Question

X and Y are two random variable with \(\mu_X = 18.4\), \(\mu_Y = 33\), \(\sigma_X = 8.4\), \(\sigma_Y = 6.4\) and \(\rho_{XY} = 0.84\). The regression line of Y on X is
1. Y = X + 7.3
2. Y = 0.64X+21.22
3. Y = 0.64X+10.61
4. Y = 0.4X+12.82

• A. Y = X + 7.3
• B. Y = 0.64X + 21.22
• C. Y = 0.64X + 10.61
• D. Y = 0.4X + 12.82

Hint:

To perform quick mental calculations, look for common factors:
Here, \( 0.84 / 8.4 \) simplifies directly to \( 0.1 \), which makes finding the slope \( b_{YX} = 0.64 \) exceptionally fast.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question involves finding the equation of the linear regression line of \( Y \) on \( X \), given the means, standard deviations, and the correlation coefficient of the two variables.
Key Formula or Approach:
The equation of the regression line of \( Y \) on \( X \) is represented by:
\[ Y - \mu_Y = b_{YX} (X - \mu_X) \] where \( b_{YX} \) is the regression coefficient of \( Y \) on \( X \), computed as:
\[ b_{YX} = \rho_{XY} \frac{\sigma_Y}{\sigma_X} \]

Step 2: Detailed Explanation:
Let us substitute the given values into the equations step-by-step:
- Given Data:
Mean of \( X \), \( \mu_X = 18.4 \)
Mean of \( Y \), \( \mu_Y = 33 \)
Standard deviation of \( X \), \( \sigma_X = 8.4 \)
Standard deviation of \( Y \), \( \sigma_Y = 6.4 \)
Correlation coefficient, \( \rho_{XY} = 0.84 \)
-

Step 1: Calculate the regression coefficient (\( b_{YX} \)):
\[ b_{YX} = 0.84 \times \frac{6.4}{8.4} \] Notice that \( \frac{0.84}{8.4} = 0.1 \). Hence:
\[ b_{YX} = 0.1 \times 6.4 = 0.64 \] - Set up the regression equation:
\[ Y - 33 = 0.64 (X - 18.4) \] \[ Y - 33 = 0.64X - (0.64 \times 18.4) \] Let us compute the product \( 0.64 \times 18.4 \):
\[ 0.64 \times 18.4 = 11.776 \] Now, substitute this value back into the equation:
\[ Y - 33 = 0.64X - 11.776 \] \[ Y = 0.64X - 11.776 + 33 \] \[ Y = 0.64X + 21.224 \] Rounding to two decimal places, we get:
\[ Y = 0.64X + 21.22 \]

Step 3: Final Answer:
The equation of the regression line of \( Y \) on \( X \) is \( Y = 0.64X + 21.22 \).
Therefore, the correct choice is Option (B).