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Chapter 6: Problem 103

In Exercises 6.103 and 6.104 , 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 distance of a commute for a worker in Atlanta, using data in Commute Atlanta with \(\bar{x}=\) 18.156 miles, \(s=13.798,\) and \(n=500\)

Short Answer

Expert verified
The exact values of the confidence intervals would depend on the exact computations, but as a rough approximation, the results from both methods should be similar and within an acceptable range of one another. Any large discrepancies between the results of the two methods would imply that an error might have occurred during calculations.

Step by step solution

01

Bootstrap Confidence Interval

Use a technology such as StatKey. Input the given data and set the percentile bootstrap confidence interval to 95%. The software will generate a range for the confidence interval.
02

T-Distribution Confidence Interval

Utilize the formula \(\bar{x} \pm t* (\frac{s}{\sqrt{n}})\), where \(\bar{x}\) = 18.156 miles, s = 13.798, and n = 500. The symbol 't*' represents the t-score, which, for a 95% confidence level and degrees of freedom (n-1 = 499), can be gathered from the t-distribution table or a calculator.
03

Calculate Standard Error

Calculate the standard error using the formula \(\frac{s}{\sqrt{n}}\) that will be used in the confidence interval formula.
04

Compute Confidence Interval

Substitute the obtained values into the formula to calculate the low and high bounds of the confidence interval.
05

Compare the Results

Compare the confidence intervals produced from both methods. If they are similar, it validates the accuracy of both methods. If not, there might be an error in one method or the difference might be due to the difference between exact percentile based confidence interval and approximation using t-distribution.

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Most popular questions from this chapter

Systolic Blood Pressure and Survival Status Use technology and the ICUAdmissions dataset to find a \(95 \%\) confidence interval for the difference in systolic blood pressure (Systolic) upon admission to the Intensive Care Unit at the hospital based on survival of the patient (Status with 0 indicating the patient lived and 1 indicating the patient died.) Interpret the answer in context. Is "No difference" between those who lived and died a plausible option for the difference in mean systolic blood pressure? Which group had higher systolic blood pressures on arrival?

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