To find the coefficient of determination ($r^2$) and the percentage of the total variation explained by the linear relationship, follow these steps:

Part 1: Coefficient of Determination ($r^2$)

The coefficient of determination, $r^2$, is calculated by squaring the correlation coefficient $r$.

Given: $r = 0.049$

Calculating $r^2$: $r^2 = (0.049)^2 = 0.0024$

So, the coefficient of determination, rounded to four decimal places, is: $r^2 = 0.0024$

Part 2: Percentage of Total Variation Explained

To find the percentage of the total variation explained by the linear relationship, convert $r^2$ to a percentage by multiplying by 100%.

$\text{Explained variation} = r^2 \times 100\% = 0.0024 \times 100\% = 0.24\%$

So, the percentage of the total variation explained by the linear relationship, rounded to two decimal places, is: $\text{Explained variation} = 0.24\%$

Summary of Answers:

1. Coefficient of Determination: $r^2 = 0.0024$
2. Percentage of Explained Variation: $0.24\%$

Would you like further details or have any questions on these steps?

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1. What is the significance of a low $r^2$ value?
2. How does $r$ differ from $r^2$ in interpreting data relationships?
3. What does an $r^2$ of 1 indicate about a dataset?
4. Can $r^2$ ever be negative, and why or why not?
5. How would outliers affect the correlation coefficient $r$ and the determination coefficient $r^2$?

Tip: The coefficient of determination $r^2$ indicates the strength of the linear relationship; closer to 1 means a stronger relationship, while closer to 0 indicates a weaker one.