MAT 243 Project Three Summary Report
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Southern New Hampshire University *
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Feb 20, 2024
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MAT 243 Project Three Summary Report
Danielle Dahlberg
Danielle.Dahlberg@snhu.edu
Southern New Hampshire University
1.
Introduction
As the data analyst for the Sacramento Kings, my role is to delve into the statistical relationships between various factors and the total number of wins. The objective is to extract valuable insights that can guide informed decision-making. The data set that forms the basis of this analysis is derived from multiple NBA teams from 1995-2015.
The results from this analysis will enable the Sacramento Kings to comprehend the influence of specific variables on their performance and pinpoint areas that require improvement. The findings will serve as a foundation for data-driven decisions pertaining to team strategy, player recruitment, and overall performance enhancement.
The analytical approach for this project encompasses several statistical analyses to decipher the relationship between the total number of wins and various factors. The analyses include:
1.
Scatterplot and Correlation
: This will provide a visual representation and measure the correlation strength between the total number of wins and average relative skill, as well as average points scored.
2.
Simple Linear Regression
: This will predict the total number of wins using the average relative skill as the predictor variable.
3.
Multiple Regression
: This will predict the total number of wins using average points scored and average relative skill as predictor variables.
4.
Multiple Regression with additional variables
: This will predict the total number of wins using average points scored, average relative skill, average points differential, and average relative skill differential as predictor variables.
This comprehensive approach ensures a thorough analysis of the factors influencing the team’s performance.
2. Data Preparation
The data set that forms the foundation of this analysis encompasses the following variables:
1.
Average Points Differential (avg_pts_differential)
: This variable signifies the average points differential of a team, computed by deducting the average opponent points from the average points scored by the team. It provides an insight into the team’s scoring performance relative to their opponents. A positive value suggests that the team, on average, scores more points than their opponents, while a negative value indicates the contrary.
2.
Average Elo Rating (avg_elo_n)
: This variable denotes the average Elo rating of a team,
a metric that measures the team’s relative skill level. Elo ratings, which are based on the outcomes of previous games, serve as an indicator of the team’s overall performance and competitiveness. A higher Elo rating implies a stronger team, whereas a lower rating suggests a relatively weaker team.
3. Simple Linear Regression: Scatterplot and Correlation for the Total Number of Wins and Average Relative Skill
Data visualization techniques are used to study relationship trends between two variables by graphically representing the data points. Scatterplots are commonly used for this purpose. By plotting the data points on a scatterplot, we can visually observe any patterns or trends between the variables.
The correlation coefficient is used to measure the strength and direction of the association between two variables. It ranges from -1 to 1, where -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no correlation. The sign of the coefficient indicates the direction of the association, while the magnitude represents the strength.
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Based on the scatterplot and the Pearson correlation coefficient of 0.9127, there is a strong positive correlation between the total number of wins and the average relative skill. As the average relative skill increases, the total number of wins also tends to increase. This association is statistically significant.
The P-value of 0.0 indicates that the correlation coefficient is statistically significant. With a 0.01 level of significance, we reject the null hypothesis that there is no correlation between the total number of wins and the average relative skill. Instead, we conclude that there is a significant positive correlation between these two variables.
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 (total number of wins) using the predictor variable (average relative skill). It assumes a linear relationship between the predictor variable and the response variable and estimates the coefficients of the
equation that best fits the data. The equation for this model is: total_wins = 39.8590 + 0.1096 * avg_elo_differential. Hypothesis test for the overall F-test:
a. Null Hypothesis (
H
0
): The average relative skill has no impact on the total number of wins.
b. Alternative Hypothesis (
H
a
): The average relative skill has an impact on the total number of wins.
c. Level of Significance (
α
): The level of significance is 1% (α = 0.01).
Table 1: Hypothesis Test for the Overall F-Test
Statistic
Value
Test Statistic
3071.00
P-value
0.0000
d. Since the P-value of 0.000 is less than 0.01 significance level, we reject the null hypothesis. This means that the average relative skill does have a significant impact on the total number of wins in the regular season.
Based on the results of the overall F-test, we can conclude that the average relative skill can significantly predict the total number of wins in the regular season. Examples of this are as follows:
The predicted total number of wins in a regular season for a team with an average relative
skill of 1550 is: total_wins = 39.8590 + 0.1096 * 1550 = 213.599
Rounding down to the nearest integer, the predicted number of wins is 213.
The predicted number of wins in a regular season for a team with an average relative skill
of 1450 is: total_wins = 39.8590 + 0.1096 * 1450 = 190.209
Rounding down to the nearest integer, the predicted number of wins is 190.
5. Multiple Regression: Scatterplot and Correlation for the Total Number of Wins and Average Points Scored
The scatterplot represents the relationship between the total number of wins and average points scored. Each point on the scatterplot represents a particular team, with the x-coordinate indicating the average points scored and the y-coordinate representing the total number of wins. By visualizing the data in a scatterplot, you can observe the overall pattern and the direction of the relationship between the two variables.
The P-value for this model is 0.0000 which is less than the 1% level of significance (α = 0.01). Therefore, we can conclude that the correlation coefficient is statistically significant. This means that it is very unlikely that the observed correlation occurred by chance, indicating a significant linear relationship between the total number of wins and average points scored.
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6. Multiple Regression: Predicting the Total Number of Wins using Average Points Scored and Average Relative Skill
A multiple linear regression model is used to predict the response variable by using multiple predictor variables. It estimates the relationship between the response variable and the predictor variables by fitting a line or a plane through the data points. The equation for this model is: total_wins = 4.0513 + 0.3669 * avg_pts + 0.1031 * avg_elo_differential
Hypothesis test for the overall F-test:
a. Null Hypothesis (
H
0
): All coefficients for the predictor variables are equal to zero. This means that the predictor variables have no effect on the total number of wins.
b. Alternative Hypothesis (
H
a
): At least one coefficient for the predictor variables is not equal to zero. This means that at least one predictor variable has an effect on the total number of wins.
c. Level of Significance (
α
): The level of significance is 1% (α = 0.01). Table 2: Hypothesis Test for the Overall F-Test
Statistic
Value
Test Statistic
1724.00
P-value
0.0000
d. Since the p-value 0.0000 is less than the significance level 0.01 we reject the null hypothesis. Therefore, we can conclude that at least one of the predictor variables is statistically significant in predicting the total number of wins in the season.
Based on the results of the individual t-tests, both predictor variables (avg_pts and avg_elo_differential) are statistically significant at a 1% level of significance. Their p-values are both 0.000, indicating strong evidence against the null hypothesis that their coefficients are zero.
The coefficient of determination (R
²
) is 0.849, which means that approximately 84.9% of the variability in the total number of wins can be explained by the predictor variables (average points
scored and average relative skill).
The predicted total number of wins for a team averaging 75 points per game with a relative skill level of 1350 is:
total_wins = 4.0513 + 0.3669 * 75 + 0.1031 * 1350
total_wins = 4.0513 + 27.5175 + 139.2285
total_wins ≈ 170.7978
Therefore, the predicted total number of wins for this team is approximately 171.
The predicted total number of wins for a team averaging 100 points per game with an average relative skill level of 1600 is:
total_wins = 4.0513 + 0.3669 * 100 + 0.1031 * 1600
total_wins = 4.0513 + 36.6900 + 164.9600
total_wins ≈ 205.7013
Therefore, the predicted total number of wins for this team is approximately 206.
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 using multiple predictor variables by fitting a linear equation to the data. The model assumes a linear relationship between the response variable and the predictor variables. The equation for the model is: total wins = 34.5753 + 0.2597 * avg_pts - 0.0134 * avg_elo_n + 1.6206 * avg_pts_differential + 0.0525 * avg_elo_differential
Hypothesis test for the overall F-test:
a. Null Hypothesis
(H
0
): All the coefficients of the predictor variables are zero.
b. Alternative Hypothesis(
H
a
): At least one of the coefficients of the predictor variables is not zero.
c. Level of Significance (
α
): The level of significance is 1% (
α = 0.01)
Table 3: Hypothesis Test for Overall F-Test
Statistic
Value
Test Statistic
1102.00
P-value
0.0000
d. Since the p-value 0.0000 is less than the significance level 0.01, we reject the null hypothesis. This means that at least one of the predictor variables is statistically significant in predicting the number of wins in the season.
Based on the results of the overall F-test, at least one of the predictors is statistically significant in predicting the total number of wins in the season.
The results of individual t-tests for the parameters of each predictor variable are as follows:
Average Points Scored:
The t-value is 6.070 and the P-value is 0.000, which is less than the 1% level of significance. Therefore, the average points scored is statistically significant in predicting the total number of wins.
Average Relative Skill:
The t-value is -0.769 and the P-value is 0.442, which is greater than the 1% level of significance. Therefore, the average relative skill is not statistically significant in predicting the total number of wins.
Average Points Differential:
The t-value is 12.024 and the P-value is 0.000, which is less than the 1% level of significance. Therefore, the average points differential is statistically significant in predicting the total number of wins.
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Average Relative Skill Differential:
The t-value is 2.915 and the P-value is 0.004, which is less than the 1% level of significance. Therefore, the average relative skill differential is statistically significant in predicting the total number of wins.
The coefficient of determination (R
²
) is 0.878, indicating that 87.8% of the variance in total wins can be explained by the average points scored, average relative skill, average points differential, and average relative skill differential.
The predicted total number of wins for a team averaging 75 points per game with a relative skill level of 1350, an average point differential of -5, and an average relative skill differential of -30 is:
total wins = 34.5753 + 0.2597 * 75 - 0.0134 * 1350 + 1.6206 * (-5) + 0.0525 * (-30) = 34.5753 + 19.4775 - 18.09 - 8.103 + (-1.575) = 26.3848
Therefore, the predicted total number of wins in a regular season for this team is approximately 26.38.
The predicted total number of wins for a team averaging 100 points per game with a relative skill level of 1600, an average point differential of +5, and an average relative skill differential of +95 is:
total wins = 34.5753 + 0.2597 * 100 - 0.0134 * 1600 + 1.6206 * 5 + 0.0525 * 95 = 34.5753 + 25.97 - 21.44 + 8.103 + 4.9875 = 52.1958
Therefore, the predicted total number of wins in a regular season for this team is approximately 52.20.
8. Conclusion
In conclusion, the statistical analyses conducted in this report provide valuable insights for the Sacramento Kings. The results indicate strong relationships between the total number of wins
and various factors such as average relative skill, average points scored, average points differential, and average relative skill differential. These findings have practical importance as they can guide data-driven decisions pertaining to team strategy, player recruitment, and overall performance enhancement.
The analysis revealed a strong positive correlation between the total number of wins and average relative skill. As the average relative skill increases, the total number of wins also tends to increase. This association is statistically significant and can be used to predict the total number
of wins based on the average relative skill.
Furthermore, the multiple regression analysis showed that average points scored, average points differential, and average relative skill differential are also significant predictors of the total
number of wins. The coefficient of determination indicates that approximately 84.9% of the variability in total wins can be explained by these predictor variables.
The practical importance of these analyses lies in the ability to make data-driven decisions for the Sacramento Kings. By understanding the influence of specific variables on their performance, the team can pinpoint areas that require improvement and make informed decisions
regarding team strategy and player recruitment. The predictions generated from the regression models can also provide insight into the expected number of wins based on certain variables, allowing the team to set realistic goals and expectations.
9. Citations
zyBooks
. (n.d.). Learn.zybooks.com. https://learn.zybooks.com/zybook/MAT-243-X2785-OL-
TRAD-UG.23EW2/chapter/5/section/1
Chapters 5-7
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