MAT 243 Project Three (2)
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Southern New Hampshire University *
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243
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Mathematics
Date
Feb 20, 2024
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docx
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Uploaded by jynae123
MAT 243 Project Three Summary Report
Southern New Hampshire University
1.
Introduction
The dataset I am exploring is the performances between basketball teams determined by the average relative skill levels (avg_elo_n), the average points scored (avg_pts), the average point differentials (avg_pts_differential), and the average relative skill differentials (avg_elo_differential). These results will be used to provide some insight for the coaches and the teams to work on their performance and skills that could win them more games during the regular season. The type of analysis I will be using to run this project is simple linear regressions, multiple linear regressions, as well as hypothesis testing.
2. Data Preparation
The variable avg, pts, and differential represent a basketball team’s performance during a regular season. In simpler terms, I’m comparing which teams outperform or fall behind their opponents based on their average point differential. The variable avg, elo, n represents the team’s performance between the two teams based on their average relative skill differential.
3. Simple Linear Regression: Scatterplot and Correlation for the Total Number of Wins and Average Relative Skill
Data visualization techniques are used to study relationships and trends between two variables
by using charts, graphs, or maps. This is an easier way to see and point out the trends, outliers, and patterns in a data set. The correlation coefficient used to get the strength and direction of the association between two variables is called the Pearson correlation. The Pearson correlation measures the strength and the direction of a relationship between two variables
with values ranging from -1 to 1. A coefficient of 1 indicates a positive linear relationship while a
-1 indicates a negative linear relationship.
The scatter plot tells me how the two variables are relative to each other. Looking at my scatter plot, it shows that there is a positive trend. This indicates that as the average relative skill increases, the total number of wins increases as well. My Pearson correlation coefficient is 0.9072 which is significantly positive. This means that this confirms that there is a strong positive association between average relative skill and the total number of wins. The correlation coefficient is statistically significant since I have a p- p-value of 0.0.
4. Simple Linear Regression: Predicting the Total Number of Wins using Average Relative Skill
A simple linear regression model is used to predict the response variable using a regression line to predict values of Y given the values of X. The equation for my model is total_wins = -128.2475 + 0.1121 * (avg_elo_n).
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a.
Null Hypothesis(H0
): There is no significant linear relationship between avg_elo_n and the total_wins.
b.
Alternative Hypothesis (Ha): There is a significant linear relationship between avg_elo_n and the total_wins.
c.
Level of Significance
: The level of significance is 0.01 to 1%.
d.
Report the test statistic and the P-value in a formatted table as shown below:
Table 1: Hypothesis Test for the Overall F-Test
Statistic
Value
Test Statistic
2865.0000
P-value
8.06e-234(0.0000)
Based on the results of the overall F-test, the average relative skill can predict the total number of wins in the regular season. For a team that has an average relative skill of 1550, the predicted total number of wins they have in a regular season is -128.2475 + (0.1121*1550) = 45.50 or 46 wins. The predicted number of wins in a regular season for a team that has an average relative skill of 1450 is -128.2475 + (0.1121* 1450) = 34 wins.
5. Multiple Regression: Scatterplot and Correlation for the Total Number of Wins and Average Points Scored
The scatterplot tells me about the relationship between the total number of wins and the average points scored by basketball teams in a regular season. The Pearson correlation coefficient tells me about the association between the total wins and the average points scored. My Pearson correlation coefficient is 0.4777. This coefficient has a positive association. This means that as the team’s average points scored in a regular season increases, the total number of wins also increases. The correlation coefficient based on the p-value (0.0) is statistically significant when using a 1% level of significance.
6. Multiple Regression: Predicting the Total Number of Wins using Average Points Scored and Average Relative Skill
In general, a multiple linear regression model is used to predict the response variable by using several explanatory variables to predict the outcome of a response.
The equation for my model is predicted total_wins = -152.5736 + (0.1055 * avg_elo_n) + (0.3497 * avg_pts).
a.
Null Hypothesis(H0): The predictors do not influence the total wins.
b.
Alternative Hypothesis (Ha): The predictors influence the total wins.
c.
Level of Significance: The level of significance is 1%.
d.
Report the test statistic and the P-value in a formatted table as shown below:
Table 2: Hypothesis Test for the Overall F-Test Statistic
Value
Test Statistic
1580.00
P-value
4.41e-243(0.0000)
Based on the results of the overall F-test, I reject the null hypothesis and accept the alternative hypothesis. This means that the predictors are statistically significant when predicting the total number of wins in the season. The results of individual t-tests for the parameters of each predictor variable at a 1% significance level are statistically significant. Both predictors have a significant impact on the total number of wins in a regular season.
The coefficient of determination is 0.837. This means that approximately 83.7% of the variability in total wins is determined by both the average relative skill and the average points scored. For a team averaging 75 points per game with a relative skill level of 1350 is -152.5736 +
(0.1055 * 1352) + (0.3497 * 100) = 16 wins. The predicted total number of wins in a regular season for a team that
averages 100 points per game with an average relative skill level of 1600 is -152.5736 + (0.1055 * 1600) + (0.3497 * 100) = 51 wins.
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.
7. Multiple Regression: Predicting the Total Number of Wins using Average Points Scored, Average Relative Skill, Average Points Differential, and Average Relative Skill Differential
A multiple linear regression model is used to predict the response variable by using several explanatory variables to predict the outcome of a response. The equation for my model is: Total_wins = 34.5753 + (0.2597 *
1
+ -0.0134* 2
) + (1.6206* 3
+ 0.0525 *
4
).
a.
Null Hypothesis (H0): None of the predictors is statistically significant in predicting the total number of wins.
b.
Alternative Hypothesis (Ha): At least one of the predictors is statistically significant in predicting the total number of wins.
c.
Level of Significance:
1% or alpha = 0.01
d.
Report the test statistic and the P-value in a formatted table as shown below:
Table 3: Hypothesis Test for Overall F-Test
Statistic
Value
Test Statistic
1102.00
P-value
3.07e-278(0.0000)
Based on the results of the overall F-test, at least one of the predictors is statistically significant in predicting the number of wins in the season based on the p- p-value (1102.00 < 0.01). This means I reject the null hypothesis and accept the alternative hypothesis. The coefficient of determination is 0.878. This means that there is an 87.8% chance the predictor variables explain significant variability in the number of wins. For a team averaging 75 points per game with a relative skill level of 1350, the predicted total number of wins in a regular season is:
Total number of wins = 34.5753 + (0.2597) * (75) + (-0.0134) * (1350) +(1.6206) * (-5) + (0.0525) * (30
0 = 26 wins. The predicted total number of wins in a regular season for a team that is averaging 100 points per game with a relative skill level of 1600, average point differential of +5, and average relative skill differential of +95, is: Total number of wins = 34.5753 + (0.2597) * (100) + (-0.0134) * (1600) + (1.6206) * (5) + (0.0525) * (95) = 52 wins.
8. Conclusion
In conclusion, the simple regression analysis revealed a significant and positive association between my team's average relative skill and their total number of wins meaning that teams with a higher relative skill level tend to get more wins like the coach expected. The multiple linear regression showed me that the team’s average points scored in a regular season increased, which increased the total number of wins like the coach expected. The practical importance of this analysis could encourage the teams to work on their skills to increase their average points scored because it can potentially lead to more wins in a game.