Question

overtime hours worked a random sample of 12 registered nurses in a large hospital showed that they worked on average 44.8 hours per week. the standard deviation of the sample was 2.5. estimate the mean of the population with 95% confidence. assume the variable is normally distributed. round intermediate answers to at least three decimal places. round your final answers to one decimal place.

Explanation:

Step1: Determine the degrees of freedom and critical value

The degrees of freedom \(df=n - 1=12-1 = 11\). For a 95% confidence interval, \(\alpha=1 - 0.95=0.05\), and \(\frac{\alpha}{2}=0.025\). Using the t - distribution table or a calculator, the critical value \(t_{\frac{\alpha}{2}}\) with \(df = 11\) is \(t_{0.025}=2.201\).

Step2: Calculate the margin of error

The formula for the margin of error \(E=t_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}\). Given \(s = 2.5\), \(n = 12\), and \(t_{\frac{\alpha}{2}}=2.201\).

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Step3: Calculate the confidence interval

The confidence interval for the population mean \(\mu\) is \(\bar{x}-E<\mu<\bar{x} + E\). Given \(\bar{x}=44.8\).

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Answer:

\(43.2<\mu<46.4\)