Final answer:
To calculate the sample size needed for a 99% confidence interval with a margin of error of 5.4 and a population standard deviation of 20.5, we use the margin of error formula, which results in a necessary sample size of at least 98 individuals when rounded up.
Step-by-step explanation:
To determine how large a sample must be drawn so that a 99% confidence interval for the population mean (μ) will have a margin of error equal to 5.4 when the population standard deviation (σ) is 20.5, we can use the formula for the margin of error (EBM):
EBM = Z * (σ / √n)
Where:
- Z is the Z-value corresponding to the desired confidence level (for a 99% confidence interval, Z is approximately 2.576)
- σ is the population standard deviation, in this case, 20.5
- n is the sample size
- EBM is the desired margin of error, in this case, 5.4
Rearranging the formula to solve for n gives us:
n = (Z × σ / EBM)^2
Plugging in the values we get:
n = (2.576 × 20.5 / 5.4)^2 ≈ 97.456
Since we cannot have a fraction of a person, we would round up to the nearest whole number. Therefore, a sample of at least 98 individuals needs to be drawn for a 99% confidence interval to have a margin of error of 5.4.