Question

when a population is finite, the formula that determines the standard error of the mean $sigma_{overline{x}}$ needs to be adjusted. if $n$ is the size of the population and $n$ is the size of the sample (where $ngeq0.05n$), then the standard error of the mean is $sigma_{overline{x}}=\frac{sigma}{sqrt{n}}sqrt{\frac{n - n}{n - 1}}$. the expression $sqrt{\frac{n - n}{n - 1}}$ is called the finite population correction factor. use the finite population correction factor to construct the confidence interval for the population mean described below.
$c = 0.80$
$overline{x}=44.9$
$sigma = 5.1$
$n = 700$
$n = 64$
the $80%$ confidence interval for the population mean is $(square,square)$.
(round to two decimal places as needed.)

Explanation:

Step1: Find the value of \( z_{\alpha/2} \)

For a confidence level \( c = 0.80 \), the significance level \( \alpha=1 - c=1 - 0.80 = 0.20 \). Then \( \alpha/2=\frac{0.20}{2}=0.10 \).
Looking up in the standard - normal table (or using a calculator), \( z_{\alpha/2}=z_{0.10}\approx1.28 \).

Step2: Calculate the standard error of the mean \( \sigma_{\bar{x}} \)

Given \( \sigma = 5.1 \), \( n = 64 \), \( N = 700 \).
First, calculate the finite - population correction factor \( \sqrt{\frac{N - n}{N - 1}}=\sqrt{\frac{700 - 64}{700 - 1}}=\sqrt{\frac{636}{699}}\approx\sqrt{0.910}\approx0.954 \).
The formula for \( \sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\sqrt{\frac{N - n}{N - 1}} \).
Substitute \( \sigma = 5.1 \), \( n = 64 \) into \( \frac{\sigma}{\sqrt{n}} \): \( \frac{5.1}{\sqrt{64}}=\frac{5.1}{8}=0.6375 \).
Then \( \sigma_{\bar{x}}=0.6375\times0.954\approx0.61 \).

Step3: Calculate the margin of error \( E \)

The margin of error \( E = z_{\alpha/2}\sigma_{\bar{x}} \).
Substitute \( z_{\alpha/2}=1.28 \) and \( \sigma_{\bar{x}}\approx0.61 \) into the formula: \( E=1.28\times0.61 = 0.7808 \).

Step4: Calculate the confidence interval

The confidence interval for the population mean \( \mu \) is given by \( \bar{x}-E\lt\mu\lt\bar{x} + E \).
Given \( \bar{x}=44.9 \), then \( \bar{x}-E=44.9-0.7808 = 44.12 \) and \( \bar{x}+E=44.9 + 0.7808=45.68 \).

Answer:

\((44.12,45.68)\)