Questions: For the following data set: x 8 4 6 12 -9 -3 5 -11 y 4 3 31 36 0 4 -3 -16 Part 1 of 2 (a) Use a TI-84 calculator to compute the coefficient of determination. Round the answer to at least three decimal places. The coefficient of determination is

Solution Steps

To compute the coefficient of determination (R²) for the given data set, we need to perform a linear regression analysis. The coefficient of determination indicates how well the regression line approximates the real data points. We can use Python's numpy and scipy libraries to perform this calculation.

We are given the following data sets for \( x \) and \( y \):

\[ x = [8, 4, 6, 12, -9, -3, 5, -11] \] \[ y = [4, 3, 31, 36, 0, 4, -3, -16] \]

To find the relationship between \( x \) and \( y \), we perform a linear regression analysis. The regression line can be expressed as:

\[ y = mx + b \]

where \( m \) is the slope and \( b \) is the y-intercept.

The coefficient of determination \( R^2 \) is calculated to assess how well the regression line fits the data. It is defined as:

\[ R^2 = r^2 \]

where \( r \) is the correlation coefficient obtained from the regression analysis.

From the calculations, we find:

\[ R^2 \approx 0.5132 \]

Final Answer