sigma_"Y"^2` = 36
∴ `sigma_"Y" = 6`
Since `"b"_"XY" = "r" xx sigma_"X"/sigma_"Y"`
`(- 1)/6 = - 0.5 xx sigma_"X"/6`
∴ `sigma_"X" = (-6)/(- 6 xx 0.5) = 2`
∴ `sigma_"X"^2` = Var(X) = 4
APPEARS IN
संबंधित प्रश्न
The equations given of the two regression lines are 2x + 3y - 6 = 0 and 5x + 7y - 12 = 0.
Find:
(a) Correlation coefficient
(b) `sigma_x/sigma_y`
Identify the regression equations of X on Y and Y on X from the following equations :
2x + 3y = 6 and 5x + 7y – 12 = 0
Find the equation of the regression line of y on x, if the observations (x, y) are as follows :
(1,4),(2,8),(3,2),(4,12),(5,10),(6,14),(7,16),(8,6),(9,18)
Also, find the estimated value of y when x = 14.
If Σx1 = 56 Σy1 = 56, Σ`x_1^2` = 478,
Σ`y_1^2` = 476, Σx1y1 = 469 and n = 7, Find
(a) the regression equation of y on x.
(b) y, if x = 12.
Compute the product moment coefficient of correlation for the following data:
n = 100, `bar x` = 62, `bary` = 53, `sigma_x` = 10, `sigma_y` = 12
`Sigma (x_i - bar x) (y_i - bary) = 8000`
Information on v:ehicles [in thousands) passing through seven different highways during a day (X) and number of accidents reported (Y) is given as follows :
`Sigmax_i` = 105, `Sigmay_i` = 409, n = 7, `Sigmax_i^2` = 1681, `Sigmay_i^2` = 39350 `Sigmax_iy_i` = 8075
Obtain the linear regression of Y on X.
The two lines of regressions are x + 2y – 5 = 0 and 2x + 3y – 8 = 0 and the variance of x is 12. Find the variance of y and the coefficient of correlation.
For the given lines of regression, 3x – 2y = 5 and x – 4y = 7, find:
(a) regression coefficients byx and bxy
(b) coefficient of correlation r (x, y)
Calculate the Spearman’s rank correlation coefficient for the following data and interpret the result:
| X | 35 | 54 | 80 | 95 | 73 | 73 | 35 | 91 | 83 | 81 |
| Y | 40 | 60 | 75 | 90 | 70 | 75 | 38 | 95 | 75 | 70 |
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
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 byx = – 1.2 and bxy = – 0.3 then find r.
Regression equation of X on Y is ______
Regression equation of X on Y is_________
Choose the correct alternative:
If the lines of regression of Y on X is y = `x/4` and X on Y is x = `y/9 + 1` then the value of r is
Choose the correct alternative:
y = 5 – 2.8x and x = 3 – 0.5 y be the regression lines, then the value of byx is
State whether the following statement is True or False:
If equation of regression lines are 3x + 2y – 26 = 0 and 6x + y – 31= 0, then mean of X is 7
State whether the following statement is True or False:
bxy is the slope of regression line of y on x
The age in years of 7 young couples is given below. Calculate husband’s age when wife’s age is 38 years.
| Husband (x) | 21 | 25 | 26 | 24 | 22 | 30 | 20 |
| Wife (y) | 19 | 20 | 24 | 20 | 22 | 24 | 18 |
If n = 5, Σx = Σy = 20, Σx2 = Σy2 = 90, Σxy = 76 Find the regression equation of x on y
If n = 6, Σx = 36, Σy = 60, Σxy = –67, Σx2 = 50, Σy2 =106, Estimate y when x is 13
The regression equation of x on y is 40x – 18y = 214 ......(i)
The regression equation of y on x is 8x – 10y + 66 = 0 ......(ii)
Solving equations (i) and (ii),
`barx = square`
`bary = square`
∴ byx = `square/square`
∴ bxy = `square/square`
∴ r = `square`
Given variance of x = 9
∴ byx = `square/square`
∴ `sigma_y = square`
If `bar"X"` = 40, `bar"Y"` = 6, σx = 10, σy = 1.5 and r = 0.9 for the two sets of data X and Y, then the regression line of X on Y will be:
For certain bivariate data on 5 pairs of observations given:
∑x = 20, ∑y = 20, ∑x2 = 90, ∑y2 = 90, ∑xy = 76 then bxy = ______.
The management of a large furniture store would like to determine sales (in thousands of ₹) (X) on a given day on the basis of number of people (Y) that visited the store on that day. The necessary records were kept, and a random sample of ten days was selected for the study. The summary results were as follows:
`sumx_i = 370 , sumy_i = 580, sumx_i^2 = 17200 , sumy_i^2 = 41640, sumx_iy_i = 11500, n = 10`
Complete the following activity to find, the equation of line of regression of Y on X and X on Y for the following data:
Given:`n=8,sum(x_i-barx)^2=36,sum(y_i-bary)^2=40,sum(x_i-barx)(y_i-bary)=24`
Solution:
Given:`n=8,sum(x_i-barx)=36,sum(y_i-bary)^2=40,sum(x_i-barx)(y_i-bary)=24`
∴ `b_(yx)=(sum(x_i-barx)(y_i-bary))/(sum(x_i-barx)^2)=square`
∴ `b_(xy)=(sum(x_i-barx)(y_i-bary))/(sum(y_i-bary)^2)=square`
∴ regression equation of Y on :
`y-bary=b_(yx)(x-barx)` `y-bary=square(x-barx)`
`x-barx=b_(xy)(y-bary)` `x-barx=square(y-bary)`
Out of the two regression lines x + 2y – 5 = 0 and 2x + 3y = 8, find the line of regression of y on x.
For a bivariate data `barx = 10`, `bary = 12`, V(X) = 9, σy = 4 and r = 0.6
Estimate y when x = 5
Solution: Line of regression of Y on X is
`"Y" - bary = square ("X" - barx)`
∴ Y − 12 = `r.(σ_y)/(σ_x)("X" - 10)`
∴ Y − 12 = `0.6 xx 4/square ("X" - 10)`
∴ When x = 5
Y − 12 = `square(5 - 10)`
∴ Y − 12 = −4
∴ Y = `square`
XYZ company plans to advertise some vacancies. The Manager is asked to suggest the monthly salary for these vacancies based on the years of experience. To do so, the Manager studies the years of service and the monthly salary drawn by the existing employees in the company.
Following is the data that the Manager refers to:
| Years of service (X) | 11 | 7 | 9 | 5 | 8 | 6 | 10 |
| Monthly salary (in ₹ 1000)(Y) | 10 | 8 | 6 | 5 | 9 | 7 | 11 |
- Find the regression equation of monthly salary on the years of service.
- If a person with 13 years of experience applies for a job in this company, what monthly salary will be suggested by the Manager?
