Understanding the Regression Line of Y on X
This problem asks us to find the equation for the line of regression of y on x. This line helps us predict the value of variable 'y' when the value of variable 'x' is known.
Key Concepts and Formula
The general formula for the equation of the line of regression of y on x is:
$$y - \bar{y} = b_{yx}(x - \bar{x})$$Where:
- $y$ is the dependent variable.
- $x$ is the independent variable.
- $\bar{y}$ is the mean of the y-values.
- $\bar{x}$ is the mean of the x-values.
- $b_{yx}$ is the coefficient of regression of y on x. This coefficient indicates how much y changes for a one-unit change in x.
Given Information
From the question, we have the following values:
- Mean of x, $\bar{x} = 32$
- Mean of y, $\bar{y} = 38$
- Coefficient of regression of x on y, $b_{xy} = -0.2337$ (Note: This value is not used for the regression of y on x)
- Coefficient of regression of y on x, $b_{yx} = -0.6643$
Step-by-Step Calculation
We need to substitute the given values into the formula for the regression line of y on x:
- Substitute the means and the $b_{yx}$ coefficient:
Start with the formula: $y - \bar{y} = b_{yx}(x - \bar{x})$
Plug in the values: $y - 38 = -0.6643(x - 32)$
- Distribute the regression coefficient:
Multiply $-0.6643$ by both terms inside the parentheses:
$y - 38 = (-0.6643 \times x) - (-0.6643 \times 32)$
$y - 38 = -0.6643x + 21.2576$
Calculation check: $0.6643 \times 32 = 21.2576$
- Isolate 'y' to find the final equation:
Add $38$ to both sides of the equation to solve for $y$:
$y = -0.6643x + 21.2576 + 38$
$y = -0.6643x + 59.2576$
Final Equation of the Regression Line
The calculated equation for the line of regression of y on x is:
y = -0.6643x + 59.2576
This matches one of the provided options.
