Question

which of the following critical values is appropriate for a 98% confidence level where ( n = 7 ), ( sigma = 27 ), and the population appears to be normally distributed?

a. ( t_{alpha/2}=2.575 )
b. ( z_{alpha/2}=2.33 )
c. ( z_{alpha/2}=2.05 )
d. ( t_{alpha/2}=1.96 )

Explanation:

Step1: Determine the distribution type

Since the population standard deviation \(\sigma\) is known (\(\sigma = 27\)), we use the \(z\) - distribution. The \(t\) - distribution is used when \(\sigma\) is unknown. So we can eliminate options A and D.

Step2: Calculate \(\alpha\)

The confidence level \(C=0.98\). We know that \(C = 1-\alpha\), so \(\alpha=1 - 0.98=0.02\). Then \(\frac{\alpha}{2}=\frac{0.02}{2}=0.01\).

Step3: Find \(z_{\alpha/2}\)

We want to find \(z\) such that \(P(Z>z_{\alpha/2})=\frac{\alpha}{2}=0.01\). Looking up in the standard normal table (or using a calculator with a normal - distribution function), \(z_{0.01}\approx2.33\) (because \(P(Z > 2.33)\approx0.01\) and \(P(Z<2.33)\approx0.99\)).

Answer:

B. \(z_{\alpha/2}=2.33\)