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 body temperature, in \({ }^{\circ} \mathrm{F}\), using the data in BodyTemp50 with \(n=50, \bar{x}=98.26,\) and \(s=\) 0.765

Short answer

The answer is the comparison between the standard errors calculated using bootstrap method and Central Limit Theorem. These exact numbers aren't given in the question and so would depend on the result of your calculations using the given data set (BodyTemp50) and methods described.

Step-by-step solution

Generate bootstrap samples

A bootstrap sample is created by randomly selecting n observations from the dataset - BodyTemp50, and the process is repeated many times (say 10,000 times}. For each bootstrap sample compute the mean and create a bootstrap distribution of sample means.

Calculate standard error from bootstrap distribution

The standard error for the bootstrap distribution is the standard deviation of the bootstrap sample means. Calculate it.

Calculate standard error using Central Limit Theorem (CLT)

According to CLT, standard error = sample standard deviation / sqrt(n). So calculate standard error as s/ sqrt(n). Given, n=50 and s=0.765.

Compare the two estimates

Finally, compare the standard error obtained from the bootstrap method with the one calculated using CLT. This will help in understanding the performance of bootstrap method as compared to the theoretical approach.

Key concepts

Standard Error

When it comes to data analysis and statistics, the standard error (SE) is a critical concept that quantifies the variability of a sampling distribution. In simpler terms, it measures how far the sample mean (average) of your data is likely to be from the true population mean. This is vital for making inferences about the larger population from which the sample was drawn.

The standard error is calculated using the formula: \[ SE = \frac{s}{\sqrt{n}} \] where 's' represents the sample standard deviation, and 'n' is the size of the sample. Smaller standard errors indicate that our sample mean is closer to the true population mean, leading to more precise estimates. Understanding and calculating the SE is crucial for determining the accuracy and reliability of statistical estimates and is especially relevant when working with smaller samples where individual variability can have a larger impact on the result.

Central Limit Theorem

One of the most powerful concepts in statistics is the Central Limit Theorem (CLT). It states that, given a sufficiently large sample size, the distribution of the sample means will be approximately normally distributed, regardless of the shape of the population distribution. This key tenet allows statisticians to make reliable inferences about population parameters.

The CLT is pivotal for understanding why we can use the normal distribution as a basis for creating confidence intervals and hypothesis tests even when dealing with non-normal populations. In practice, a sample size of 30 or more is often considered sufficient for the CLT to hold true. Additionally, the CLT provides the foundation for calculating the standard error of the mean, reinforcing its importance in the assessment of the variability within sampling distributions.

Sampling Distribution

A sampling distribution isn't about a single sample; instead, it's a probability distribution of a statistic that is created from numerous samples of the same size randomly taken from the same population. Imagine shooting an arrow at a target many times; the pattern of hits represents a sampling distribution of your archery skills.

When you calculate the mean of each sample and then plot those means, the emerging pattern is the sampling distribution of the sample mean. This distribution helps us to predict the frequency of obtaining certain sample means and is key for estimating population parameters. It's this concept that allows us to understand—and quantify—the variability we'd expect to see when we take multiple samples from the same population.

Bootstrap Samples

Bootstrap samples bring a clever twist to the concept of sampling. By repeatedly resampling from an original sample with replacement, we can create numerous 'bootstrap' samples. Each of these samples is the same size as the original and is treated as if it was drawn directly from the population.

Bootstrap methods give us a way to understand the variability of our estimates without having to make assumptions about the original population's distribution. After calculating statistics for many bootstrap samples—like the mean—we can form a bootstrap distribution. This distribution provides an empirical estimate of the standard error and allows us to construct confidence intervals, offering insight into how precise our sample-based estimates are likely to be. The bootstrap approach is a powerful non-parametric tool, often used when theoretical distributions are difficult to apply or unknown.