Hypothesis Testing

This section incorporates several hypothesis tests to reveal temporal patterns and to facilitate the investigation of specific queries, such as the evaluation of the performance of specific teams and the assessment of whether players from other nations of origin demonstrate superior metrics.

Hypothesis Tests

How bad are the Sacramento Kings?

In the first hypothesis test, a comparison is made between the Sacramento Kings and the remainder of the teams in the NBA. Historically, the Sacramento Kings have exhibited subpar performances in the realm of basketball. The team's last appearance in the NBA playoffs dates back to the 2005-06 season. The inquiry revolves around determining whether the Kings' performance is inferior enough compared to the broader NBA to warrant their reassignment to the G-league, the NBA's developmental league. The null hypothesis (Ho) posits that the Kings' performance aligns with the NBA's standard of performance, while the alternative hypothesis (Ha) suggests that the Kings' performance is more representative of a lower-level league than of an NBA team. Consequently, a one-sided t-test will be conducted, employing point differential as the performance metric. If it is determined that the Kings' point differentials derive from a significantly distinct distribution compared to the NBA as a whole, the null hypothesis will be rejected. 

Below is a snippet of the data employed for this analysis, with each NBA team identified by a unique Team ID, enabling the separation of the Sacramento Kings from the rest of the NBA.

The test is conducted with 1,643 degrees of freedom (the number of Sacramento Kings' games in our data minus 1). The t-statistic stands at approximately -0.693, resulting in an area under the t-curve less than or equal to our t-statistic of roughly 0.244. Given that the standard rejection level (alpha) for this type of test is typically set at 0.05 or lower, there is insufficient evidence to reject the null hypothesis at any reasonable alpha level. The Kings are historically an abysmally-performing NBA team. Therefore, they appear consistently in the lower tail of the NBA team performance distribution. However, with a reasonable rejection rate, we cannot argue that the Kings do not deserve a place in the NBA.

Are European NBA players actually better free throw shooters?

The second hypothesis test aims to investigate whether NBA players born and trained in Europe exhibit a genuine advantage in free throw shooting. Within the NBA community, it is commonly believed that European-born and -trained players bring a distinctive focus on particular skills, often emphasizing fundamental aspects of the sport. In this context, the null hypothesis (Ho) posits that there is no difference in free throw accuracy between European-born players and all other (mostly American) NBA players. Conversely, the alternative hypothesis (Ha) proposes that European-born players are better at shooting free throws than other (mostly American) NBA players. 

The data used for this analysis is divided into two datasets. The first dataset contains player statistics, with each entry representing a player associated with a specific team and year, encompassing performance metrics during that period. Of particular interest are the variables FT (free throws made) and FTA (free throws attempted). The second dataset comprises biographical information for each NBA player. By merging these datasets, we can simultaneously consider free throw metrics and country of origin information. Below are the snippets of the data used for this analysis.

In the initial t-test attempt, we introduced weighting for means and variances based on the number of free throws each player attempted. This approach aimed to prevent Player 1's 100% (2 out of 2) metric from being treated equivalently to Player 2's 80% (800 out of 1000), offering more nuanced comparisons. The results were striking, yielding a t-statistic of 11.1. Under no reasonable alpha value could we retain the null hypothesis. However, this method proved overly simplistic for this intricate phenomenon. 

While we acknowledge the importance of distinguishing between players with minimal free throw attempts and more seasoned peers, a basic weighting mechanism introduced distortions. For instance, consider Steve Nash (non-European) and Dirk Nowitzki (European), both esteemed free throw shooters with percentages of 90.4% and 87.9%, respectively. Yet, Nash attempted 3,384 free throws in his career, while Nowitzki attempted 8,081. The previous weighting system disproportionately favored Nowitzki's 87.9% over Nash's 90.4%, which seems inequitable since 3,384 attempts should sufficiently represent true free throw ability. 

In response, we adopt a different approach. Instead of using weights, we opt to exclude observations (players) whose total attempted free throws fall below a certain arbitrary threshold. Unfortunately, the exact number of free throw attempts required for an accurate metric remains uncertain, and raising this threshold reduces our sample size. To explore this, we iterate through a range of cutoff values, ranging from 10 to 5000, and assess the outcomes.

This result is very confusing. As the cutoff value increases, the resulting t-statistic goes from negative to extremely positive and back again. Part of the reason for this is likely because the sample size of European players drops to an insignificant number by the time the cutoff reaches about 2,000 attempted shots. In the following plot, we can see how dramatic that reduction in sample size is.

As this plot demonstrates, there are proportionally more players that have taken fewer free throw attempts. The problem with running this test is that career free-throw percentage is an imperfect proxy for a player's inherent free throw accuracy. There is certainly a correlation between free throw ability and career longevity (and therefore free throws attempted). Omitting observations distorts the distribution of our sample in addition to reducing the sample sizes. The less-skilled players are going to be the first to be omitted regardless of the cutoff number selected. However, we have no guarantee that the proportion of lower-skilled reduction is identical between the two cohorts. European players who do not find success in the NBA are going to be more likely to pursue a career in the European basketball leagues. In effect, they would be self-selecting out of our dataset at a higher rate than similarly skilled, say, American players. Therefore, the jury is still out on this question, and we would need more relevant metrics to arrive at a decision.

Are 7-footers worse at free throws?

In the third test, an analysis was conducted to determine whether players standing at 7 feet tall, often referred to as big men, exhibit lower free throw efficiency compared to their shorter counterparts. Big men are frequently associated with poorer shooting abilities, making this comparison intriguing. The same dataset as before was utilized, including player attributes such as height. The null hypothesis (Ho) posited that 7-footers perform free throws at a level equal to that of any other NBA player, while the alternative hypothesis (Ha) proposed that 7-footers fare worse in this regard. A t-test was employed due to the unavailability of data spanning the entirety of NBA history. The resulting t-statistic, with 141 degrees of freedom, is approximately -79 which yields a p-value very close to zero. No reasonable alpha value would suffice to retain the null hypothesis. Consequently, the null hypothesis suggesting that 7-footers perform as well as other NBA players in free throws was rejected.

Assuming the player lasts for an actual career, do players drafted in the first round actually end up being better?

The NBA draft is a huge component of the NBA, and is often where the worst teams in the league in a given year have a chance to upgrade their teams with the top players entering the draft. Many resources are spent trying to pick the right player for their team, but this can be hard to predict. There are 2 rounds for 30 teams in each draft every year, thus 60 players with the potential to change the course of a franchise. Players can also be signed outside of the draft, and are thus undrafted. Number 1 overall picks can end up having a poor NBA career, while afterthoughts picked in the second round can end up being the best player on their team, and even in the NBA. We generally expect that players picked early on will have a better career, but is that really true? Using our yearly player attribute data, we can use an attribute called net rating to determine this question. The one filter we wanted to make for this question was to have a similar average year for each year a player played. Players tend to be worse their first few years in the league, and as an older player. Therefore, we created a plot that shows the average rating for each year in a player's career, shown below. Note that the population size for each cohort is labeled. For example, in our data, there are 2,463 players who completed a rookie year in the NBA and there is only 1 player who completed 22 years in the NBA.

Observing the plot, it becomes evident that net ratings are notably lower in the initial three years of a player's career. This aligns with the common understanding that young players often require a few seasons to reach their full potential. Additionally, there is a scarcity of players who manage to extend their careers beyond the 15-16 year mark. Consequently, we decided not to consider the net rating from a player's first three seasons or from their 17th season onward. 

The null hypothesis (Ho) posits that career non-first-round draft picks perform on par with career first-round draft picks, while the alternative hypothesis (Ha) suggests that first-round picks are correctly assessed and generally evolve into superior players in the long run. 

We employed a one-tailed t-test with 733 degrees of freedom, yielding a t-statistic of approximately 1.567 and a p-value of approximately 0.941. With an alpha level of 0.05, there is no basis to reject the null hypothesis, indicating that career non-first-round draft picks are statistically just as good as career first-round draft picks.