Choose the one alternative that best completes the statement or answers the question.
-A random sample of 40 part-time workers has a mean annual earnings of $\$ 3120$ and a population standard deviation of $\$ 677$. Compute the $95 \%$ confidence interval for $\mu$.
A) $( \$ 2910 , \$ 3330 )$
B) $( \$ 210 , \$ 110 )$
C) $( \$ 4812 , \$ 5342 )$
D) $( \$ 1987 , \$ 2346 )$

Question

Quizplus · Accepted Answer

To compute the 95% confidence interval for the mean ($\mu$) when the population standard deviation ($\sigma$) is known, we use the formula for the confidence interval based on the Z-distribution since the sample size is large enough (n=40). The formula is $\bar{x} \pm Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $Z_{\alpha/2}$ is the Z-score corresponding to the desired confidence level (for 95% confidence, $Z_{\alpha/2} = 1.96$), $\sigma$ is the population standard deviation, and $n$ is the sample size. Plugging in the given values, we find that the interval $(\$2910, \$3330)$ is the correct answer, which corresponds to choice A.

Choose the One Alternative That Best Completes the Statement or Answers