Question

Suppose, the correlation between height and weight is +0.80. What proportion of the variability in weight can be explained by the relationship with height

• A. 20%
• B. 36%
• C. 64%
• D. 80%

Hint:

To find the percentage of explained variation (variance), simply square the correlation coefficient (\( r \)) and multiply by 100.
For \( r = 0.8 \), the explained variation is \( 0.8^2 \times 100 = 64\% \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The correlation coefficient (\( r \)) measures the strength and direction of the linear relationship between two variables.
To determine the proportion of variance in one variable that can be explained by the other, we must calculate the coefficient of determination (\( R^2 \)).
Key Formula or Approach:
The coefficient of determination (\( R^2 \)) is calculated by squaring the correlation coefficient (\( r \)):
\[ R^2 = r^2 \]

Step 2: Detailed Explanation:
Given the correlation coefficient between height and weight is \( r = +0.80 \).
To find the proportion of explained variability, we calculate the coefficient of determination:
\[ R^2 = (0.80)^2 = 0.64 \]
To express this proportion as a percentage, we multiply by 100:
\[ \text{Percentage of explained variability} = 0.64 \times 100\% = 64\% \]
This means that 64% of the total variation in weight can be explained by its linear relationship with height.
The remaining 36% of the variability is unexplained by height and is due to other factors (such as genetics, diet, and exercise) or random error.
Therefore, Option (C) is the correct answer.

Step 3: Final Answer:
The proportion of explained variability is 64%.