Question

Use StatKey or other technology to generate a bootstrap distribution of sample means and find the standard error for that distribution. Compare the result to the standard error given by the Central Limit Theorem, using the sample standard deviation as an estimate of the population standard deviation. Mean commute time in Atlanta, in minutes, using the data in CommuteAtlanta with \(n=500\), \(\bar{x}=29.11,\) and \(s=20.72\)

Short answer

The bootstrap method and the Central Limit Theorem should both yield similar values for the standard error, showing the reliability of the Central Limit Theorem for estimating the standard error.

Step-by-step solution

Generate a Bootstrap Distribution

By using a technology tool (StatKey or other available software), one can generate a bootstrap distribution of the sample means. This is done by resampling the original sample data with replacement, thus creating many bootstrap samples. For each created sample, calculate the sample mean and make a list of these means

Calculate the Standard Error of the Bootstrap Distribution

Once a bootstrap distribution has been generated, the standard error (SE) of that distribution can be calculated. The standard error is the standard deviation of the sample means from all the bootstrap samples.

Calculate Standard Error using Central Limit Theorem (CLT)

According to the Central Limit Theorem, the standard error of the sample means can be calculated with the formula SE = \(s/\sqrt{n}\), where s is the sample standard deviation and n is the sample size.

Compare the Two Standard Errors

Finally, compare the calculated standard error from the bootstrap distribution with the standard error obtained by the Central Limit Theorem. The answers should be similar, demonstrating the Central Limit Theorem's prediction of the standard deviation of the distribution of sample means (standard error).

Key concepts

Understanding Standard Error

The concept of standard error is pivotal in statistics as it helps to measure the variability or dispersion of sample means around the true population mean. Simply put, the standard error gives us an idea of how far the average of our sample data is likely to be from the actual population average. This concept is particularly significant when dealing with larger samples because it allows researchers to estimate the accuracy of the sample mean compared to the population mean.

The formula for calculating the standard error (SE) is given as SE = \(\frac{s}{\sqrt{n}}\), where \(s\) represents the sample standard deviation, and \(n\) is the sample size. This equation shows us that as the sample size increases, the standard error decreases. That means with larger samples, we can be more confident that our sample mean is close to the population mean.

Decoding the Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle of probability theory which states that the distribution of sample means will tend to be normal (or bell-shaped), regardless of the shape of the population distribution, provided the sample size is sufficiently large. This theorem is the cornerstone that allows us to make inferences about populations from sample data.

According to the CLT, if you were to take a large number of samples from a population and calculate the mean of each sample, the sample means would form a normal distribution. This matters a lot because it enables us to apply statistical techniques that assume normality even if the population itself is not normally distributed. A crucial aspect of the CLT is that the standard deviation of the sample means (the standard error) can be estimated even if the population standard deviation is unknown, highlighting the utility of the sample standard deviation in inferential statistics.

Sample Means: The Building Blocks of Inferences

Sample means provide the foundation for many statistical inferences. When you collect data from a sample, calculating the mean of this sample offers a point estimate of the population mean. If you were to repeatedly take samples from the same population and calculate their means, these means would constitute a distribution of their own - the sampling distribution of the mean.

This sampling distribution holds valuable information. It can tell us how the mean of our sample relates to the population mean, guiding us towards conclusions about the population as a whole. When evaluating the sampling distribution, we can determine the probability of obtaining a sample mean as extreme or more extreme than our observed mean, which is a central task in hypothesis testing.

Resampling with Replacement: The Engine Behind Bootstrapping

Resampling with replacement is a non-parametric approach to statistical inference, which lies at the heart of bootstrapping. In the context of creating a bootstrap distribution, this method involves taking repeated samples from the original data set (allowing the same data point to be selected more than once) to generate numerous new samples. Each of these 'bootstrap samples' is then used to calculate a statistic of interest - often the mean.

Through the process of resampling with replacement, we create a distribution (the bootstrap distribution) of the statistic being studied. This serves as an empirical representation of the sampling distribution and allows us to estimate the standard error and construct confidence intervals for our statistic without making stringent assumptions about the population. Bootstrapping is particularly useful when dealing with small samples or when the underlying population distribution is unknown, making it a powerful tool for researchers in various fields of study.