Question

Number of Fouls in a Season by NBA Players The variable Fouls in the dataset NBAPlayers2015 shows the total number of fouls during the \(2014-2015\) season for all players in the \(\mathrm{NBA}\) (National Basketball Association) who played at least 24 minutes per game that season. We use this group as a sample of all NBA players in all seasons who play regularly. Use this information to test whether there is evidence that NBA players who play regularly have a mean number of fouls in a season less than 160 (or roughly 2 fouls per game).

Short answer

The answer requires calculations that aren't given in the problem, like the sample mean and standard deviation of fouls. However, the outlined steps guide how to perform a one-sample t-test given those values and test the hypotheses that the mean number of fouls per NBA player per season is less than 160.

Step-by-step solution

Define the Null and Alternative Hypotheses

The null hypothesis (\(H_0\)) is that the population mean (\(\mu\)) foul rate for regular NBA players is 160 fouls per season. The alternative hypothesis (\(H_a\)) is that the population mean foul rate is less than 160. In mathematical terms: \(H_0: \mu = 160\), \(H_a: \mu < 160\).

Choose a Significance Level

The common significance level is \(\alpha = 0.05\). This means there is a 5% chance of rejecting the null hypothesis when it's true.

Calculate Sample Mean and Standard Deviation

Use the given dataset to calculate the sample mean (\(\bar{x}\)) and sample standard deviation (\(s\)). This will help in calculating the t-score later.

Conduct T-test and Evaluate the P-value

With the sample mean, standard deviation, sample size, and assumed population mean, calculate the t-value using the formula: \(t = (\bar{x} - \mu) / (s/ \sqrt{n})\). Then, look up the associated p-value for the calculated t-value in a t-distribution table or by using statistical software.

Interpret the Results

If the p-value is less than the significance level (0.05), we reject the null hypothesis in favor of the alternative. This would mean that there's evidence that NBA players who play regularly commit less than 160 fouls per season on average. If the p-value is greater than the significance level, we fail to reject the null hypothesis, suggesting there's not enough evidence to suggest that the average number of fouls is less than 160.