Statistical Analysis: T-Test Application on NBA Power Forwards' PER

This research report focuses on applying a one-sample t-test to analyze the Player Efficiency Rating (PER) scores of NBA power forwards who have played more than 50 games. The study aims to determine if the population mean PER score of these players exceeds 15. Data was collected from Statcrunch, focusing on 342 players from the 2013-2014 season, with a final sample of 66 power forwards meeting the criteria. The null hypothesis, stating that the mean PER score is less than or equal to 15, was tested against the alternative hypothesis that it is greater than 15. The results indicated that the null hypothesis could not be rejected at a 0.05 significance level, suggesting that there isn't sufficient evidence to conclude that the mean PER score is greater than 15. The report includes descriptive statistics, scatter diagrams, and histograms to visualize the data and support the findings.

Running head: RESEARCH PROJECT ON APPLICATION OF T-TEST
RESEARCH PROJECT ON APPLICATION OF T-TEST
Name of the Student
Name of the University
Author Note

2RESEARCH PROJECT ON APPLICATION OF T-TEST
Table of Contents
Introduction:...............................................................................................................................3
Research Question:.....................................................................................................................3
Null Hypothesis (H0):............................................................................................................4
Alternative Hypothesis (H1):.................................................................................................4
Population:.................................................................................................................................4
Data collection:..........................................................................................................................4
Study Design:.............................................................................................................................5
Results & Findings:....................................................................................................................6
Discussion and Conclusion:.......................................................................................................9
Reference list:...........................................................................................................................11

3RESEARCH PROJECT ON APPLICATION OF T-TEST
Introduction:
The aim of this particular research is to find a suitable research topic where the
research question/s can be answered using the method of the t-test. The three types of t-tests
that can be applied based on the research question are one sample t-test, the matched pair test
and the two-sample t-test. The one sample t-test is applied where a single sample of the data
is obtained from a single population and some parameter (like mean) of the population is
estimated or some conclusion about the population is made from the same parameter of the
sample data. The matched paired t-test (also known as dependent sample t-test) is chosen
when data in two sets are collected for the same people or elements may be in different time
or for two different methods (O'Mahony, 2017). Then the two sets are compared to see if the
desired properties are the same or different in the population. The two-sample t-test which is
also known as independent sample t-test is applied when the two samples are collected from
two different populations and then mean of those two samples are compared to estimate if the
means of two population is different or same (Brown et al., 2017). The statistical records of
different teams play under the National Basketball Association, United States is quite
amazing and data has been collected from Statcrunch Website. Power Forwards plays an
important role in Basketball as they create more opportunities to score and hence the research
aims to estimate some statistics about Power Forwards (PF) who played in NBA
championship till date (Carter, Schnepel & Steigerwald, 2017). Now, here, based on the
collected data set of records of NBA basketball teams of the United States one sample t-test is
applied to answer the research question as given below.
Research Question:
The research aims to estimate the mean PER score of the all team players who played
in the position of Power Forwards (PF) for more than 50 games. It is presumed before doing

4RESEARCH PROJECT ON APPLICATION OF T-TEST
the analysis that the mean PER score of all PFs who played more than 50 games is more than
15. The hypothesis statements are as follows.
Null Hypothesis (H0):
The population mean PER score of NBA power forwards who played more than 50 games is
less than or equals to 15.
Alternative Hypothesis (H1):
The population mean PER score of NBA power forwards who played more than 50 games is
more than 15.
Population:
The population is the PER scores of all the Power Forwards who played in NBA
annual Championship series till now.
Variable: Although there are several qualitative and quantitative variables in the whole
dataset, the concerned variables which are used are the position, PER score and a number of
games played by different players. Using the conditions given in the research question the
combined variable PER scores of Power Forwards who played more than 50 games are
created in the excel spreadsheet.
Data collection:
A small sample of 342 players’ statistics is collected from Statcrunch website, where
the players who played in the period of 2013-2014 is given. The collected data is a secondary
data and the records are authentic as many of the statisticians used different datasets from
here for their statistical methods and found significant results matching with common
predictions. In the dataset there are several column names starting from the player name,
position, age, team, games, Minutes, PER and statistics of different attributes in the game

5RESEARCH PROJECT ON APPLICATION OF T-TEST
("Advanced NBA Statistics for 2013-2014 Season on StatCrunch", 2014). At first as per the
research question, the players who played in the position of Power Forward are extracted by
using excel sorting technique and it is found that there is a total of 74 Power Forwards are
there in the dataset. Then, again the Power Forwards who played for more than 50 games are
selected and then their PER scores are filtered out. It has been found that there is a total of 66
PFs are there who played more than 50 games. Hence, the final sample size which is used for
the t-test method is 66.
Study Design:
Now, as the population size is unknown and the population standard deviation is also
not known hence the method of t-test can be applied to assume the population is
approximately normally distributed (Kim, 2015). The null and the alternative hypothesis
statements can be written in terms of mathematical equations as follows.
H0: μ ≤15
H1: μ>15
Where, μ is the hypothesized mean i.e. the mean of the PER scores of NBA power forwards
who played more than 50 games is more than 15.
Now, from the more than sign in the alternative hypothesis it can be said that the test is a
right tail t-test. The degrees of freedom of the right tail t-test is sample size(n) -1 = 66-1 = 65.
Now, the significance level of the test is considered as 0.05 or in other words 95% assurance
about the conclusion of the test can be given (Mertler & Reinhart, 2016).
Now, the sample mean i.e. the mean PER score of 66 Power Forwards is calculated using
excel as xbar = 15.66 and standard deviation(s) of the same is 4.30.
Now, the test statistics is t = (xbar - μ)/(s/sqrt(n)) = 1.247.

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Now, if the p value i.e. the area under the t distribution curve with the given degrees of
freedom and in the right half of the test statistics t is less than the significance level then the
null hypothesis can be rejected otherwise it will be concluded that not sufficient evidence is
found to reject the null hypothesis (Manly & Alberto, 2016).
Here, the p value is found more than 0.05 which is our considered significance level and the
null hypothesis has not been rejected.
Results & Findings:
The results as calculated in excel using proper excel functions are given below.
Sample Size(n) 66
Sample
mean(xbar)
15.66061
Sample sd(s) 4.304412
Null hypothesis The mean PER score of NBA power forwards who played more than
50 games is less than or equals to 15
Alternative
hypothesis
The mean PER score of NBA power forwards who played more than
50 games is more than 15
standard error 0.529836
t stat 1.246811
One sample right tailed t test
Considered
Significance level
0.05
df 65
p value 0.10847
Decision As the p value is more than considered significance level hence the

7RESEARCH PROJECT ON APPLICATION OF T-TEST
null hypothesis can't be rejected
Conclusion The mean PER score of NBA power forwards who played more than
50 games is less than or equals to 15
Descriptive Statistics:
PER score of PF who
played more than 50
games
Mean 15.66061
Standard Error 0.529836
Median 15
Mode 11.1
Standard
Deviation
4.304412
Sample
Variance
18.52796
Kurtosis 0.960602
Skewness 0.775332
Range 21.8
Minimum 7.5
Maximum 29.3
Sum 1033.6
Count 66

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The descriptive depicts that the scores are largely deviated from mean as the sample variance
is large 18.528. The sample distribution of PER scores is slightly positively skewed as the
skewness is 0.775.
Apart from this, scatter diagram of the number of games played by 66 Power Forwards and
their corresponding PER scores are also shown below.
45 50 55 60 65 70 75 80 85
0
5
10
15
20
25
30
35
PER score of PF who played more than 50 games
Now, another finding of the Sum of PER scores for difference positions for the collected
sample of 342 players are shown below.
C PF PF-SF PG SF SF-PF SG SG-PG SG-SF
0
200
400
600
800
1000
1200
Sum of PER by Position

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The clustered chart represents that sum of the PER scores is maximum for the Power Forward
position.
Now, the distribution of the chosen sample of 66 Power Forwards who played more than 50
games are represented by the following histogram.
less than 7 7 to 12 12 to 17 17 to 22 22 to 27 27 to 32
0
5
10
15
20
25
30
35
Histogram of PER scores of 66 sample Power
Forwards played more than 50 games
The histogram depicts that the most of the Power Forwards who played more than 50 games
scored between 12 and 17 points. Also, the sample distribution is slightly positively skewed
but can be approximately normally distrusted if the number of intervals is increased.
Discussion and Conclusion:
Hence, it can be concluded that the objective of the research paper has been
successfully met as by using proper method of one sample t test, it has been found that all the
Power Forwards who played more than 50 games in NBA championship have the mean PER
score less than or equals to 15. The population of the research variable is considered to be
normally distributed with mean equals to the sample mean and unknown standard deviation.
Hence, with 95% confidence level it can be said that there is a 5% chance that the obtained

10RESEARCH PROJECT ON APPLICATION OF T-TEST
decision does not hold. However, with the increase in the confidence level the confidence
interval of population mean increases and a more precise decision can be made. Another way
of obtaining a more precise decision is to increase the sample size (minimum sample size for
1 sample t-test is 40). As, the sample size increases the sampling distribution of the sample
mean gets closer to the normally distributed population and the conclusion can be more
assured. Besides this, various graphs between sample variables are formed for better
understanding of different statistical measures of the sample data. Finally, it can be said that
the research about some sample data (that is collected from an authentic source) is performed
by using method of t tests and produced an appropriate result with minimum systematic error.

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Reference list:
Advanced NBA Statistics for 2013-2014 Season on StatCrunch. (2014). Retrieved
from https://www.statcrunch.com/app/index.php?dataid=1096769
Brown, D. G., Rao, S., Weir, T. L., O’Malia, J., Bazan, M., Brown, R. J., & Ryan, E.
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Cao-Lormeau, V. M., Blake, A., Mons, S., Lastère, S., Roche, C., Vanhomwegen,
J., ... & Vial, A. L. (2016). Guillain-Barré Syndrome outbreak associated with Zika virus
infection in French Polynesia: a case-control study. The Lancet, 387(10027), 1531-1539.
Carter, A. V., Schnepel, K. T., & Steigerwald, D. G. (2017). Asymptotic behavior of
at-test robust to cluster heterogeneity. Review of Economics and Statistics, 99(4), 698-709.
Kim, T. K. (2015). T test as a parametric statistic. Korean journal of anesthesiology,
68(6), 540-546.
Manly, B. F., & Alberto, J. A. N. (2016). Multivariate statistical methods: a primer.
Chapman and Hall/CRC.
Mertler, C. A., & Reinhart, R. V. (2016). Advanced and multivariate statistical
methods: Practical application and interpretation. Routledge.
O'Mahony, M. (2017). Sensory evaluation of food: statistical methods and
procedures. Routledge.