Questions: The following table contains data regarding golf scores and the average lengths of drives (in yards) for a sample of five golfers. Golf Scores and Average Drive Lengths Golf Scores, x 67 69 67 66 71 Length of Drive (in Yards), y 276 242 251 290 222 Calculate the sum of squared errors, SSE, based on a regression analysis of the golf data. Round your answer to two decimal places, if necessary.

Solution Steps

The means of the golf scores and drive lengths are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 68.0 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 256.2 \]

The correlation coefficient \( r \) is computed to assess the strength of the linear relationship between the two variables:

\[ r = -0.9187 \]

The slope \( \beta \) is calculated using the formula:

\[ \text{Numerator for } \beta: \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 86909 - 5 \cdot 68.0 \cdot 256.2 = -199.0 \]

\[ \text{Denominator for } \beta: \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 23136 - 5 \cdot 68.0^2 = 16.0 \]

Thus, the slope \( \beta \) is:

\[ \beta = \frac{-199.0}{16.0} = -12.4375 \]

The intercept \( \alpha \) is calculated as:

\[ \alpha = \bar{y} - \beta \bar{x} = 256.2 - (-12.4375) \cdot 68.0 = 1101.95 \]

The equation of the line of best fit is given by:

\[ y = 1101.95 - 12.4375x \]

Using the regression equation, the predicted drive lengths for the golf scores are:

\[ \text{Predicted Drive Lengths} = [268.6375, 243.7625, 268.6375, 281.0750, 218.8875] \]

The sum of squared errors (SSE) is calculated as follows:

\[ \text{SSE} = \sum (y - \hat{y})^2 = 457.74 \]

Final Answer