Question
The line of best fit for a dataset is y=−2/3x+4.2
. If this model is appropriate for the dataset, what must be true of the correlation coefficient and the residuals?(1 point)
Responses
The correlation coefficient is close to −1
, and the residuals are all relatively small.
The correlation coefficient is close to negative Start Fraction 2 over 3 End Fraction , and the residuals are all relatively small.
The correlation coefficient is close to −1
, and the residuals widely vary in size.
The correlation coefficient is close to negative Start Fraction 2 over 3 End Fraction , and the residuals widely vary in size.
The correlation coefficient is close to 1, and the residuals widely vary in size.
The correlation coefficient is close to 1, and the residuals widely vary in size.
The correlation coefficient is close to 1, and the residuals are all relatively small.
The correlation coefficient is close to 1, and the residuals are all relatively small.
. If this model is appropriate for the dataset, what must be true of the correlation coefficient and the residuals?(1 point)
Responses
The correlation coefficient is close to −1
, and the residuals are all relatively small.
The correlation coefficient is close to negative Start Fraction 2 over 3 End Fraction , and the residuals are all relatively small.
The correlation coefficient is close to −1
, and the residuals widely vary in size.
The correlation coefficient is close to negative Start Fraction 2 over 3 End Fraction , and the residuals widely vary in size.
The correlation coefficient is close to 1, and the residuals widely vary in size.
The correlation coefficient is close to 1, and the residuals widely vary in size.
The correlation coefficient is close to 1, and the residuals are all relatively small.
The correlation coefficient is close to 1, and the residuals are all relatively small.
Answers
GPT-4o mini
In your scenario, you have a line of best fit described by the equation \( y = -\frac{2}{3}x + 4.2 \). The negative slope suggests a negative correlation between the variables. Let's analyze the choices based on the properties of correlation coefficients and residuals:
1. **Correlation Coefficient**: The correlation coefficient \( r \) would be negative, as indicated by the negative slope of the line. A slope of \(-\frac{2}{3}\) implies a moderate negative correlation, so the correlation coefficient is expected to be close to \(-\frac{2}{3}\), not approaching \(-1\) or \(1\).
2. **Residuals**: If the model is appropriate for the dataset, the residuals (the differences between observed and predicted values) should be small, indicating that the predicted values are close to the actual values.
Considering these points, the most accurate statement is:
**The correlation coefficient is close to negative \(\frac{2}{3}\), and the residuals are all relatively small.**
1. **Correlation Coefficient**: The correlation coefficient \( r \) would be negative, as indicated by the negative slope of the line. A slope of \(-\frac{2}{3}\) implies a moderate negative correlation, so the correlation coefficient is expected to be close to \(-\frac{2}{3}\), not approaching \(-1\) or \(1\).
2. **Residuals**: If the model is appropriate for the dataset, the residuals (the differences between observed and predicted values) should be small, indicating that the predicted values are close to the actual values.
Considering these points, the most accurate statement is:
**The correlation coefficient is close to negative \(\frac{2}{3}\), and the residuals are all relatively small.**