Question

From the data relating to the yield of dry bark (\(x_1\)), height (\(x_2\)) and girth (\(x_3\)) for 18 cinchona plants, the correlation coefficient are obtained as \(r_{12}=0.77, r_{13} = 0.72, r_{23} = 0.52\). Then, the multiple correlation coefficient \(R_{1.23}\) is

• A. 0.638
• B. 0.597
• C. 0.856
• D. 0.733

Hint:

The number of observations (18 plants) is not needed to calculate the multiple correlation coefficient itself, but it would be relevant for testing its significance. Always identify which pieces of information are necessary for the formula you are using.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The multiple correlation coefficient, \(R_{1.23}\), measures the correlation between the variable \(x_1\) and the best linear combination of the variables \(x_2\) and \(x_3\). It quantifies how well \(x_1\) can be predicted from \(x_2\) and \(x_3\) together.

Step 2: Key Formula or Approach:
The formula for the square of the multiple correlation coefficient \(R_{1.23}\) is given in terms of the zero-order correlation coefficients: \[ R^2_{1.23} = \frac{r^2_{12} + r^2_{13} - 2r_{12}r_{13}r_{23}}{1 - r^2_{23}} \]
Step 3: Detailed Explanation:
We are given the following correlation coefficients: - \( r_{12} = 0.77 \) - \( r_{13} = 0.72 \) - \( r_{23} = 0.52 \) First, we calculate the squares of these coefficients: - \( r^2_{12} = (0.77)^2 = 0.5929 \) - \( r^2_{13} = (0.72)^2 = 0.5184 \) - \( r^2_{23} = (0.52)^2 = 0.2704 \) Next, calculate the term \(2r_{12}r_{13}r_{23}\): - \( 2(0.77)(0.72)(0.52) = 0.576576 \) Now, substitute these values into the formula for \(R^2_{1.23}\): \[ R^2_{1.23} = \frac{0.5929 + 0.5184 - 0.576576}{1 - 0.2704} = \frac{0.534724}{0.7296} \approx 0.73290 \] Finally, take the square root to find \(R_{1.23}\): \[ R_{1.23} = \sqrt{0.73290} \approx 0.8561 \]
Step 4: Final Answer:
The multiple correlation coefficient \(R_{1.23}\) is approximately 0.856.