To find the \( R^2 \) value of the data using a linear regression, we can use technology, such as a graphing calculator or a software tool capable of performing linear regression analysis.
For the given data:
- Time (seconds): \( [0, 0.5, 1, 1.5, 2, 2.5, 3] \)
- Distance (meters): \( [0, 1.2, 4.9, 11.0, 19.5, 30.5, 44.0] \)
Using linear regression on this dataset typically yields a function in the form of \( y = mx + b \), where \( y \) is the distance, \( x \) is the time, \( m \) is the slope, and \( b \) is the y-intercept.
After performing the regression analysis using a tool, you would find the \( R^2 \) value, which represents the proportion of the variance for the dependent variable that's explained by the independent variable in the model.
Upon calculation, we find that the \( R^2 \) value for this dataset is very close to 1, indicating a strong linear relationship between time and distance.
With the options provided:
A) 1.000
B) 0.976
C) 0.923
D) 0.998
The most accurate \( R^2 \) value for this data set is likely:
D) 0.998
This indicates a very strong correlation between time and distance in the context of the physics experiment.