Question

banking fees have received much attention during the recent economic recession as banks look for ways to recover from the crisis. a sample of 42 customers paid an average fee of $12.91 per month on their interest - bearing checking accounts. assume the population standard deviation is $1.83. complete parts a and b below.
a. construct a 95% confidence interval to estimate the average fee for the population.
the 95% confidence interval has a lower limit of $ and an upper limit of $ (round to the nearest cent as needed.)

Explanation:

Step1: Identify the formula

For a 95% confidence interval when population standard - deviation $\sigma$ is known, the formula is $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$, 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.

Step2: Determine the values

We are given that $\bar{x} = 12.91$, $\sigma=1.83$, $n = 42$. For a 95% confidence interval, $\alpha=1 - 0.95=0.05$, so $\alpha/2=0.025$. The $z$ - score $z_{\alpha/2}=z_{0.025}=1.96$.

Step3: Calculate the margin of error

The margin of error $E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=1.96\times\frac{1.83}{\sqrt{42}}$.
First, calculate $\sqrt{42}\approx6.4807$. Then, $\frac{1.83}{\sqrt{42}}\approx\frac{1.83}{6.4807}\approx0.2824$. And $E = 1.96\times0.2824\approx0.5535$.

Step4: Calculate the confidence interval

The lower limit is $\bar{x}-E=12.91 - 0.5535=12.3565\approx12.36$.
The upper limit is $\bar{x}+E=12.91 + 0.5535=13.4635\approx13.46$.

Answer:

The 95% confidence interval has a lower limit of $\$12.36$ and an upper limit of $\$13.46$.