Question

1. essential question how can you evaluate the goodness of fit of a line of best fit for a paired data set?
2. vocabulary describe the difference between interpolation and extrapolation.
3. error analysis a student says that a correlation coefficient of -0.93 indicates that the two quantities of a data set have a weak correlation. explain the error the student made.

Explanation:

1. The goodness - of - fit of a line of best fit for a paired data set can be evaluated using the correlation coefficient ($r$) and the coefficient of determination ($r^{2}$). A value of $r$ close to 1 or - 1 indicates a strong linear relationship, and $r^{2}$ represents the proportion of the variance in the dependent variable that is predictable from the independent variable. Residual analysis can also be used, where the differences between the observed and predicted values are examined for randomness.
2. Interpolation is the process of estimating a value within the range of known data points. For example, if we have data points for ages 10, 20, and 30, estimating the value at age 15 is interpolation. Extrapolation is estimating a value outside the range of known data points. Using the previous example, estimating the value at age 35 is extrapolation. Extrapolation is often less reliable as it assumes the existing trend continues beyond the observed data.
3. The correlation coefficient ($r$) ranges from - 1 to 1. A value close to 1 indicates a strong positive linear correlation, a value close to - 1 indicates a strong negative linear correlation, and a value close to 0 indicates a weak or no linear correlation. Since - 0.93 is close to - 1, it indicates a strong negative correlation, not a weak one.

Answer:

1. Evaluate using correlation coefficient ($r$), coefficient of determination ($r^{2}$), and residual analysis.
2. Interpolation estimates within known data range; extrapolation estimates outside known data range.
3. The student is wrong because - 0.93 is close to - 1, indicating a strong negative correlation, not a weak one.