Estimating Mean PER Scores Of Power Forwards In NBA Championship Using One-Sample T-Test

Research Question

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.

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 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.

The population mean PER score of NBA power forwards who played more than 50 games is more than 15.

Alternative Hypothesis

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.

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 (“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.

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).

Population

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.

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.

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 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

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.

Now, another finding of the Sum of PER scores for difference positions for the collected sample of 342 players are shown below.

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.

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.

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 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.

Reference list:

Advanced NBA Statistics for 2013-2014 Season on StatCrunch. (2014). Retrieved from https://www.statcrunch.com/app/index.php?dataid=1096769

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