Task #1. Examine the relationship between predictor variables (e.g., Age, Eagles, Birdies, Pars, or Bogies) and Earnings per Round.
A simple linear regression was performed for the predictor variables of age, eagles, birdies, pars, and bogies versus earnings per round. The results of the analysis were shown in tables 1, 2, 3, 4, and 5.
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Table 1. Simple Linear Regression of Age and Earnings Per Round
| Model Summary - Earnings per Round | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Model | R | R² | Adjusted R² | RMSE | |||||
| H₀ | 0.000 | 0.000 | 0.000 | 20708.229 | |||||
| H₁ | 0.195 | 0.038 | 0.033 | 20360.398 | |||||
| ANOVA | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Model | Sum of Squares | df | Mean Square | F | p | ||||||||
| H₁ | Regression | 3.257e +9 | 1 | 3.257e +9 | 7.857 | 0.006 | |||||||
| Residual | 8.208e +10 | 198 | 4.145e +8 | ||||||||||
| Total | 8.534e +10 | 199 | |||||||||||
| Note. The intercept model is omitted, as no meaningful information can be shown. | |||||||||||||
| Coefficients | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Model | Unstandardized | Standard Error | Standardized | t | p | ||||||||
| H₀ | (Intercept) | 23228.695 | 1464.293 | 15.863 | < .001 | ||||||||
| H₁ | (Intercept) | 43456.771 | 7358.522 | 5.906 | < .001 | ||||||||
| Age | -605.722 | 216.090 | -0.195 | -2.803 | 0.006 | ||||||||
Residuals vs. Predicted
Q-Q Plot Standardized Residuals
Table 2. Simple Linear Regression of Eagles and Earnings Per Round
| Model Summary - Earnings per Round | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Model | R | R² | Adjusted R² | RMSE | |||||
| H₀ | 0.000 | 0.000 | 0.000 | 20733.457 | |||||
| H₁ | 0.268 | 0.072 | 0.067 | 20024.946 | |||||
| ANOVA | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Model | Sum of Squares | df | Mean Square | F | p | ||||||||
| H₁ | Regression | 6.090e +9 | 1 | 6.090e +9 | 15.187 | < .001 | |||||||
| Residual | 7.860e +10 | 196 | 4.010e +8 | ||||||||||
| Total | 8.469e +10 | 197 | |||||||||||
| Note. The intercept model is omitted, as no meaningful information can be shown. | |||||||||||||
| Coefficients | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Model | Unstandardized | Standard Error | Standardized | t | p | ||||||||
| H₀ | (Intercept) | 23063.111 | 1473.463 | 15.652 | < .001 | ||||||||
| H₁ | (Intercept) | 12903.439 | 2970.156 | 4.344 | < .001 | ||||||||
| Eagles | 1693.279 | 434.504 | 0.268 | 3.897 | < .001 | ||||||||
Residuals vs. Predicted
Q-Q Plot Standardized Residuals
Table 3. Simple Linear Regression of Birdies and Earnings Per Round
| Model Summary - Earnings per Round | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Model | R | R² | Adjusted R² | RMSE | |||||
| H₀ | 0.000 | 0.000 | 0.000 | 20733.457 | |||||
| H₁ | 0.140 | 0.020 | 0.015 | 20580.930 | |||||
| ANOVA | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Model | Sum of Squares | df | Mean Square | F | p | ||||||||
| H₁ | Regression | 1.665e +9 | 1 | 1.665e +9 | 3.931 | 0.049 | |||||||
| Residual | 8.302e +10 | 196 | 4.236e +8 | ||||||||||
| Total | 8.469e +10 | 197 | |||||||||||
| Note. The intercept model is omitted, as no meaningful information can be shown. | |||||||||||||
| Coefficients | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Model | Unstandardized | Standard Error | Standardized | t | p | ||||||||
| H₀ | (Intercept) | 23063.111 | 1473.463 | 15.652 | < .001 | ||||||||
| H₁ | (Intercept) | 14042.632 | 4779.089 | 2.938 | 0.004 | ||||||||
| Birdies | 35.360 | 17.835 | 0.140 | 1.983 | 0.049 | ||||||||
Residuals vs. Predicted
Q-Q Plot Standardized Residuals
Table 4. Simple Linear Regression of Pars and Earnings Per Round
| Model Summary - Earnings per Round | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Model | R | R² | Adjusted R² | RMSE | |||||
| H₀ | 0.000 | 0.000 | 0.000 | 20733.457 | |||||
| H₁ | 0.028 | 0.001 | -0.004 | 20778.405 | |||||
| ANOVA | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Model | Sum of Squares | df | Mean Square | F | p | ||||||||
| H₁ | Regression | 6.416e +7 | 1 | 6.416e +7 | 0.149 | 0.700 | |||||||
| Residual | 8.462e +10 | 196 | 4.317e +8 | ||||||||||
| Total | 8.469e +10 | 197 | |||||||||||
| Note. The intercept model is omitted, as no meaningful information can be shown. | |||||||||||||
| Coefficients | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Model | Unstandardized | Standard Error | Standardized | t | p | ||||||||
| H₀ | (Intercept) | 23063.111 | 1473.463 | 15.652 | < .001 | ||||||||
| H₁ | (Intercept) | 24954.509 | 5123.759 | 4.870 | < .001 | ||||||||
| Pars | -2.338 | 6.064 | -0.028 | -0.385 | 0.700 | ||||||||
Residuals vs. Predicted
Q-Q Plot Standardized Residuals
Table 5. Simple Linear Regression of Bogies and Earnings Per Round
| Model Summary - Earnings per Round | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Model | R | R² | Adjusted R² | RMSE | |||||
| H₀ | 0.000 | 0.000 | 0.000 | 20733.457 | |||||
| H₁ | 0.118 | 0.014 | 0.009 | 20641.610 | |||||
| ANOVA | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Model | Sum of Squares | df | Mean Square | F | p | ||||||||
| H₁ | Regression | 1.175e +9 | 1 | 1.175e +9 | 2.757 | 0.098 | |||||||
| Residual | 8.351e +10 | 196 | 4.261e +8 | ||||||||||
| Total | 8.469e +10 | 197 | |||||||||||
| Note. The intercept model is omitted, as no meaningful information can be shown. | |||||||||||||
| Coefficients | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Model | Unstandardized | Standard Error | Standardized | t | p | ||||||||
| H₀ | (Intercept) | 23063.111 | 1473.463 | 15.652 | < .001 | ||||||||
| H₁ | (Intercept) | 31498.703 | 5287.894 | 5.957 | < .001 | ||||||||
| Bogies | -44.786 | 26.972 | -0.118 | -1.660 | 0.098 | ||||||||
Residuals vs. Predicted
Q-Q Plot Standardized Residuals
Summary and Analysis of Results
The linear regression for age variable showed that age significantly predicted earnings per round with F (1, 198) =7.857, p <0.001. Pearson’s coefficient of -0.195 showed a negative relationship but the nature of the prediction was weak. The relationship had the following equation
Earnings per round = 43456.771 + (-605.722 * Age)
The linear regression for eagles’ variable showed that eagles significantly predicted earnings per round with F (1, 196) =15.187, p <0.001. Pearson’s coefficient of 0.268 showed a positive relationship but the nature of the prediction was weak. The relationship had the following equation
Earnings per round = 12903.439 + (1693.279* Eagles)
The linear regression for birdies variable showed that birdies significantly predicted earnings per round with F (1, 196) = 3.931, p <0.001. Pearson’s coefficient of 0.140 showed a positive relationship but the nature of the prediction was weak. The relationship had the following equation
Earnings per round = 14042.632 + (35.360* Birdies)
The linear regression for pars variable showed that pars weakly predicted earnings per round with F (1, 196) = 0.149, p = 0.700. Pearson’s coefficient of 0.028 was close to zero showing a very weak prediction in the negative relationship. The relationship had the following equation
Earnings per round = 24954.509 + (-2.338* Pars)
The linear regression for bogies variable showed that bogies weakly predicted earnings per round with F (1, 196) = 2.757, p = 0.098 Pearson’s coefficient of 0.118 was close to zero showing a weak prediction in the positive relationship. The relationship had the following equation
Earnings per round = 31498.703 + (-44.786* bogies)
Task #2. Examine performance measures (i.e., putting, driving yard, driving accuracy, greens in regulation, sand save percentage) and identify important predictors of Earnings per Round.
Multiple regression analysis was performed and it yielded the following results for all the variables of putting, driving yard, driving accuracy, greens in regulation, and sand save percentage versus earnings per round.
Table 6. Multiple Regression
| Model Summary - Earnings per Round | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
Durbin-Watson |
|||||||||||||||
|
Model |
R |
R² |
Adjusted R² |
RMSE |
Autocorrelation |
Statistic |
p |
||||||||
| 1 |
0.659 |
0.434 |
0.417 |
16069.150 |
0.495 |
0.937 |
< .001 |
||||||||
| ANOVA | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
Model |
Sum of Squares |
df |
Mean Square |
F |
p |
||||||||
| 1 | Regression |
3.282e +10 |
5 |
6.564e +9 |
25.419 |
< .001 |
|||||||
| Residual |
4.286e +10 |
166 |
2.582e +8 |
||||||||||
| Total |
7.568e +10 |
171 |
|||||||||||
| Coefficients | |||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
Collinearity Statistics |
|||||||||||||||||
|
Model |
Unstandardized |
Standard Error |
Standardized |
t |
p |
Tolerance |
VIF |
||||||||||
| 1 | (Intercept) |
312336.800 |
119073.293 |
2.623 |
0.010 |
||||||||||||
| Putting Average |
-428752.084 |
56542.827 |
-0.474 |
-7.583 |
< .001 |
0.875 |
1.143 |
||||||||||
| Yards/Drive |
920.506 |
206.201 |
0.405 |
4.464 |
< .001 |
0.414 |
2.418 |
||||||||||
| Driving Accuracy |
706.181 |
362.162 |
0.171 |
1.950 |
0.053 |
0.442 |
2.262 |
||||||||||
| Greens in Regulation |
2088.410 |
671.358 |
0.237 |
3.111 |
0.002 |
0.588 |
1.699 |
||||||||||
| Sand Save Pct |
401.193 |
221.685 |
0.111 |
1.810 |
0.072 |
0.903 |
1.107 |
||||||||||
| Collinearity Diagnostics | |||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
Variance Proportions |
|||||||||||||||||||
|
Model |
Dimension |
Eigenvalue |
Condition Index |
(Intercept) |
Putting Average |
Yards/Drive |
Driving Accuracy |
Greens in Regulation |
Sand Save Pct |
||||||||||
| 1 |
1 |
5.980 |
1.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
||||||||||
|
2 |
0.012 |
22.067 |
0.000 |
0.000 |
0.000 |
0.034 |
0.002 |
0.760 |
|||||||||||
|
3 |
0.006 |
31.229 |
0.001 |
0.001 |
0.015 |
0.322 |
0.004 |
0.077 |
|||||||||||
|
4 |
7.892e -4 |
87.053 |
0.023 |
0.034 |
0.000 |
0.002 |
0.634 |
0.044 |
|||||||||||
|
5 |
2.922e -4 |
143.061 |
0.010 |
0.121 |
0.798 |
0.565 |
0.295 |
0.001 |
|||||||||||
|
6 |
6.515e -5 |
302.988 |
0.966 |
0.844 |
0.187 |
0.077 |
0.064 |
0.117 |
|||||||||||
| Casewise Diagnostics | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
|
Case Number |
Std. Residual |
Earnings per Round |
Predicted Value |
Residual |
Cook's Distance |
||||||
|
1 |
3.669 |
125589.000 |
69700.608 |
55888.392 |
0.253 |
||||||
|
2 |
3.872 |
120936.000 |
61789.594 |
59146.406 |
0.266 |
||||||
|
3 |
4.589 |
124746.000 |
53005.847 |
71740.153 |
0.199 |
||||||
|
4 |
3.710 |
110271.000 |
51829.521 |
58441.479 |
0.093 |
||||||
Residuals vs. Predicted
Q-Q Plot Standardized Residuals
Summary and Analysis of Results
The value of R-squared increased gradually as the performance measures were added one at a time suggesting an improvement in the nature of correlation when all performance measures were considered. The variables are good predictors of earnings per round as they had a Pearson’s Coefficient of 0.659. The relationship also had a model of F (5, 166) = 25.419, p <0.001 showing a very strong relationship. The equation was given by:
Earnings per round = 312,336.800 + (-428,752.084*Putting average) + (920.506*Driving yard) + (706.181*driving accuracy) + (2088.410*Greens in regulation) + (401.193*Sand save percentage).
