Question

in a random sample of eleven people, the mean driving distance to work was 19.3 miles and the standard deviation was 5.5 miles. assume the population is normally distributed and use the t - distribution to find the margin of error and construct a 95% confidence interval for the population mean μ. interpret the results.

identify the margin of error.
miles
(round to one decimal place as needed.)

Explanation:

Step1: Determine the degrees of freedom

The degrees of freedom \(df=n - 1\), where \(n = 11\). So \(df=11-1 = 10\).

Step2: Find the critical value \(t_{\alpha/2}\)

For a 95% confidence interval, \(\alpha=1 - 0.95=0.05\), and \(\alpha/2=0.025\). Using the t - distribution table or a calculator, for \(df = 10\) and \(\alpha/2=0.025\), \(t_{\alpha/2}=2.228\).

Step3: Calculate the margin of error \(E\)

The formula for the margin of error when using the t - distribution is \(E=t_{\alpha/2}\frac{s}{\sqrt{n}}\), where \(s = 5.5\) (sample standard deviation) and \(n = 11\).

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Answer:

\(3.7\)