Final answer:
To find the minimum sample size required to estimate an unknown population mean, use the formula n = (Z * σ / E)², where Z is the z-value corresponding to the desired confidence level, σ is the population standard deviation, and E is the margin of error. Substitute the given values and calculate the required sample size. In this case, the minimum sample size is approximately 1609.
Step-by-step explanation:
To find the minimum sample size required to estimate an unknown population mean, we can use the formula:
n = (Z * σ / E)²
where:
- n is the sample size
- Z is the z-value corresponding to the desired confidence level
- σ is the population standard deviation
- E is the margin of error
In this case, the margin of error is 143.0 and the confidence level is 90.0 percent.
We are also given that σ = 574.8. Now we can substitute these values into the formula:
n = (Z * σ / E)² = (Z * 574.8 / 143.0)²
To calculate the required sample size, we need to determine the z-value associated with a 90.0 percent confidence level. Using a standard normal distribution table or calculator, we find that the z-value is approximately 1.645.
Substituting this value into the formula:
n = (1.645 * 574.8 / 143.0)²
n ≈ 1608.67
The minimum sample size required to estimate the unknown population mean with a margin of error of 143.0 and a confidence level of 90.0 percent is approximately 1609.