Question

select the correct answer. the branch manager of a clothing store is analyzing the average total bill of sale for his location. the national manager has communicated that the overall population mean is $45.90 with a standard deviation of $10.34. the branch manager has a sample of 400 total bills of sale for his location. by the central limit theorem, which interval can the branch manager be 95% certain that the sample mean will fall within? a. $45.85 and $45.95 b. $44.87 and $46.93 c. $45.38 and $46.42 d. $44.35 and $47.45

Explanation:

Step1: Identify the formula for confidence interval

For a large - sample (n = 400, which is large as n>30), the confidence interval for the population mean $\mu$ is given by $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $z_{\alpha/2}$ is the z - score, $\sigma$ is the population standard deviation, and n is the sample size. For a 95% confidence interval, $\alpha=1 - 0.95 = 0.05$, and $\alpha/2=0.025$. The $z -$score $z_{\alpha/2}=z_{0.025}=1.96$.

Step2: Calculate the margin of error

We are given that $\bar{x} = 45.90$, $\sigma = 10.34$, and n = 400. The margin of error $E=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=1.96\times\frac{10.34}{\sqrt{400}}$. First, $\sqrt{400}=20$. Then $\frac{10.34}{20}=0.517$. And $1.96\times0.517 = 1.01332$.

Step3: Calculate the confidence interval

The lower limit of the confidence interval is $\bar{x}-E=45.90 - 1.01332=44.8867\approx44.87$. The upper limit of the confidence interval is $\bar{x}+E=45.90 + 1.01332=46.9133\approx46.93$.

Answer:

B. $44.87$ and $46.93$