Question

in a random sample of five people, the mean driving distance to work was 19.1 miles and the standard deviation was 7.5 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. (round to one decimal place as needed.)

Explanation:

Step1: Determine the degrees of freedom

The degrees of freedom \(df=n - 1\), where \(n = 5\). So \(df=5-1 = 4\).

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 \(df = 4\) and \(\alpha/2=0.025\), \(t_{\alpha/2}=2.776\).

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

The formula for the margin of error when using the t - distribution is \(E=t_{\alpha/2}\times\frac{s}{\sqrt{n}}\), where \(s = 7.5\) (sample standard deviation) and \(n = 5\).

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Answer:

\(9.3\)