MAT 243 SNHU Simple Linear Regression & Multiple Regression Discussion

Summary Report: Once you have completed all the steps in your Python script, you will create a summary report to present your findings. Use the provided template to create your report. You must complete each of the following sections:

Project Three Jupyter Script
Project Three: Simple Linear Regression and
Multiple Regression
This notebook contains step-by-step directions for Project Three. It is very important to run through
the steps in order. Some steps depend on the outputs of earlier steps. Once you have completed the
steps in this notebook, be sure to write your summary report.
You are a data analyst for a basketball team and have access to a large set of historical data that you
can use to analyze performance patterns. The coach of the team and your management have
requested that you come up with regression models that predict the total number of wins for a team
in the regular season based on key performance metrics. Although the data set is the same that you
used in the previous projects, the data set used here has been aggregated to study the total number
of wins in a regular season based on performance metrics shown in the table below. These
regression models will help make key decisions to improve the performance of the team. You will
use the Python programming language to perform the statistical analyses and then prepare a report
of your findings to present for the team’s management. Since the managers are not data analysts,
you will need to interpret your findings and describe their practical implications.
There are four important variables in the data set that you will utilize in Project Three.
Variable
What does it represent
total_wins
Total number of wins in a regular season
avg_pts
Average points scored in a regular season
avg_elo_n
Average relative skill of each team in a regular season
avg_pts_differential
Average point differential between the team and their opponents in a regular
season
avg_elo_differential
Average relative skill differential between the team and their opponent in a
regular season
The average relative skill (represented by the variable avg_elo_n in the data set) is simply the
average of a team’s relative skill in a regular season. Relative skill is measured using the ELO rating.
This measure is inferred based on the final score of a game, the game location, and the outcome of
the game relative to the probability of that outcome. The higher the number, the higher the relative
skill of a team.
Reminder: It may be beneficial to review the summary report document for Project Three prior to
starting this Python script. That will give you an idea of the questions you will need to answer with
the outputs of this script.
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Project Three Jupyter Script
Step 1: Data Preparation
This step uploads the data set from a CSV file and transforms the data into a form that will be used to
create regression models. The data will be aggregated to calculate the number of wins for teams in a
basketball regular season between the years 1995 and 2015.
Click the block of code below and hit the Run button above.
In [1]: import numpy as np
import pandas as pd
import scipy.stats as st
import matplotlib.pyplot as plt
from IPython.display import display, HTML
# dataframe for this project
nba_wins_df = pd.read_csv(‘nba_wins_data.csv’)
display(HTML(nba_wins_df.head().to_html()))
print(“printed only the first five observations…”)
print(“Number of rows in the dataset =”, len(nba_wins_df))
year_id fran_id
avg_pts
avg_opp_pts avg_elo_n
avg_opp_elo_n avg_pts_differen
0 1995
Bucks
99.341463
103.707317
1368.604789 1497.311587
-4.365854
1 1995
Bulls
101.524390 96.695122
1569.892129 1488.199352
4.829268
2 1995
Cavaliers 90.451220
89.829268
1542.433391 1498.848261
0.621951
3 1995
Celtics
102.780488 104.658537
1431.307532 1495.936224
-1.878049
4 1995
Clippers
96.670732
1309.053701 1517.260260
-9.158537
105.829268
printed only the first five observations…
Number of rows in the dataset = 618
Step 2: Scatterplot and Correlation for the Total Number of
Wins and Average Relative Skill
Your coach expects teams to win more games in a regular season if they have a higher average
relative skill compared to their opponents. This is because the chances of winning are higher if a
team can maintain high average relative skill. Therefore, it is expected that the total number of wins
and the average relative skill are correlated. Calculate the Pearson correlation coefficient and its Pvalue. Make the following edits to the code block below:
1. Replace ??DATAFRAME_NAME?? with the name of the dataframe used in this project.
See Step 1 for the name of dataframe used in this project.
1. Replace ??RELATIVE_SKILL?? with the name of the variable for average relative skill.
See the table included in the Project Three instructions above to pick the variable name.
Enclose this variable in single quotes. For example, if the variable name is var1 then
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Project Three Jupyter Script
replace ??RELATIVE_SKILL?? with ‘var1’.
1. Replace ??WINS?? with the name of the variable for the total number of wins in a regular
season. Remember to enclose the variable in single quotes. See the table included in the
Project Three instructions above to pick the variable name. Enclose this variable in single
quotes. For example, if the variable name is var2 then replace ??WINS?? with ‘var2’.
The code block below will print a scatterplot of the total number of wins against the average relative
skill.
After you are done with your edits, click the block of code below and hit the Run button above.
In [2]: import scipy.stats as st
# —- TODO: make your edits here —plt.plot(nba_wins_df[‘avg_pts’], nba_wins_df[‘total_wins’], ‘o’)
plt.title(‘Total Number of Wins by Average Relative Skill’, fontsize=20)
plt.xlabel(‘Average Relative Skill’)
plt.ylabel(‘Total Number of Wins’)
plt.show()
# —- TODO: make your edits here —correlation_coefficient, p_value = st.pearsonr(nba_wins_df[‘avg_pts’], nba
_wins_df[‘total_wins’])
print(“Correlation between Average Relative Skill and the Total Number of
Wins “)
print(“Pearson Correlation Coefficient =”, round(correlation_coefficient,
4))
print(“P-value =”, round(p_value,4))
Correlation between Average Relative Skill and the Total Number of Wins
Pearson Correlation Coefficient = 0.4777
P-value = 0.0
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Project Three Jupyter Script
Step 3: Simple Linear Regression: Predicting the Total
Number of Wins using Average Relative Skill
The coach of your team suggests a simple linear regression model with the total number of wins as
the response variable and the average relative skill as the predictor variable. He expects a team to
have more wins in a season if it maintains a high average relative skill during that season. This
regression model will help your coach predict how many games your team might win in a regular
season. Create this simple linear regression model. Make the following edits to the code block
below:
1. Replace ??RESPONSE_VARIABLE?? with the variable name that is being predicted. See
the table included in the Project Three instructions above to pick the variable name. Do not
enclose this variable in quotes. For example, if the variable name is var1 then replace ??
RESPONSE_VARIABLE?? with var1.
1. Replace ??PREDICTOR_VARIABLE?? with the variable name that is the predictor. See
the table included in Project Three instructions above to pick the variable name. Do not
enclose this variable in quotes. For example, if the variable name is var2 then replace ??
PREDICTOR_VARIABLE?? with var2.
For example, if the variable names are var1 for the response variable and var2 for the predictor
variable, then the expression in the code block below should be: model = smf.ols(‘var1 ~ var2’,
nba_wins_df).fit()
After you are done with your edits, click the block of code below and hit the Run button above.
In [3]: import statsmodels.formula.api as smf
# Simple Linear Regression
# —- TODO: make your edits here –model1 = smf.ols(‘total_wins ~ avg_pts’, nba_wins_df).fit()
print(model1.summary())
OLS Regression Results
==========================================================================
====
Dep. Variable:
total_wins
R-squared:
0
.228
Model:
OLS
Adj. R-squared:
0
.227
Method:
Least Squares
F-statistic:
1
82.1
Date:
Sun, 21 Aug 2022
Prob (F-statistic):
1.52
e-36
Time:
17:37:16
Log-Likelihood:
-23
85.4
No. Observations:
618
AIC:
4
775.
Df Residuals:
616
BIC:
4
784.
Df Model:
1
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Project Three Jupyter Script
Covariance Type:
nonrobust
==========================================================================
====
coef
std err
t
P>|t|
[0.025
0.
975]
—————————————————————————-Intercept
-85.5476
9.305
-9.194
0.000
-103.820
-67
.275
avg_pts
1.2849
0.095
13.495
0.000
1.098
1
.472
==========================================================================
====
Omnibus:
24.401
Durbin-Watson:
1
.768
Prob(Omnibus):
0.000
Jarque-Bera (JB):
11
.089
Skew:
-0.033
Prob(JB):
0.0
0391
Kurtosis:
2.347
Cond. No.
1.97
e+03
==========================================================================
====
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is cor
rectly specified.
[2] The condition number is large, 1.97e+03. This might indicate that ther
e are
strong multicollinearity or other numerical problems.
Step 4: Scatterplot and Correlation for the Total Number of
Wins and Average Points Scored
Your coach expects teams to win more games in a regular season if they score more points on
average during the season. This is because the chances of winning are higher if a team maintains
high average points scored. Therefore, it is expected that the total number of wins and the average
points scored are correlated. Calculate the Pearson correlation coefficient and its P-value. Make the
following edits to the code block below:
1. Replace ??DATAFRAME_NAME?? with the name of the dataframe used in this project.
See Step 1 for the name of dataframe used in this project.
1. Replace ??POINTS?? with the name of the variable for average points scored in a regular
season. See the table included in the Project Three instructions above to pick the variable
name. Enclose this variable in single quotes. For example, if the variable name is var1 then
replace ??POINTS?? with ‘var1’.
1. Replace ??WINS?? with the name of the variable for the total number of wins in a regular
season. Remember to enclose the variable in single quotes. See the table included in the
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Project Three Jupyter Script
Project Three instructions above to pick the variable name. Enclose this variable in single
quotes. For example, if the variable name is var2 then replace ??WINS?? with ‘var2’.
The code block below will print a scatterplot of the total number of wins against the average points
scored in a regular season.
After you are done with your edits, click the block of code below and hit the Run button above.
In [4]: import scipy.stats as st
# —- TODO: make your edits here —plt.plot(nba_wins_df[‘avg_elo_n’], nba_wins_df[‘total_wins’], ‘o’)
plt.title(‘Total Number of Wins by Average Points Scored’, fontsize=20)
plt.xlabel(‘Average Points Scored’)
plt.ylabel(‘Total Number of Wins’)
plt.show()
# —- TODO: make your edits here —correlation_coefficient, p_value = st.pearsonr(nba_wins_df[‘avg_elo_n’], n
ba_wins_df[‘total_wins’])
print(“Correlation between Average Points Scored and the Total Number of W
ins “)
print(“Pearson Correlation Coefficient =”, round(correlation_coefficient,
4))
print(“P-value =”, round(p_value,4))
Correlation between Average Points Scored and the Total Number of Wins
Pearson Correlation Coefficient = 0.9072
P-value = 0.0
Step 5: Multiple Regression: Predicting the Total Number of
Wins using Average Points Scored and Average Relative
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Project Three Jupyter Script
Skill
Instead of presenting a simple linear regression model to the coach, you can suggest a multiple
regression model with the total number of wins as the response variable and the average points
scored and the average relative skill as predictor variables. This regression model will help your
coach predict how many games your team might win in a regular season based on metrics like the
average points scored and average relative skill. This model is more practical because you expect
more than one performance metric to determine the total number of wins in a regular season. Create
this multiple regression model. Make the following edits to the code block below:
1. Replace ??RESPONSE_VARIABLE?? with the variable name that is being predicted. See
the table included in the Project Three instructions above. Do not enclose this variable in
quotes. For example, if the variable name is var0 then replace ??
RESPONSE_VARIABLE?? with var0.
1. Replace ??PREDICTOR_VARIABLE_1?? with the variable name for average points
scored. Hint: See the table included in the Project Three instructions above. Do not enclose
this variable in quotes. For example, if the variable name is var1 then replace ??
PREDICTOR_VARIABLE_1?? with var1.
1. Replace ??PREDICTOR_VARIABLE_2?? with the variable name for average relative skill.
Hint: See the table included in the Project Three instructions above. Do not enclose this
variable in quotes. For example, if the variable name is var2 then replace ??
PREDICTOR_VARIABLE_2?? with var2.
For example, if the variable names are var0 for the response variable and var1, var2 for the
predictor variables, then the expression in the code block below should be: model = smf.ols(‘var0 ~
var1 + var2’, nba_wins_df).fit()
After you are done with your edits, click the block of code below and hit the Run button above.
In [5]: import statsmodels.formula.api as smf
# Multiple Regression
# —- TODO: make your edits here –model2 = smf.ols(‘total_wins ~ avg_pts + avg_elo_n’, nba_wins_df).fit()
print(model2.summary())
OLS Regression Results
==========================================================================
====
Dep. Variable:
total_wins
R-squared:
0
.837
Model:
OLS
Adj. R-squared:
0
.837
Method:
Least Squares
F-statistic:
1
580.
Date:
Sun, 21 Aug 2022
Prob (F-statistic):
4.41e
-243
Time:
17:38:26
Log-Likelihood:
-19
04.6
No. Observations:
618
AIC:
3
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Project Three Jupyter Script
815.
Df Residuals:
829.
Df Model:
Covariance Type:
615
BIC:
3
2
nonrobust
==========================================================================
====
coef
std err
t
P>|t|
[0.025
0.
975]
—————————————————————————-Intercept
-152.5736
4.500
-33.903
0.000
-161.411
-143
.736
avg_pts
0.3497
0.048
7.297
0.000
0.256
0
.444
avg_elo_n
0.1055
0.002
47.952
0.000
0.101
0
.110
==========================================================================
====
Omnibus:
89.087
Durbin-Watson:
1
.203
Prob(Omnibus):
0.000
Jarque-Bera (JB):
160
.540
Skew:
-0.869
Prob(JB):
1.38
e-35
Kurtosis:
4.793
Cond. No.
3.19
e+04
==========================================================================
====
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is cor
rectly specified.
[2] The condition number is large, 3.19e+04. This might indicate that ther
e are
strong multicollinearity or other numerical problems.
Step 6: Multiple Regression: Predicting the Total Number of
Wins using Average Points Scored, Average Relative Skill,
Average Points Differential and Average Relative Skill
Differential
The coach also wants you to consider the average points differential and average relative skill
differential as predictor variables in the multiple regression model. Create a multiple regression
model with the total number of wins as the response variable, and average points scored, average
relative skill, average points differential and average relative skill differential as predictor variables.
This regression model will help your coach predict how many games your team might win in a
regular season based on metrics like the average score, average relative skill, average points
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Project Three Jupyter Script
differential and average relative skill differential between the team and their opponents.
You are to write this code block yourself.
Use Step 5 to help you write this code block. Here is some information that will help you write this
code block. Reach out to your instructor if you need help.
1. The dataframe used in this project is called nba_wins_df.
2. The variable **avg_pts** represents average points scored by each team in a regular
season.
3. The variable **avg_elo_n** represents average relative skill of each team in a regular
season.
4. The variable **avg_pts_differential** represents average points differential between each
team and their opponents in a regular season.
5. The variable **avg_elo_differential** represents average relative skill differential between
each team and their opponents in a regular season.
6. Print the model summary.
Write your code in the code block section below. After you are done, click this block of code and hit
the Run button above. Reach out to your instructor if you need more help with this step.
In [6]: # Write your code in this code block section
model2 = smf.ols(‘total_wins ~ avg_pts + avg_elo_n + avg_pts_differential’
,
nba_wins_df).fit()
print(model2.summary())
OLS Regression Results
==========================================================================
====
Dep. Variable:
total_wins
R-squared:
0
.876
Model:
OLS
Adj. R-squared:
0
.876
Method:
Least Squares
F-statistic:
1
449.
Date:
Sun, 21 Aug 2022
Prob (F-statistic):
5.03e
-278
Time:
17:39:07
Log-Likelihood:
-18
19.8
No. Observations:
618
AIC:
3
648.
Df Residuals:
614
BIC:
3
665.
Df Model:
3
Covariance Type:
nonrobust
==========================================================================
==============
coef
std err
t
P>|t|
[0.0
25
0.975]
————————————————————————–
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Project Three Jupyter Script
————-Intercept
-35.8921
9.252
-3.879
0.000
-54.0
62
-17.723
avg_pts
0.2406
0.043
5.657
0.000
0.1
57
0.324
avg_elo_n
0.0348
0.005
6.421
0.000
0.0
24
0.045
avg_pts_differential
1.7621
0.127
13.928
0.000
1.5
14
2.011
==========================================================================
====
Omnibus:
181.805
Durbin-Watson:
0
.975
Prob(Omnibus):
0.000
Jarque-Bera (JB):
506
.551
Skew:
-1.452
Prob(JB):
1.01e
-110
Kurtosis:
6.352
Cond. No.
7.51
e+04
==========================================================================
====
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is cor
rectly specified.
[2] The condition number is large, 7.51e+04. This might indicate that ther
e are
strong multicollinearity or other numerical problems.
End of Project Three
Download the HTML output and submit it with your summary report for Project Three. The HTML
output can be downloaded by clicking File, then Download as, then HTML. Do not include the
Python code within your summary report.
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MAT 243 Project Three Summary Report
[Full Name]
[SNHU Email]
Southern New Hampshire University
Notes:
• Replace the bracketed text on page one (the cover page) with your personal information.
• You will use your selected team for all three projects
1. Introduction
Discuss the statement of the problem in terms of the statistical analyses that are being performed. Be
sure to address the following:
•
•
•
What is the data set that you are exploring?
How will your results be used?
What type of analyses will you be running in this project?
Answer the questions in a paragraph response. Remove all questions and this note before
submitting! Do not include Python code in your report.
2. Data Preparation
There are some important variables that are used in this project. Identify and explain these variables.
See the introductory section and Step 1 of the Python script to address the following:
•
•
What does the variable avg_pts_differential represent? How would you explain it to someone
who does not understand the data?
What does the variable avg_elo_n represent? How would you explain it to someone who does
not understand the data?
Answer the questions in a paragraph response. Remove all questions and this note before
submitting! Do not include Python code in your report.
3. Scatterplot and Correlation for the Total Number of Wins and Average Relative Skill
You constructed a scatterplot of the total number of wins and the average relative skill to study their
correlation. You also calculated the Pearson correlation coefficient along with its P-value.
See Step 2 in the Python script to address the following items:
•
•
•
•
In general, how are data visualization techniques used to study relationship trends between two
variables?
How is the correlation coefficient used to get the strength and direction of the association
between two variables?
In this activity, you generated a scatterplot of the total number of wins and the average relative
skill. Include a screenshot of this plot in your report.
What do the scatterplot and the Pearson correlation coefficient tell you about the association
between total number of wins and average relative skill?
•
Is the correlation coefficient statistically significant based on the P-value? Use a 1% level of
significance.
Answer the questions in a paragraph response. Remove all questions and this note before
submitting! Do not include Python code in your report.
4. Simple Linear Regression: Predicting the Total Number of Wins using Average Relative Skill
You created a simple linear regression model for the total number of wins in a regular season using the
average relative skill as the predictor variable.
See Step 3 in the Python script to address the following items:
•
•
•
In general, how is a simple linear regression model used to predict the response variable using
the predictor variable?
What is the equation for your model?
What are the results of the overall F-test? Summarize all important steps of this hypothesis test.
This includes:
a. Null Hypothesis (statistical notation and its description in words)
b. Alternative Hypothesis (statistical notation and its description in words)
c. Level of Significance
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
Test Statistic
P-value
•
•
•
Value
X.XX
*Round off to 2 decimal places.
X.XXXX
*Round off to 4 decimal places.
e. Conclusion of the hypothesis test and its interpretation based on the P-value
Based on the results of the overall F-test, can average relative skill predict the total number of
wins in the regular season?
What is the predicted total number of wins in a regular season for a team that has an average
relative skill of 1550? Round your answer down to the nearest integer.
What is the predicted number of wins in a regular season for a team that has an average relative
skill of 1450? Round your answer down to the nearest integer.
Answer the questions in a paragraph response. Remove all questions and this note (but not the
table) before submitting! Do not include Python code in your report.
5. Scatterplot and Correlation for the Total Number of Wins and Average Points Scored
You constructed a scatterplot of total number of wins and average points scored. You also calculated the
Pearson correlation coefficient along with its P-value.
See Step 4 in the Python script to answer the following questions:
•
•
•
In this activity, you generated a scatterplot of the total number of wins and average points
scored. Include a screenshot of this plot in your report.
What do the scatterplot and the Pearson correlation coefficient tell you about the association
between total number of wins and average points scored?
Is the correlation coefficient statistically significant based on the P-value? Use a 1% level of
significance.
Answer the questions in a paragraph response. Remove all questions and this note before
submitting! Do not include Python code in your report.
6. Multiple Regression: Predicting the Total Number of Wins using Average Points Scored and Average
Relative Skill
You created a multiple regression model with the total number of wins as the response variable, with
average points scored and average relative skill as predictor variables.
See Step 5 in the Python script to answer the following questions:
•
•
•
In general, how is a multiple linear regression model used to predict the response variable using
predictor variables?
What is the equation for your model?
What are the results of the overall F-test? Summarize all important steps of this hypothesis test.
This includes:
a. Null Hypothesis (statistical notation and its description in words)
b. Alternative Hypothesis (statistical notation and its description in words)
c. Level of Significance
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
Test Statistic
P-value
•
•
•
Value
X.XX
*Round off to 2 decimal places.
X.XXXX
*Round off to 4 decimal places.
e. Conclusion of the hypothesis test and its interpretation based on the P-value
Based on the results of the overall F-test, is at least one of the predictors statistically significant
in predicting the total number of wins in the season?
What are the results of individual t-tests for the parameters of each predictor variable? Is each
of the predictor variables statistically significant based on its P-value? Use a 1% level of
significance.
Report and interpret the coefficient of determination.
•
•
What is the predicted total number of wins in a regular season for a team that is averaging 75
points per game with a relative skill level of 1350?
What is the predicted total number of wins in a regular season for a team that is averaging 100
points per game with an average relative skill level of 1600?
Answer the questions in a paragraph response. Remove all questions and this note (but not the
table) before submitting! Do not include Python code in your report.
7. Multiple Regression: Predicting the Total Number of Wins using Average Points Scored, Average
Relative Skill, Average Points Differential, and Average Relative Skill Differential
You created a multiple regression model with the total number of wins as the response variable, with
average points scored, average relative skill, average points differential, and average relative skill
differential as predictor variables.
See Step 6 in the Python script to answer the following questions:
•
•
•
In general, how is a multiple linear regression model used to predict the response variable using
predictor variables?
What is the equation for your model?
What are the results of the overall F-test? Summarize all important steps of this hypothesis test.
This includes:
a. Null Hypothesis (statistical notation and its description in words)
b. Alternative Hypothesis (statistical notation and its description in words)
c. Level of Significance
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
Test Statistic
P-value
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•
•
•
Value
X.XX
*Round off to 2 decimal places.
X.XXXX
*Round off to 4 decimal places.
e. Conclusion of the hypothesis test and its interpretation based on the P-value
Based on the results of the overall F-test, is at least one of the predictors statistically significant
in predicting the number of wins in the season?
What are the results of individual t-tests for the parameters of each predictor variable? Is each
of the predictor variables statistically significant based on its P-value? Use a 1% level of
significance.
Report and interpret the coefficient of determination.
What is the predicted total number of wins in a regular season for a team that is averaging 75
points per game with a relative skill level of 1350, average point differential of -5 and average
relative skill differential of -30?
•
What is 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?
Answer the questions in a paragraph response. Remove all questions and this note (but not the
table) before submitting! Do not include Python code in your report.
8. Conclusion
Describe the results of the statistical analyses clearly, using proper descriptions of statistical terms and
concepts. Fully describe what these results mean for your scenario.
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•
Briefly summarize your findings in plain language.
What is the practical importance of the analyses that were performed?
9. Citations
You were not required to use external resources for this report. If you did not use any resources, you
should remove this entire section. However, if you did use any resources to help you with your
interpretation, you must cite them. Use proper APA format for citations.
Insert references here in the following format:
Author’s Last Name, First Initial. Middle Initial. (Year of Publication). Title of book: Subtitle of book,
edition. Place of Publication: Publisher.