Question

construct a 95% confidence interval to estimate the population mean using the data below.
\\(\bar{x}=44\\) \\(\sigma = 12\\) \\(n = 47\\)

with 95% confidence, when \\(n = 47\\) the population mean is between a lower limit of \\(\square\\) and an upper limit of \\(\square\\). (round to two decimal places as needed.)

Explanation:

Step1: Determine the z - score

For a 95% confidence interval, the z - score $z_{\alpha/2}=1.96$.

Step2: Calculate the margin of error E

The formula for the margin of error when the population standard deviation $\sigma$ is known is $E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$. Substitute $\sigma = 12$, $n = 47$, and $z_{\alpha/2}=1.96$ into the formula.
$E=1.96\times\frac{12}{\sqrt{47}}\approx1.96\times\frac{12}{6.8557}\approx1.96\times1.7504\approx3.43$.

Step3: Calculate the lower and upper limits

The lower limit of the confidence interval is $\bar{x}-E$ and the upper limit is $\bar{x} + E$. Given $\bar{x}=44$, the lower limit is $44 - 3.43=40.57$ and the upper limit is $44+3.43 = 47.43$.

Answer:

Lower limit: $40.57$, Upper limit: $47.43$