select the correct answer. the branch manager of a clothing store is analyzing the average total bill of…

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. $45.38 and $46.42 b. $45.85 and $45.95 c. $44.87 and $46.93 d. $44.35 and $47.45

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. $45.38 and $46.42 b. $45.85 and $45.95 c. $44.87 and $46.93 d. $44.35 and $47.45

Answer

Answer:

A. $45.38 and $46.42

Explanation:

Step1: Identify z - score for 95% confidence

For 95% confidence interval, the z - score $z = 1.96$.

Step2: Calculate the standard error

The standard deviation of the population $\sigma=$10.34$ and sample size $n = 400$. The standard error $SE=\frac{\sigma}{\sqrt{n}}=\frac{10.34}{\sqrt{400}}=\frac{10.34}{20}=0.517$.

Step3: Calculate the margin of error

Margin of error $E = z\times SE=1.96\times0.517\approx1.01$.

Step4: Calculate the confidence interval

The population mean $\mu = 45.90$. The lower limit is $\mu - E=45.90 - 1.01=45.38$ and the upper limit is $\mu+E=45.90 + 1.01 = 46.42$.