Question
An experiment in a physics class involves dropping a golf ball and recording the distance (in meters) it falls for different times (in sec) after it was released. The data are given in the table. Use technology to find the 𝑅² value of the data using a linear regression.
Time: 0 0.5 1 1.5 2 2.5 3
Distance: 0 1.2 4.9 11.0 19.5 30.5 44.0
A) 1.000
B) 0.976
C) 0.923
D) 0.998
Time: 0 0.5 1 1.5 2 2.5 3
Distance: 0 1.2 4.9 11.0 19.5 30.5 44.0
A) 1.000
B) 0.976
C) 0.923
D) 0.998
Answers
GPT-4o mini
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.
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.