To determine the minimum sample size required to estimate an unknown population mean [tex]\(\mu\)[/tex] with a given margin of error, confidence level, and population standard deviation, we follow these steps:
1. Identify the given information:
- Margin of Error ([tex]\(E\)[/tex]): \[tex]$135
- Confidence Level: \(95\%\)
- Population Standard Deviation (\(\sigma\)): \$[/tex]500
2. Determine the z-score associated with the 95% confidence level:
- For a 95% confidence level, the corresponding z-score is approximately [tex]\(1.96\)[/tex]. This value can be found using z-tables or statistical software.
3. Set up the formula to compute the sample size ([tex]\(n\)[/tex]):
- The formula to calculate the sample size for estimating a population mean is:
[tex]\[
n = \left(\frac{z \cdot \sigma}{E}\right)^2
\][/tex]
- Plugging in the values, we have:
[tex]\[
n = \left(\frac{1.96 \cdot 500}{135}\right)^2
\][/tex]
4. Calculate the sample size:
- First, we calculate the quantity inside the parentheses:
[tex]\[
\frac{1.96 \cdot 500}{135} \approx 7.259259259
\][/tex]
- Next, square this value to find [tex]\(n\)[/tex]:
[tex]\[
n \approx (7.259259259)^2 \approx 52.695
\][/tex]
5. Round up to the nearest whole number since sample size must be an integer:
- Even though the calculated sample size is [tex]\(52.695\)[/tex], the actual minimum sample size cannot be less than a whole number. Therefore, we round up to the next whole number.
6. Final Answer:
- The minimum sample size required is [tex]\(53\)[/tex].
Thus, the minimum sample size required to estimate an unknown population mean [tex]\(\mu\)[/tex] with a margin of error of \[tex]$135, a confidence level of 95%, and a population standard deviation of \$[/tex]500 is 53.
