For the variables X and Y, we collect 4 observations with ∑x = 10, ∑y = 14, ∑x 2= 30, ∑y 2= 54, ∑xy = 31. The regression line y on x is
Understanding Linear Regression and the Regression Line
Linear regression is a statistical method used to model the relationship between two variables, typically denoted as X and Y. When we want to predict the value of Y based on the value of X, we find the regression line of Y on X. This line is the best-fitting straight line through the data points that minimizes the sum of the squared differences between the observed Y values and the predicted Y values.
The equation of a linear regression line of Y on X is usually represented as:
\(\hat{y} = a + bx\)
where:
- \(\hat{y}\) is the predicted value of Y for a given value of X.
- \(a\) is the y-intercept (the value of \(\hat{y}\) when \(x = 0\)).
- \(b\) is the slope of the line (the change in \(\hat{y}\) for a one-unit increase in X).
To find the equation of the regression line, we need to calculate the values of the slope (\(b\)) and the y-intercept (\(a\)) using the given data.
Calculating the Slope (b) for Regression Line y on x
The formula for the slope (\(b\)) of the regression line of Y on X using sums of the variables is:
\(b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}\)
where:
- \(n\) is the number of observations.
- \(\sum x\) is the sum of the X values.
- \(\sum y\) is the sum of the Y values.
- \(\sum xy\) is the sum of the products of corresponding X and Y values.
- \(\sum x^2\) is the sum of the squared X values.
Calculating the Y-Intercept (a) for Regression Line y on x
The formula for the y-intercept (\(a\)) is based on the mean values of X and Y and the calculated slope (\(b\)):
\(a = \bar{y} - b\bar{x}\)
where:
- \(\bar{x}\) is the mean of X (\(\bar{x} = \frac{\sum x}{n}\)).
- \(\bar{y}\) is the mean of Y (\(\bar{y} = \frac{\sum y}{n}\)).
Applying the Formulas with Given Observations
We are given the following information from 4 observations:
- Number of observations, \(n = 4\)
- Sum of X values, \(\sum x = 10\)
- Sum of Y values, \(\sum y = 14\)
- Sum of squared X values, \(\sum x^2 = 30\)
- Sum of squared Y values, \(\sum y^2 = 54\) (Note: This value is not needed for the regression line y on x but is provided in the question)
- Sum of the products of X and Y values, \(\sum xy = 31\)
First, let's calculate the means of X and Y:
\(\bar{x} = \frac{\sum x}{n} = \frac{10}{4} = 2.5\)
\(\bar{y} = \frac{\sum y}{n} = \frac{14}{4} = 3.5\)
Next, let's calculate the slope (\(b\)) using the formula:
\(b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}\)
\(b = \frac{(4)(31) - (10)(14)}{(4)(30) - (10)^2}\)
\(b = \frac{124 - 140}{120 - 100}\)
\(b = \frac{-16}{20}\)
\(b = -0.8\)
Now, let's calculate the y-intercept (\(a\)) using the formula:
\(a = \bar{y} - b\bar{x}\)
\(a = 3.5 - (-0.8)(2.5)\)
\(a = 3.5 - (-2.0)\)
\(a = 3.5 + 2.0\)
\(a = 5.5\)
Forming the Regression Line Equation
Now that we have the slope (\(b = -0.8\)) and the y-intercept (\(a = 5.5\)), we can write the equation for the regression line y on x:
\(\hat{y} = a + bx\)
\(\hat{y} = 5.5 + (-0.8)x\)
This can be written as:
\(\hat{y} = -0.8x + 5.5\)
Or, using 'y' to represent the predicted value:
\(y = -0.8x + 5.5\)
Comparing with the Options
Let's compare our calculated regression line equation with the given options:
- Option 1: \(y = -0.8x - 5.5\) (Does not match)
- Option 2: \(y = 0.8x - 5.5\) (Does not match)
- Option 3: \(y = -0.8x + 5.5\) (Matches our result)
- Option 4: \(y = 0.8x + 5.5\) (Does not match)
The calculated regression line \(y = -0.8x + 5.5\) matches Option 3.
| Calculation Step | Formula / Value | Result |
|---|---|---|
| Number of observations (n) | Given | 4 |
| Sum of X (\(\sum x\)) | Given | 10 |
| Sum of Y (\(\sum y\)) | Given | 14 |
| Sum of X\(^2\) (\(\sum x^2\)) | Given | 30 |
| Sum of XY (\(\sum xy\)) | Given | 31 |
| Mean of X (\(\bar{x}\)) | \(\frac{\sum x}{n}\) | \(\frac{10}{4} = 2.5\) |
| Mean of Y (\(\bar{y}\)) | \(\frac{\sum y}{n}\) | \(\frac{14}{4} = 3.5\) |
| Slope (b) | \(\frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}\) | \(\frac{(4)(31) - (10)(14)}{(4)(30) - (10)^2} = \frac{124 - 140}{120 - 100} = \frac{-16}{20} = -0.8\) |
| Y-intercept (a) | \(\bar{y} - b\bar{x}\) | \(3.5 - (-0.8)(2.5) = 3.5 + 2.0 = 5.5\) |
| Regression Line y on x | \(y = a + bx\) | \(y = 5.5 + (-0.8)x \implies y = -0.8x + 5.5\) |
Revision Table: Key Linear Regression Concepts
| Concept | Description | Formula (for y on x) |
|---|---|---|
| Regression Line Equation | Equation modeling the linear relationship between Y and X | \(y = a + bx\) |
| Slope (b) | Change in Y per unit change in X; direction and steepness of the line | \(\frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}\) |
| Y-intercept (a) | Value of Y when X is 0; where the line crosses the Y-axis | \(\bar{y} - b\bar{x}\) |
| Mean of X (\(\bar{x}\)) | Average value of X | \(\frac{\sum x}{n}\) |
| Mean of Y (\(\bar{y}\)) | Average value of Y | \(\frac{\sum y}{n}\) |
Additional Information on Regression Analysis
Linear regression is a fundamental tool in statistics for understanding the relationship between variables. The regression line y on x helps us predict the dependent variable (Y) based on the independent variable (X). Similarly, we could calculate the regression line x on y, which predicts X based on Y, but the formulas for slope and intercept would be different.
The strength and direction of the linear relationship can also be measured by the correlation coefficient (r), while the coefficient of determination (r\(^2\)) tells us the proportion of the variance in Y that is predictable from X. These measures are often calculated alongside the regression line to provide a more complete picture of the relationship.
Assumptions are made when performing linear regression, such as linearity of the relationship, independence of errors, homogeneity of variances (homoscedasticity), and normality of errors. Violations of these assumptions can affect the validity of the regression results.
Similar Questions
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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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