To determine the appropriate line of best fit for the given data, we should analyze the trend in the scatter plot. The winning times in a 50-meter race over eight consecutive years appear to decrease, suggesting a negative correlation between years and winning times. This means that the winning times are likely getting faster as years progress.
The options provided for the line of best fit include both positive and negative slopes. Since we expect a downward trend, we should consider only the options with a negative slope:
- \( \hat{y} = -0.29x + 5.67 \)
- \( \hat{y} = -0.45x + 7.69 \)
Now, we can analyze the slopes:
- A slope of \(-0.29\) implies that for each additional year, the winning time decreases by 0.29 seconds.
- A slope of \(-0.45\) implies that for each additional year, the winning time decreases by 0.45 seconds.
Given the data generally trends downward, either option could be valid depending on the extent of the decrease noticed in the plot. However, since the decrease appears to be significant, a steeper slope, such as \(-0.45\), might be more appropriate.
Therefore, the suggested line of best fit from the provided options would be:
\[ \hat{y} = -0.45x + 7.69 \]
This line captures the trend of decreasing winning times more sharply compared to the other negative option.