Find the minimum sample size needed to be 99% confident that the sample’s standard deviation is within 10% of the population standard deviation.

To find the minimum sample size needed to be 99% confident that the sample's standard deviation is within 10% of the population standard deviation, we can use the formula:

n = \left( \dfrac{{2.576 \cdot \sigma}}{{E}} \right)^2

where:
- n = sample size
- \sigma = population standard deviation
- E = maximum error (10% of the population standard deviation)
- 2.576 is the z-score for 99% confidence interval

Plugging in the values: E = 0.10 , and the z-score of 99% confidence interval is 2.576, we get:

n=\left(\dfrac{{2.576 \cdot\sigma}}{{0.10 \cdot\sigma}}\right)^2=\left(\dfrac{{2.576}}{{0.10}}\right)^2=663.5776

Therefore, the minimum sample size needed is 664.

\boxed{n=664}