To determine how many donuts should be sampled to estimate the mean weight of cream filling within 0.2 grams of the true mean with 99% confidence, we use the formula for sample size nnn for a population mean:
n=(z×σE)2n=\left(\frac{z\times \sigma}{E}\right)^2n=(Ez×σ)2
where:
- zzz is the z-score corresponding to the desired confidence level (for 99% confidence, z≈2.576z\approx 2.576z≈2.576)
- σ\sigma σ is the population standard deviation (estimated as 1 gram)
- EEE is the desired margin of error (0.2 grams)
Plugging in the values:
n=(2.576×10.2)2=(2.5760.2)2=(12.88)2=166.05n=\left(\frac{2.576\times 1}{0.2}\right)^2=\left(\frac{2.576}{0.2}\right)^2=(12.88)^2=166.05n=(0.22.576×1)2=(0.22.576)2=(12.88)2=166.05
Since sample size must be an integer and large enough to meet the margin of error, round up:
n=167n=167n=167
Therefore, the manufacturer should sample at least 167 donuts to be 99% confident that the sample mean weight of cream filling is within 0.2 grams of the true mean
. If the confidence level or margin of error changes, the sample size would need to be recalculated accordingly.
