If 4x – 5y + 33 = 0 and 20x – 9y = 107 are two lines of regression, then what are the values of x̅ and y̅ respectively?
13 and 17
Finding Mean x and y from Regression Lines
The question asks for the values of the mean of x (\(\bar{x}\)) and the mean of y (\(\bar{y}\)) given two regression lines. A fundamental property of regression lines is that they always intersect at the point representing the means of the two variables. That is, the point of intersection of the line of regression of y on x and the line of regression of x on y is (\(\bar{x}\), \(\bar{y}\)).
Therefore, to find \(\bar{x}\) and \(\bar{y}\), we need to solve the given system of linear equations:
- Equation 1: \(4x - 5y + 33 = 0 \implies 4x - 5y = -33\)
- Equation 2: $20x - 9y = 107$
We can solve this system using the elimination method.
Solving the System of Linear Equations
Our goal is to eliminate one variable (either x or y) to solve for the other. Let's eliminate x.
- Multiply Equation 1 by 5 to make the coefficient of x the same as in Equation 2:
\(5 \times (4x - 5y) = 5 \times (-33)\)
$20x - 25y = -165$ (Let's call this Equation 3)
- Now we have the system:
Equation 2: $20x - 9y = 107$
Equation 3: $20x - 25y = -165$
- Subtract Equation 3 from Equation 2:
$(20x - 9y) - (20x - 25y) = 107 - (-165)$
$20x - 9y - 20x + 25y = 107 + 165$
$16y = 272$
- Solve for y:
\(y = \frac{272}{16}\)
$y = 17$
Finding the Value of x
Now that we have the value of y, substitute $y = 17$ into either original equation to find x. Let's use Equation 1:
$4x - 5y = -33$
$4x - 5(17) = -33$
$4x - 85 = -33$
$4x = -33 + 85$
$4x = 52$
- Solve for x:
\(x = \frac{52}{4}\)
$x = 13$
Resulting Mean Values
The intersection point of the two regression lines is $(x, y) = (13, 17)$. Therefore, the mean of x is \(\bar{x} = 13\) and the mean of y is \(\bar{y} = 17\). The question asks for \(\bar{x}\) and \(\bar{y}\) respectively.
So, the values of \(\bar{x}\) and \(\bar{y}\) are 13 and 17.
Let's verify these values with the second equation:
$20x - 9y = 107$
$20(13) - 9(17) = 260 - 153 = 107$. The values satisfy the second equation as well.
Revision Table: Key Concepts
| Concept | Explanation |
|---|---|
| Regression Lines | Linear models describing the relationship between two variables. One line is y on x, the other is x on y. |
| Intersection of Regression Lines | The point where the two regression lines cross. This point is always \((\bar{x}, \bar{y})\), the means of the two variables. |
| Mean (\(\bar{x}, \bar{y}\)) | The average value of the x and y variables in the dataset from which the regression lines were derived. |
| System of Linear Equations | A set of two or more linear equations involving the same variables, whose solution is the set of values that satisfy all equations simultaneously. |
Additional Information on Regression Analysis
Regression analysis is a statistical method used to estimate the relationships between a dependent variable and one or more independent variables. In the case of two variables, we have simple linear regression.
- There are generally two regression lines for bivariate data: the regression line of y on x and the regression line of x on y.
- The line of regression of y on x is used to predict the value of y given a value of x. Its equation is typically written as $y = a + bx$, where $b$ is the regression coefficient of y on x.
- The line of regression of x on y is used to predict the value of x given a value of y. Its equation is typically written as $x = c + dy$, where $d$ is the regression coefficient of x on y.
- The formulas for the regression coefficients involve the correlation coefficient ($r$), the standard deviations (\(\sigma_x, \sigma_y\)), and the means (\(\bar{x}, \bar{y}\)).
- The fact that both lines pass through the point \((\bar{x}, \bar{y})\) is a crucial property used in solving problems like this one.
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- Including relevant lagged values of the dependent variable on the right hand side of a regression equation could lead to which one of the following ?
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i. Coefficients estimates may be misleading.
ii. Hypothesis tests could reach the wrong conclusions.
iii. Forecasts made from the model could be biased.
iv. Standard errors may be inappropriate. - Regression analysis is of particular interest
- Which of the following can be linked with those providing method for estimating black money in India ?
I. Kaldor’s approach
II. NIPFP method
III. O.P. Chopra’s method
IV. Arjun Sengupta
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