How well do professional golfers putt from different distances? Researchers collected data on the percent of putts made for various distances to the hole (in feet). Model: ln(percent made) vs. Distance Predictor Coef SE Coef T P Constant 4.6649 0.0825 56.511 0.0000 Distance -0.1091 0.0067 -16.24 0.0000 S = 0.160 R-sq = 0.939 R-Sq (adj) = 0.936 Calculate and interpret the residual for the point where the golfers made 31% of putts from 14 feet away. Enter all calculations below accurate to at least 2 decimal places.
The Correct Answer and Explanation is:
Solution and Explanation
We are given a log-linear regression model:ln(percent made)=4.6649−0.1091⋅Distance\ln(\text{percent made}) = 4.6649 – 0.1091 \cdot \text{Distance}ln(percent made)=4.6649−0.1091⋅Distance
We are told that at 14 feet, the actual percent made is 31%, or as a decimal, 0.31.
We need to calculate the residual, which is defined as:Residual=Actual−Predicted\text{Residual} = \text{Actual} – \text{Predicted}Residual=Actual−Predicted
But note: since the model uses the natural logarithm of percent made, we must work in the log scale first, and then return to the original percent scale to compute the residual correctly.
Step 1: Calculate the predicted ln(percent made)
Use the regression equation:ln(percent made)=4.6649−0.1091⋅14\ln(\text{percent made}) = 4.6649 – 0.1091 \cdot 14ln(percent made)=4.6649−0.1091⋅14ln(percent made)=4.6649−1.5274=3.1375\ln(\text{percent made}) = 4.6649 – 1.5274 = 3.1375ln(percent made)=4.6649−1.5274=3.1375
Step 2: Exponentiate to get the predicted percent made
Predicted percent made=e3.1375\text{Predicted percent made} = e^{3.1375}Predicted percent made=e3.1375
Using a calculator:e3.1375≈23.04e^{3.1375} \approx 23.04e3.1375≈23.04
So the predicted percent made is approximately 23.04%.
Step 3: Calculate the residual
The actual percent made is 31%, and the predicted is 23.04%.Residual=Actual−Predicted=31−23.04=7.96%\text{Residual} = \text{Actual} – \text{Predicted} = 31 – 23.04 = \boxed{7.96\%}Residual=Actual−Predicted=31−23.04=7.96%
Interpretation
The residual of 7.96% means that professional golfers made 7.96 percentage points more putts from 14 feet than what the model predicted. In other words, at 14 feet, the golfers performed better than expected based on the overall trend.
Summary
- Predicted percent made at 14 feet: 23.04%
- Actual percent made: 31%
- Residual: 7.96%
- The model underestimated golfer performance at this distance.
This type of residual analysis helps evaluate the accuracy of a model and identify distances where performance significantly deviates from the predicted trend.
