For the data set the regression coefficient b yx (y on x) equals to:x: 1 2 3 4 y: 4 5 3 2
-0.8
Understanding Regression Coefficient \(b_{yx}\)
The question asks us to calculate the regression coefficient of y on x, denoted as \(b_{yx}\), for the given dataset. The regression coefficient \(b_{yx}\) represents the average change in the dependent variable y for a one-unit change in the independent variable x. It is a key component in linear regression, helping us understand the linear relationship between two variables.
Formula for Regression Coefficient \(b_{yx}\)
The formula to calculate the regression coefficient \(b_{yx}\) for a set of n data points \((x_i, y_i)\) is:
\[b_{yx} = \frac{n \Sigma xy - (\Sigma x)(\Sigma y)}{n \Sigma x^2 - (\Sigma x)^2}\]
Where:
- \(n\) is the number of data points.
- \(\Sigma xy\) is the sum of the products of corresponding x and y values.
- \(\Sigma x\) is the sum of all x values.
- \(\Sigma y\) is the sum of all y values.
- \(\Sigma x^2\) is the sum of the squares of all x values.
- \((\Sigma x)^2\) is the square of the sum of all x values.
Calculating Necessary Components from the Dataset
The given dataset is:
| x | y |
|---|---|
| 1 | 4 |
| 2 | 5 |
| 3 | 3 |
| 4 | 2 |
Let's calculate the required sums:
- \(n = 4\) (the number of data points)
- \(\Sigma x = 1 + 2 + 3 + 4 = 10\)
- \(\Sigma y = 4 + 5 + 3 + 2 = 14\)
- \(\Sigma xy = (1 \times 4) + (2 \times 5) + (3 \times 3) + (4 \times 2) = 4 + 10 + 9 + 8 = 31\)
- \(\Sigma x^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30\)
Calculating the Regression Coefficient \(b_{yx}\)
Now, substitute these values into the formula for \(b_{yx}\):
\[b_{yx} = \frac{(4)(31) - (10)(14)}{(4)(30) - (10)^2}\]
\[b_{yx} = \frac{124 - 140}{120 - 100}\]
\[b_{yx} = \frac{-16}{20}\]
\[b_{yx} = -0.8\]
Conclusion on Regression Coefficient \(b_{yx}\)
The calculated regression coefficient \(b_{yx}\) is -0.8.
Comparing this value with the given options:
- Option 1: -0.86
- Option 2: 0.8
- Option 3: 0.86
- Option 4: -0.8
The calculated value of -0.8 matches Option 4.
Revision Table: Key Regression Concepts
| Concept | Description | Formula (for y on x) |
|---|---|---|
| Regression Coefficient \(b_{yx}\) | Measures the change in y per unit change in x. Represents the slope of the regression line. | \(b_{yx} = \frac{n \Sigma xy - (\Sigma x)(\Sigma y)}{n \Sigma x^2 - (\Sigma x)^2}\) |
| Regression Line (y on x) | The line that best fits the data points, representing the linear relationship \(y = a_{yx} + b_{yx}x\). | \(y - \bar{y} = b_{yx}(x - \bar{x})\) or \(y = \bar{y} - b_{yx}\bar{x} + b_{yx}x\) where \(a_{yx} = \bar{y} - b_{yx}\bar{x}\) |
| Regression Coefficient \(b_{xy}\) | Measures the change in x per unit change in y (regression of x on y). | \(b_{xy} = \frac{n \Sigma xy - (\Sigma x)(\Sigma y)}{n \Sigma y^2 - (\Sigma y)^2}\) |
Additional Information on Linear Regression
Linear regression is a statistical method used to model the linear relationship between a dependent variable (in this case, y) and one or more independent variables (in this case, x). The equation of a simple linear regression line (y on x) is typically written as \(y = a + bx\), where:
- \(y\) is the predicted value of the dependent variable.
- \(x\) is the independent variable.
- \(a\) is the y-intercept (the value of y when x is 0). This is often denoted as \(a_{yx}\) for y on x regression.
- \(b\) is the regression coefficient (the slope of the line). This is \(b_{yx}\) for y on x regression.
The goal of linear regression is to find the values of \(a\) and \(b\) that result in the line that best fits the data. The "best fit" is usually defined using the method of least squares, which minimizes the sum of the squared differences between the observed y values and the y values predicted by the line.
The regression coefficient \(b_{yx}\) indicates both the strength and direction of the linear relationship between y and x. A negative value like -0.8 suggests a negative linear relationship: as x increases, y tends to decrease. The magnitude (0.8) indicates the strength of this relationship (larger absolute value means stronger relationship).
Similar Questions
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If six hand-writings were ranked by two judges in a competition and the rankings are as follows:
Hand Writing 1 2 3 4 5 6 Judge1 6 5 4 3 2 1 Judge2 1 2 3 4 5 6
Then the value of Spearman's rank correlation coefficient isIf the difference between the rank of the 4 observations are 2.5, 0.5, -1.5, -1.5, then Spearman's rank correlation coefficient equals to:
For the two variables X and Y, the following observations are tabulated.
X: 3 4 4 Y: 10 10 9 The Spearman's correlation coefficient is:
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If x = X- \(\bar{X}\) and y = Y - \(\bar{Y}\) and the number of pairs (X, Y) is n, then the Karl Pearson's coefficient of correlation is:
Important Questions from Correlation and Regression
- For testing the significance of overall regression the test to be used is
- 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 ?
- If OLS is used in the presence of auto-correlation, which of the following will be likely consequences ?
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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