Question

a sample of 30 commuters in the area of a certain city yielded the accompanying commute times, in minutes. find a 90% confidence interval for the mean commute time of all commuters in the area of the city. the sample mean was 28.87 minutes and the sample standard deviation was 9.44 minutes. the 90% confidence interval is from minute(s) to minute(s). (round to one decimal place as needed.)

Explanation:

Step1: Determine the critical value

Since the sample size \(n = 30\) (small - sample, \(n<30\) is the traditional cut - off for small - sample, but when population standard deviation is unknown, we use the \(t\) - distribution. Here \(n = 30\), degrees of freedom \(df=n - 1=30 - 1 = 29\). For a \(90\%\) confidence interval, the significance level \(\alpha=1 - 0.90 = 0.10\), and \(\frac{\alpha}{2}=0.05\). Using a \(t\) - table or a calculator, \(t_{\frac{\alpha}{2},df}=t_{0.05,29}\approx1.699\).

Step2: Calculate the margin of error

The formula for the margin of error \(E\) when the population standard deviation \(\sigma\) is unknown is \(E = t_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}\), where \(s\) is the sample standard deviation and \(n\) is the sample size. Given \(s = 9.44\), \(n = 30\), and \(t_{\frac{\alpha}{2}}=1.699\).

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Step3: Calculate the confidence interval

The formula for the confidence interval for the population mean \(\mu\) (when \(\sigma\) is unknown) is \(\bar{x}-E<\mu <\bar{x}+E\), where \(\bar{x}\) is the sample mean. Given \(\bar{x}=28.87\) and \(E\approx2.93\).

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Answer:

The \(90\%\) confidence interval is from \(25.9\) minute(s) to \(31.8\) minute(s).