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 price of used Mustang cars online (in \$1000s) using the data in MustangPrice with \(n=25\), \(\bar{x}=15.98,\) and \(s=11.11\)

Short answer

To determine the standard error, both the bootstrap method and Central Limit Theorem were used. After generating a bootstrap distribution of sample means, the standard error was calculated. According to Central Limit Theorem, the standard error of the distribution was computed to be 2.22. In practice, these two values should be similar if the sample is representative of the population.

Step-by-step solution

Calculate Bootstrap Standard Error

First, generate a bootstrap distribution of sample means. This would involve repeatedly resampling from the observed data, each time calculating the sample mean and storing it. Repeat this process many times, say 10,000, to generate a distribution. The standard deviation of this distribution is the bootstrap estimate of the standard error of the mean. In StatKey, this would be labeled as 'standard deviation of bootstrap distribution'.

Calculate Standard Error Using CLT

The Central Limit Theorem states that if you have a population with mean \(\mu\) and standard deviation \(\sigma\) and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will approximate a normal distribution. The standard error can be calculated as \(\sigma / \sqrt{n}\), where \(n\) is the sample size and \(\sigma\) is the standard deviation of the population, estimated here by the sample standard deviation \(s\). In this case, the standard error would be \(11.11 / \sqrt{25} = 2.22\).

Compare Bootstrap And Theoretical Standard Errors

Now, compare the bootstrap standard error obtained in Step 1 with the theoretical one calculated in Step 2 based on Central Limit Theorem (CLT). This step involves simply comparing the numerical values. If the sample is representative of the population, the two values should be close.