Final answer:
To estimate the sample size needed to estimate the true proportion of apples exceeding the weight requirement, we can use the formula for sample size.
By plugging in the values and conducting the calculations, we find that the manager would need to sample at least 281 apples to be 90% confident of estimating the true proportion within ±4%
Step-by-step explanation:
To determine the sample size needed to estimate the true proportion of apples exceeding the weight requirement with a 90% confidence level and a margin of error of ±4%, we can use the formula for sample size:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-score (corresponding to the desired confidence level)
p = expected proportion of apples exceeding the weight requirement
E = margin of error (expressed as a proportion)
In this case, since the manager wants to use the expected proportion as an estimate, we can plug in the values:
n = (Z^2 * p * (1-p)) / E^2
n = (Z^2 * 0.70 * (1-0.70)) / 0.04^2
Let's assume the desired confidence level is 90%, which corresponds to a Z-score of approximately 1.645. Plugging in these values:
n = (1.645^2 * 0.70 * (1-0.70)) / 0.04^2
Simplifying the equation:
n ≈ 281
Therefore, the manager would need to sample at least 281 apples to be 90% confident in estimating the true proportion within ±4% using the expected proportion.