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: Concept Overview:
The multiple correlation coefficient, \(R_{1.23}\), assesses the relationship between \(x_1\) and the optimal linear combination of \(x_2\) and \(x_3\). It indicates the accuracy of predicting \(x_1\) using \(x_2\) and \(x_3\) together.

Step 2: Formula:
The square of the multiple correlation coefficient \(R_{1.23}\) is calculated using zero-order correlation coefficients as follows: \[ R^2_{1.23} = \frac{r^2_{12} + r^2_{13} - 2r_{12}r_{13}r_{23}}{1 - r^2_{23}} \]
Step 3: Calculation:
Given: - \( r_{12} = 0.77 \)- \( r_{13} = 0.72 \)- \( r_{23} = 0.52 \) Calculate the squares: - \( r^2_{12} = (0.77)^2 = 0.5929 \)- \( r^2_{13} = (0.72)^2 = 0.5184 \)- \( r^2_{23} = (0.52)^2 = 0.2704 \) Calculate \(2r_{12}r_{13}r_{23}\): - \( 2(0.77)(0.72)(0.52) = 0.576576 \) Substitute into the \(R^2_{1.23}\) formula: \[ R^2_{1.23} = \frac{0.5929 + 0.5184 - 0.576576}{1 - 0.2704} = \frac{0.534724}{0.7296} \approx 0.73290 \] Calculate \(R_{1.23}\) by taking the square root: \[ R_{1.23} = \sqrt{0.73290} \approx 0.8561 \]
Step 4: Result:
The multiple correlation coefficient \(R_{1.23}\) is approximately 0.856.