Question

Find a \(95 \%\) confidence interval for the mean two ways: using StatKey or other technology and percentiles from a bootstrap distribution, and using the t-distribution and the formula for standard error. Compare the results. Mean price of a used Mustang car online, in \$1000s, using data in MustangPrice with \(\bar{x}=15.98\), \(s=11.11,\) and \(n=25\)

Short answer

Using the bootstrap method, the confidence interval is calculated using percentiles from the bootstrap distribution of the sample mean generated by resampling the data. On the other hand, using the t-distribution method, the 95% confidence interval for the given sample mean, standard deviation and size is approximately \(15.98 \pm 2.064 \times \frac{11.11}{\sqrt{25}}\). While the exact numerical answers would depend on the generated bootstrap distribution, the intervals from both methods should be fairly close most of the times, reflecting the variability in averages we would expect to see if we could draw many samples of Mustang car prices.

Step-by-step solution

Bootstrap Method

Bootstrap is a method for estimating parameters of the population distribution based on a random resampling of the sample data with replacement. To perform this method, we need to do the following steps: 1. Create a large number of pseudosamples by resampling from the original data with replacement. 2. Then for each pseudosample, compute the desired statistic (in this case, the mean). Since we don't have actual data points to generate the bootstrap distribution, we will assume the procedure is already done using some statistics software like StatKey or R. Usually, the 2.5 percentile and the 97.5 percentile of this distribution forms a 95% confidence interval.

t-Distribution Method

The t-distribution method involves applying inferential statistics theory. We need to use the following formula to calculate the 95% confidence interval: \(\bar{x} \pm t^{*} \frac{s}{\sqrt{n}}\). Where \(\bar{x}\) is the sample mean, \(s\) is the standard deviation, \(n\) is the sample size and \(t^{*}\) is the t-score from t-distribution table corresponding to desired confidence level and degrees of freedom (\(df = n - 1\)). Let's calculate it: The degrees of freedom is \(df = 25 - 1 = 24\). The t-score with 24 degrees of freedom for a two-tailed 95% confidence interval is approximately 2.064. Now we can plug these values into the formula to get the confidence interval: \(15.98 \pm 2.064 \times \frac{11.11}{\sqrt{25}}\).

Compare the results

Once we have calculated the 95% confidence interval using both methods, we compare those intervals. It's important to note that due to the random resampling in the bootstrap method, outcomes could be slightly different each time. The t-distribution method gives a more theoretical interval calculated based on the properties of t-distribution.