Question

Standard Deviation of NHL Penalty Minutes Exercise 3.102 describes data on the number of penalty minutes for Ottawa Senators NHL players. The sample has a fairly large standard deviation, \(s=27.3\) minutes. Use StatKey or other technology to create a bootstrap distribution, estimate the standard error, and give a \(95 \%\) confidence interval for the standard deviation of penalty minutes for NHL players. Assume that the data in OttawaSenators can be viewed as a reasonable sample of all NHL players.

Short answer

The actual solution will rely on the data generated from the bootstrap method and software outputs, which are not specified in this task. However, the outlined steps provide a general approach to solving such problems. If specific data in OttawaSenators was provided and bootstrapping was carried out, one could provide the estimated standard error and the 95% confidence interval of the population standard deviation.

Step-by-step solution

Understand the question

The task provides the sample standard deviation (s=27.3) of penalty minutes for Ottawa Senators NHL players. However, the aim is to estimate the standard deviation of penalty minutes for all NHL players. The assumption is that data about the Ottawa Senators can be used to make a prediction about the entire NHL player population. Therefore, using the bootstrap distribution, the standard error needs to be estimated, and a 95% confidence interval for the population standard deviation needs to be determined.

Create a bootstrap distribution

To create a bootstrap distribution, the process of resampling is repeated a large number of times (typically 1000 or more) with replacement from the original sample. These samples are the exact same size as the original sample. The standard deviation is calculated for each resample and a histogram is constructed with each of these calculated standard deviations. This histogram is the bootstrap distribution of the standard deviation.

Estimate the standard error

The standard error in this context would be the standard deviation of our bootstrap distribution. It gives an estimate of the variability we might expect if we did our bootstrap sampling over again many times. This could be calculated using appropriate statistical software.

Create a 95% confidence interval

The 95% confidence interval is determined from the bootstrap distribution. In general, this involves finding the 2.5th percentile and 97.5th percentile of the bootstrap distribution (because of symmetric data), which give the lower and upper bounds of the 95% confidence interval for the standard deviation of penalty minutes for all NHL players.

Key concepts

Standard Deviation

When studying data, such as penalty minutes for NHL players, one vital statistic is the 'standard deviation', which measures the spread of the data around the mean. It gives you a sense of how far, on average, each measurement is from the mean — the higher the standard deviation, the more varied the data. In our exercise, the sample standard deviation is given as 27.3 minutes, indicating that the penalty minutes are spread out quite a bit from the average.

Understanding standard deviation is crucial because it plays a key role in forming other statistical measures, such as confidence intervals and assessing the standard error of parameter estimates, like the mean or a proportion. When we draw inferences about a larger population from a sample, the standard deviation of our sample clues us into the variability we might expect in the wider population.

Resampling

Resampling is a non-parametric approach to statistical inference. It involves repeatedly drawing samples from an observed data set and assessing a specified statistic for each sample. In the exercise, we're particularly using a method called 'bootstrap resampling' to estimate the variability of the standard deviation. This entails creating 'bootstrap samples' by randomly selecting individuals from the original data with replacement, meaning the same individual can be chosen more than once.

By performing this resampling process many times (often thousands), we create a bootstrap distribution that reflects the sampling variability. This distribution gives us insight into how our estimate would behave if we were to repeat our sampling process over and over again under the same conditions. It's a powerful tool that allows us to make statistical inferences without relying on traditional assumptions, such as normality.

Statistical Significance

The concept of 'statistical significance' is essential for hypothesis testing. It helps quantify the probability that the observed effect or difference in a study could have occurred just by chance, given that there actually is no effect or difference. The term 'p-value' is often used to express this probability.

If the p-value is lower than a predetermined threshold (usually 0.05 or 5%), we conclude that our results are statistically significant, meaning unlikely to have happened due to random variation alone. In the context of resampling, if the bootstrap distribution suggests the true parameter is reliably far away from a null hypothesized value, we can infer statistical significance regarding our estimate. However, in our case study, we are more focused on interval estimation than hypothesis testing per se.

Confidence Interval Estimation

A 'confidence interval' offers a range of values that is likely to contain a population parameter, such as the standard deviation, with a certain level of confidence (typically 95%). The interval is constructed from the bootstrap distribution, which embodies the variability in our estimate due to sampling.

To obtain a 95% confidence interval, we look for the range that captures the central 95% of bootstrap estimates. This means finding the 2.5th and 97.5th percentiles of the bootstrap distribution, as our exercise suggests. These percentiles provide the boundaries where we can say with 95% confidence that the true population standard deviation lies within this range. It is a robust method for interval estimation that does not need the assumption of normality, making it an incredibly useful technique in many practical scenarios.