Variable Y regresses with variable X with the conditions that \(\overline X = 5.50\) , \(\overline Y = 3.50\) and b = 1.50 in the linear regression model (Y = a + bX), where \(\overline Y\) and \(\overline X\) are means of the respective variables and b refers to gradient of line of Y w. r. t X. Which one of the following values of parameter 'a' of the model is correct?
-4.75
Understanding the Linear Regression Model
The question asks us to find the value of the parameter 'a' in a linear regression model. A linear regression model describes the relationship between two variables, typically denoted as Y (dependent variable) and X (independent variable), using a straight line equation.
The standard form of a simple linear regression model is given by:
\(Y = a + bX\)
Where:
- \(Y\) is the dependent variable.
- \(X\) is the independent variable.
- \(a\) is the y-intercept (the value of Y when X is 0).
- \(b\) is the slope or gradient of the line (how much Y changes for a one-unit change in X).
Key property of the Regression Line
A fundamental property of the least squares regression line is that it always passes through the point defined by the means of the variables, i.e., \((\overline X, \overline Y)\). This means that if we substitute the mean values of X and Y into the regression equation, the equation should hold true:
\(\overline Y = a + b\overline X\)
This property is very useful because it allows us to find one of the parameters (a or b) if the other parameter and the means are known.
Given Information for Parameter Calculation
We are provided with the following information:
- Mean of variable X, \(\overline X = 5.50\)
- Mean of variable Y, \(\overline Y = 3.50\)
- Gradient (slope) of the line of Y with respect to X, \(b = 1.50\)
We need to find the value of the parameter 'a'.
Step-by-Step Calculation of Parameter 'a'
Using the property that the regression line passes through the means, we substitute the given values into the equation \(\overline Y = a + b\overline X\):
\(3.50 = a + (1.50 \times 5.50)\)
First, calculate the product of 'b' and \(\overline X\):
\(1.50 \times 5.50\)
We can calculate this as:
\(1.5 \times 5.5 = 1.5 \times (5 + 0.5) = (1.5 \times 5) + (1.5 \times 0.5)\)
\(1.5 \times 5 = 7.5\)
\(1.5 \times 0.5 = 0.75\)
So, \(1.50 \times 5.50 = 7.5 + 0.75 = 8.25\)
Now, substitute this value back into the equation:
\(3.50 = a + 8.25\)
To find 'a', we need to isolate it on one side of the equation. Subtract 8.25 from both sides:
\(a = 3.50 - 8.25\)
Calculating the difference:
\(3.50 - 8.25 = - (8.25 - 3.50)\)
\(8.25 - 3.50 = 4.75\)
Therefore,
\(a = -4.75\)
Resulting Linear Regression Model
With the calculated value of 'a' and the given value of 'b', the linear regression model is:
\(Y = -4.75 + 1.50X\)
Final Value of Parameter 'a'
Based on the calculations using the given means and slope, the value of the parameter 'a' is -4.75.
Revision Table: Linear Regression Parameters
| Parameter | Symbol | Description | How it's found (simple regression) |
|---|---|---|---|
| Intercept | a | Value of Y when X=0 (Y-intercept) | \(\overline Y - b\overline X\) |
| Slope / Gradient | b | Change in Y for a unit change in X | \(\frac{\sum(X_i - \overline X)(Y_i - \overline Y)}{\sum(X_i - \overline X)^2}\) or \(\frac{Cov(X, Y)}{Var(X)}\) |
Additional Information on Linear Regression
Linear regression is a fundamental statistical technique used to model the relationship between a dependent variable and one or more independent variables. In simple linear regression, there is only one independent variable.
- Intercept (a): This is where the regression line crosses the Y-axis. It represents the predicted value of Y when the value of X is zero. However, interpreting the intercept can be meaningless if X=0 is outside the range of observed X values.
- Slope (b): This indicates the steepness and direction of the regression line. A positive slope means that as X increases, Y tends to increase. A negative slope means that as X increases, Y tends to decrease. The magnitude of the slope tells us how much Y is expected to change for every one-unit change in X.
- Least Squares Method: The parameters 'a' and 'b' in linear regression are typically estimated using the method of least squares. This method finds the line that minimizes the sum of the squared differences between the observed Y values and the Y values predicted by the regression line.
- Assumptions: Linear regression models rely on several assumptions about the data, such as linearity, independence of errors, homoscedasticity (constant variance of errors), and normality of errors. Violations of these assumptions can affect the validity of the model results.
Important Questions from Correlation and Regression
If the regression coefficient of x on y and y on x are \(- \frac{1}{2}\) and \(- \frac{1}{8}\) respectively, then what is the correlation coefficient between x and y?
It is given that X̅ = 10, Y̅ = 90, σ X= 3, σ Y= 12 and r XY = 0.8. The regression equation of X on Y is
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∑x = 130, ∑y = 220, ∑ x 2= 2288, ∑y 2= 5506 and ∑ xy = 3467.
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If two regression coefficients are -0.1 and -0.9, then correlation coefficient is:
