For a bivariate data set on (x, y), if the means, standard deviations and correlation coefficient are x̅ = 1.0, y̅ = 2.0, s x= 3.0, s y= 9.0, r = 0.8 Then the regression line of y on x is:
y = 2 + 2.4(x - 1)
Finding the Regression Line of y on x
A regression line helps us understand the relationship between two variables, x and y, and predict the value of one variable based on the other. The question asks for the regression line of y on x for a given bivariate data set.
Understanding the Regression Line of y on x
The general form of the regression line of y on x is given by:
\[ y - \bar{y} = b_{yx}(x - \bar{x}) \]
where:
- \( y \) is the dependent variable
- \( x \) is the independent variable
- \( \bar{y} \) is the mean of y
- \( \bar{x} \) is the mean of x
- \( b_{yx} \) is the regression coefficient of y on x
The regression coefficient \( b_{yx} \) measures the average change in y for a unit change in x. It is calculated using the correlation coefficient \( r \) and the standard deviations of x and y:
\[ b_{yx} = r \left(\frac{s_y}{s_x}\right) \]
where:
- \( r \) is the correlation coefficient between x and y
- \( s_y \) is the standard deviation of y
- \( s_x \) is the standard deviation of x
Applying the Given Data
We are provided with the following information for the bivariate data set on (x, y):
- Mean of x, \( \bar{x} = 1.0 \)
- Mean of y, \( \bar{y} = 2.0 \)
- Standard deviation of x, \( s_x = 3.0 \)
- Standard deviation of y, \( s_y = 9.0 \)
- Correlation coefficient, \( r = 0.8 \)
Calculating the Regression Coefficient \( b_{yx} \)
Using the formula for \( b_{yx} \) and the given values:
\[ b_{yx} = r \left(\frac{s_y}{s_x}\right) \]
\[ b_{yx} = 0.8 \left(\frac{9.0}{3.0}\right) \]
\[ b_{yx} = 0.8 \times 3 \]
\[ b_{yx} = 2.4 \]
So, the regression coefficient of y on x is 2.4.
Constructing the Regression Line Equation
Now, substitute the values of \( \bar{y} \), \( \bar{x} \), and \( b_{yx} \) into the regression line equation \( y - \bar{y} = b_{yx}(x - \bar{x}) \):
\[ y - 2.0 = 2.4(x - 1.0) \]
This equation directly gives the regression line of y on x.
Comparing with Options
Let's compare the derived equation with the given options:
- Option 1: \( y = 1 + 2.4(x - 1) \)
- Option 2: \( y = 2 + 0.27(x - 1) \)
- Option 3: \( y = 2 + 2.4(x - 1) \)
- Option 4: \( y = 1 + 0.27(x - 2) \)
Our derived equation is \( y - 2.0 = 2.4(x - 1.0) \). Rearranging this by moving the -2.0 to the right side gives \( y = 2.0 + 2.4(x - 1.0) \). This matches Option 3.
Revision Table: Key Regression Concepts
| Concept | Description | Formula |
|---|---|---|
| Regression Line of y on x | Predicts y based on x | \( y - \bar{y} = b_{yx}(x - \bar{x}) \) |
| Regression Coefficient \( b_{yx} \) | Slope of the regression line of y on x; change in y per unit change in x | \( b_{yx} = r \left(\frac{s_y}{s_x}\right) \) |
| Regression Line of x on y | Predicts x based on y | \( x - \bar{x} = b_{xy}(y - \bar{y}) \) |
| Regression Coefficient \( b_{xy} \) | Slope of the regression line of x on y; change in x per unit change in y | \( b_{xy} = r \left(\frac{s_x}{s_y}\right) \) |
| Correlation Coefficient \( r \) | Measures strength and direction of linear relationship | Varies between -1 and +1 |
Additional Information on Bivariate Data and Regression
Bivariate data involves observations on two variables for each individual or data point. Regression analysis is a powerful statistical method used to model the relationship between these variables. The regression line is the line that best fits the data points in a scatter plot, minimizing the distance between the points and the line.
There are typically two regression lines for a bivariate data set: the regression line of y on x (which predicts y given x) and the regression line of x on y (which predicts x given y). These lines are generally not the same unless the correlation is perfect (\( r = 1 \) or \( r = -1 \)).
The sign of the regression coefficient \( b_{yx} \) is the same as the sign of the correlation coefficient \( r \), indicating the direction of the relationship. A positive \( b_{yx} \) means y tends to increase as x increases, and a negative \( b_{yx} \) means y tends to decrease as x increases.
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Important Questions from Regression Analysis
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$x$ 2 3 4 5 $y$ 8 9 7 16 - Given the pairs of values of $(X, Y)$ as $(2, 60)$, $(4, 70)$ and $(6, 90)$, $m$ is the slope of the linear regression plot of $Y$ on $X$. The value of $m$ is _______.
- The data given in the table are fitted to the equation $y = mx$ using the method of least squares. The value of $m$ is ______ (rounded off to one decimal place).
$x$ 1 2 3 4 $y$ 2 6 7 10 The graph below shows a plot of annual rainfall on the X axis and number of plant species on the Y axis. Based on the pattern in the graph, which one of the following statistical methods would be most appropriate to model the relationship between annual rainfall and the number of plant species?
