Question

If the Pearson’s correlation coefficient (r) is 0.9, what is the value of coefficient of determination?
[Question ID = 1426]

• A. 0.3
• B. 0.45
• C. 0.81
• D. 0.18

Hint:

$R^2$ always ranges from $0$ to $1$ (or $0\%$ to $100\%$). Unlike $r$, which can be negative, $R^2$ is always non-negative. It tells you how well the "regression line" fits the data.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the "coefficient of determination," which is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable in a regression model.
Key Formula or Approach:
The coefficient of determination is denoted as $R^2$ (or $r^2$). It is simply the square of the Pearson correlation coefficient ($r$).
\[ \text{Coefficient of Determination } (R^2) = r^2 \]

Step 2: Detailed Explanation:

• Given Data: The Pearson correlation coefficient ($r$) = $0.9$.

• Calculation:
\[ R^2 = (0.9)^2 \]
\[ R^2 = 0.9 \times 0.9 \]
\[ R^2 = 0.81 \]

• Interpretation: An $R^2$ value of $0.81$ means that $81\%$ of the variance in the dependent variable is predictable from the independent variable. This indicates a very strong relationship between the two variables.

Step 3: Final Answer:
The value of the coefficient of determination is $0.81$.