. In the simple linear regression result, r2 =.47. What percentage of the variation in the

dependent variable is left UNEXPLAINED by the independent variable?

a. 53.0%

b. 47.0%

c. .47%

d. .53%

In the context of a simple linear regression, \( r^2 \) (the coefficient of determination) represents the proportion of the variance in the dependent variable that is explained by the independent variable. To find the percentage of the variation that is **unexplained**, we need to subtract \( r^2 \) from 1 and then convert that proportion to a percentage.

Given:
\[ r^2 = 0.47 \]

The unexplained variation is:
\[ 1 - r^2 = 1 - 0.47 = 0.53 \]

Converting to a percentage:
\[ 0.53 \times 100 = 53.0\% \]

So, the correct answer is:

a. 53.0%

### Do you want details or have any questions?

### Related Questions:
1. What is the interpretation of an \( r^2 \) value in simple linear regression?
2. How do you calculate the explained variation in a simple linear regression model?
3. What does an \( r^2 \) value of 1 indicate?
4. How can the residual sum of squares be used to determine the unexplained variation?
5. What are some limitations of using \( r^2 \) as a measure of model fit?
6. How can you improve the \( r^2 \) value in a regression model?
7. What is the difference between \( r^2 \) and adjusted \( r^2 \)?
8. Why is it important to assess both explained and unexplained variation in regression analysis?

### Tip:
When evaluating the fit of a regression model, consider both \( r^2 \) and the residual plots to understand the model's performance and potential areas for improvement.

Explore how to determine the percentage of variation left unexplained in a simple linear regression model with an r^2 value of 0.47. Learn the significance of r^2, variance, and its calculation. Suitable for advanced learners and professionals.

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