Question

which critical value is appropriate for a 99% confidence level where n = 17, σ is unknown, and the population appears to be normally distributed?
a. t_{α/2}=2.921
b. t_{α/2}=2.898
c. z_{α/2}=2.583
d. z_{α/2}=2.567

Explanation:

Step1: Determine the distribution

Since \(\sigma\) is unknown, we use the \(t -\)distribution. The formula for the confidence interval is \(\bar{x}\pm t_{\alpha/2}\frac{s}{\sqrt{n}}\).

Step2: Calculate the degrees of freedom

The degrees of freedom \(df=n - 1\). Given \(n = 17\), then \(df=17-1 = 16\).

Step3: Find \(\alpha\)

For a \(99\%\) confidence level, \(\alpha=1 - 0.99=0.01\), and \(\alpha/2=0.005\).

Step4: Look up the \(t -\)value

Using a \(t -\)table or a statistical software, for \(df = 16\) and \(\alpha/2=0.005\), the \(t -\)value \(t_{\alpha/2}\) is \(2.921\).

Answer:

A. \(t_{\alpha/2}=2.921\)