Final answer:
To determine the average weight of a subspecies of elephants to within 300 pounds at a 90% confidence level, with a population standard deviation of 2000 pounds, approximately 121 elephants would need to be weighed.
Step-by-step explanation:
The sample size needed for a confidence interval can be calculated using the formula for a margin of error in a mean estimation scenario.
Given that the standard deviation (σ) is 2000 pounds and the desired margin of error (E) is 300 pounds with a 90% confidence level, we can use the Z-score associated with the desired confidence level to calculate the necessary sample size (n).
To begin with, the Z-score for a 90% confidence interval is approximately 1.645 (This can also be looked up in a Z-table). The formula for margin of error E is:
E = Z * (σ/√n)
Rearranging the formula to solve for n gives us:
n = (Z² * σ2) / E²
Substituting the given values into the formula:
n = (1.645² * 2000²) / 300²
n = (2.708025 * 4000000) / 90000
n = 10832100 / 90000
n ≈ 120.356
Since we cannot have a fraction of an elephant, we round up to the next whole number, which gives us:
n = 121
Therefore, 121 elephants need to be weighed to ensure that the average weight will be accurate to within 300 pounds at a 90% confidence level.