MAT 243 Project Three Summary Report
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Southern New Hampshire University *
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Mathematics
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Feb 20, 2024
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MAT 243 Project Three Summary Report
MaeLadie Carter
Maeladie.carter@snhu.edu
Southern New Hampshire University
1. Introduction
This summary report will analyze and predict the total number of wins for a team in the regular season based on key performance metrics. The coach and the management department of the team have requested the use of regression models for this prediction. The data set used to perform this analysis include average points scored, average relative skill, and the average point differential between the teams playing. A multiple regression analysis will be used to determine the results.
2. Data Preparation
The variables being used are average points differential (avg_pts_differential) and average relative skill (avg_elo_n). Average point differential represents the difference in points between the different teams in the data set. It compares the points scored by each team and finds the difference whether higher or lower. Average relative skill is the average of the teams skill and compares the level of skill that the teams have to each-other.
3. Scatterplot and Correlation for the Total Number of Wins and Average Relative Skill
Correlation between Average Relative Skill and the Total Number of Wins Pearson Correlation Coefficient = 0.9072
P-value = 0.0
According to the scatterplot and the Pearson correlation coefficient, there is a strong positive correlation between the average relative skill and the total number of wins. This positive
correlation shows that as a team’s relative skill level increases, the average number of wins also
increases. Based on a P-value of 0.0, and using a 1% or 0.01 level of significance, the correlation coefficient is statistically significant
Looking at the scatterplot and the Pearson correlation coefficient, it shows that there is a strong positive correlation between the average relative skill and the total number of wins which means that as the team’s relative skill level increases then their average number of wins will increase.
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Looking at the P-value of 0.0, and a 1% level of significance, the correlation coefficient is proven to be statistically significant.
4. Simple Linear Regression: Predicting the Total Number of Wins using Average Relative Skill
Simple linear regression models are used to predict the response variable using a predictor variable by determining whether there is a significant impact of the variable examined. The equation for my model is: Y = -128.55 - 0.1121(X1). The null hypothesis H0 : 1 = 0 states that the predictor variable is not significant to the model and the alternative hypothesis Ha : 1 ≠
0 states that the predictor variable is significant to the model. The level of significance is 1%.
Table 1: Hypothesis Test for the Overall F-Test
Statistic
Value
Test Statistic
-128.25
P-value
0.000
5. Scatterplot and Correlation for the Total Number of Wins and Average Points Scored
Correlation between Average Points Scored and the Total Number of Wins Pearson Correlation Coefficient = 0.9072
P-value = 0.0
The correlation coefficient is used to get the strength and direction of the association between two variables. In the scatter plot above, when a team has a higher of average points scored, the more wins they have. The Pearson correlation coefficient in this scatter plot is 0.9072 and shows a positive correlation between average points scored and the number of wins. The P-value of 0.0 and using a significance level of 1% the correlation coefficient is statistically significant.
6. Multiple Regression: Predicting the Total Number of Wins using Average Points Scored and Average Relative Skill
Multiple linear regression models show the linear relationship between the response variables and predictor variables. The equation for my model is: Y = -152.5736 + 0.3497(X1) + 0.1055(X2). The null hypothesis H0 : 1 = 2 = 0 states that the predictor variables are not significant to the model and the alternative hypothesis Ha : at least one i ≠ 0 for I = 1,2 states that the predictor variables are significant to the model. The level of significance is 1%.
Table 2: Hypothesis Test for the Overall F-Test Statistic
Value
Test Statistic
33.903
P-value
0.0000
Since the P-value of 0.0000 is less than the significance level of 1% there is significant evidence to reject the null hypothesis. This means the average points scored by the team and the average relative skill of the team have an impact on the number of games won in a season. The results of the overall F-test show that both predictors are statistically significant. The individual t-tests results for the predictor variables are avg_pts 7.297 and avg_elo_n 47.952. Looking at each of the predictor variables P-value and using a significance level of 1% shows that each variable is statistically significant. The coefficient of determination is 0.837.
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A team that is averaging 75 points per game with a relative skill level of 1350 is has a predicted number of wins of 16 and a team that is averaging 100 points per game with an average relative skill level of 1600 is has a predicted number of wins of 51.
7. Multiple Regression: Predicting the Total Number of Wins using Average Points Scored,
Average Relative Skill, Average Points Differential, and Average Relative Skill Differential
Multiple linear regression models show the linear relationship between the response variables and predictor variables. The equation for my model is: Y = -35.89 + 1.7621(X1) + 0.0348(X2) + 0.2406(X3). The null hypothesis H0 : 1 = 2 = 3 = 0 states that the predictor variables are not significant to the model and the alternative hypothesis would be Ha : at least one i ≠ 0 for I = 1,2,3 states that the predictor variables are significant to the model. The level of significance is 1%.
Table 3: Hypothesis Test for Overall F-Test
Statistic
Value
Test Statistic
-35.8921
P-value
0.0000
The P-value of 0.0000 shows that the null hypothesis is rejected and that at least one variable is statistically significant. The t-test results are avg_pts_differential 1.7621, avg_elo_n 0.0348, and avg_pts 0.2406. The coefficient of determination is 0.876.
A team that is averaging 75 points per game with a relative skill level of 1350 and average point differential of -5 has a predicted number of wins in a regular season of 8.91. A team that is averaging 100 points per game with a relative skill level of 1600 and average point differential of
+5 has a predicted number of wins in a regular season of 46.18.
8. Conclusion
In conclusion, the results of this dataset show that if a team has a higher average number of points scored, a higher average skill level, and a higher point differential they will have more wins in the regular season. By using this model, an accurate prediction about how a team will perform in an upcoming season can be made using the results from the games that have already been played.