Question

Use Stat Key or other technology to generate a bootstrap distribution of sample differences in 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 deviations as estimates of the population standard deviations. Difference in mean commuting distance (in miles) between commuters in Atlanta and commuters in St. Louis, using \(n_{1}=500, \bar{x}_{1}=18.16,\) and \(s_{1}=13.80\) for Atlanta and \(n_{2}=500, \bar{x}_{2}=14.16,\) and \(s_{2}=10.75\) for St. Louis.

Short answer

The first step in the solution involves calculating the difference in means which results in 4 miles. After that, by inputting given numbers into the formula of standard error according to the Central Limit Theorem, one can obtain a standard error. This then needs to be compared with the bootstrap standard error, which could have been calculated if the raw data were provided.

Step-by-step solution

Bootstrapping

Bootstrapping involves drawing repeated samples from the observed data and recalculating the metric of interest. However, in this problem scenario, the raw data is not provided; hence, direct bootstrapping cannot be performed. Assuming the need to find the theoretical bootstrap standard error, it can be found by calculating the standard deviation of the difference in means.

Calculate Difference in Mean

The difference in means for two samples is given as: \( \Delta \mu = \bar{x}_{1} - \bar{x}_{2} \). Substituting the values given, we get \(\Delta \mu = 18.16 - 14.16 = 4\). So, the difference in means is 4 miles.

Calculate Standard Error - CLT

The standard error of the difference in means, according to the CLT, can be calculated using the formula: \(SE_{\Delta \mu} = \sqrt{ \frac{s^2_{1}}{n_{1}} + \frac{s^2_{2}}{n_{2}}}\). Substituting given values into the equation: \(SE_{\Delta \mu} = \sqrt{ \frac{13.80^2}{500} + \frac{10.75^2}{500}}\). After performing the calculation one can obtain the standard error according to the Central Limit Theorem.

Comparison

Finally, compare the bootstrap standard error with the SE calculated from the Central Limit Theorem.