Question

Average Penalty Minutes in the NHL In Exercise 3.102 on page \(241,\) we construct a \(95 \%\) confidence interval for mean penalty minutes given to NHL players in a season using data from players on the Ottawa Senators as our sample. Some percentiles from a bootstrap distribution of 5000 sample means are shown in Table 3.13. Use this information to find and interpret a \(98 \%\) confidence interval for the mean penalty minutes of NHL players. Assume that the players on this team are a reasonable sample from the population of all players.

Short answer

The 98% confidence interval for mean penalty minutes given to NHL players is (\(x_{1}, x_{99}\)). This means there is a 98% probability that the true mean penalty minutes lies within this interval, provided the sample is representative of the entire population of NHL players.

Step-by-step solution

Understanding the Percentiles

To find the 98% confidence interval, we first need to understand the percentiles given from the bootstrap distribution. Since we do not have actual percentile values in the question, for illustrative purposes, let's assume that we have the 1st percentile at \(x_{1}\) and the 99th percentile at \(x_{99}\). These two values cut off the bottom 1% and top 1% of the distribution respectively. Hence, the interval between these two percentiles will give a 98% confidence interval.

Construct the Confidence Interval

The 98% confidence interval would therefore be the range between the 1st percentile (\(x_{1}\)) and the 99th percentile (\(x_{99}\)). So the 98% confidence interval is given by (\(x_{1}, x_{99}\)).

Interpretation of Confidence Interval

This 98% confidence interval (\(x_{1}, x_{99}\)) implies that we can be 98% confident that the true mean penalty minutes of all NHL players falls within this interval. Assuming that the sample data is a good representation, these results apply to the entire population of NHL players.

Key concepts

Bootstrap Distribution

Understanding the concept of a bootstrap distribution is pivotal when dealing with statistical estimates. It is a modern resampling technique used to assess the uncertainty of a statistic, like a mean or median. A bootstrap distribution is created by repeatedly resampling with replacement from the original sample data and recalculating the statistic for each resample.
For clarity, imagine you have a bag of mixed colored marbles. You randomly pick a marble, note its color, and put it back in the bag. If you repeat this process many times, you start to form a distribution of colors that estimates the true color composition of all the marbles in the bag. This is analogous to bootstrapping, where each resample is like a new 'pick' from your data 'bag'.
The bootstrap distribution provides a way to approximate the sampling distribution of a statistic, which allows us to estimate the variability of the statistic and construct confidence intervals without making strong assumptions about the shape of the population distribution. This is incredibly useful when applying to real-world data that may not follow a neat mathematical distribution.

Percentiles

Percentiles are a form of descriptive statistics that summarize the relative standing of an observation within a dataset. They tell us the value below which a given percentage of observations in a group fall. For instance, the 25th percentile, also known as the first quartile, is the value that cuts off the first 25% of the data. Similarly, the median is the 50th percentile, and the 75th percentile is the third quartile.
When it comes to interpreting bootstrap distributions, percentiles play a pivotal role. If we take our bootstrap samples and calculate a percentile, we get a boundary that tells us where a certain proportion of bootstrap estimates lie. Under the right conditions, these can be used to construct confidence intervals. In our NHL example, the 1st and 99th percentiles from the bootstrap distribution delineate the boundaries of the 98% confidence interval.

NHL Statistics

NHL statistics encompass a wide range of data, including player performance metrics such as goals, assists, and, relevant to our exercise, penalty minutes. Penalizations are a substantial part of hockey's strategic gameplay, and tracking penalty minutes can offer insights into player behavior and team discipline.
Using NHL statistics for inferential statistics can be fascinating. For example, when we apply statistical sampling to analyze NHL penalties, we might be trying to generalize from a sample (like players on a specific team) to the entire NHL player population. However, it is essential to note that for successful generalization, the sample must be representative of the population. In our exercise, it is assumed that the Ottawa Senators players constitute a reasonable sample, which allows the findings from the bootstrap method to be generalized to all NHL players.

Statistical Sampling

Statistical sampling is the process of selecting a subset of individuals from a population to estimate characteristics about the entire group. Think of it as taking a sneak peek into a much larger pool without examining everyone in it. This method is crucial in many fields, including science, economics, and, as in our example, sports analytics.
In the real world, examining every member of a population is often impractical or impossible. Hence, sampling is a powerful tool. However, the way a sample is selected can profoundly affect the results. For trustworthy conclusions, the sample needs to be random and unbiased. In the context of our problem, the assumption is that the Ottawa Senators are representative of the larger NHL player population, which is key to applying the bootstrap distribution method to construct a confidence interval for the mean penalty minutes of NHL players.