Question

overtime hours worked a random sample of 17 registered nurses in a large hospital showed that they worked on average 44.4 hours per week. the standard deviation of the sample was 2.2. estimate the mean of the population with 99% confidence. assume the variable is normally distributed. round intermediate answers to at least three decimal places. round your final answers to one decimal place. < μ <

Explanation:

Step1: Determine the degrees of freedom and critical value

The degrees of freedom \(df=n - 1=17-1 = 16\). For a \(99\%\) confidence interval, \(\alpha=1 - 0.99=0.01\), and \(\frac{\alpha}{2}=0.005\). Using the t - distribution table or a calculator, \(t_{\frac{\alpha}{2},df}=t_{0.005,16}=2.921\)

Step2: Calculate the margin of error

The formula for the margin of error \(E=t_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}\). Given \(s = 2.2\), \(n = 17\), and \(t_{\frac{\alpha}{2}}=2.921\)

$$ E=2.921\times\frac{2.2}{\sqrt{17}}\approx2.921\times\frac{2.2}{4.123}\approx2.921\times0.533\approx1.558 $$

Step3: Calculate the confidence interval

The confidence interval for the population mean \(\mu\) is \(\bar{x}-E<\mu<\bar{x}+E\). Given \(\bar{x}=44.4\)

$$ 44.4-1.558<\mu<44.4 + 1.558 $$

$$ 42.8<\mu<46.0 $$

Answer:

\(42.8<\mu<46.0\)