a simple random sample will be obtained from a normally distributed population. find the minimum sample size needed to be 95% confident that the sample standard deviation (s) is within 10% of sigma. please help!
To determine the minimum sample size required for a 95% confidence level that the sample standard deviation (s) is within 10% of the population standard deviation (σ), you can use the formula:
n = ((Z * σ) / (0.1 * σ))^2
Where:
n = sample size
Z = Z-score for the desired confidence level (for 95% confidence level, Z ≈ 1.96)
σ = population standard deviation
To find the Z-score for a 95% confidence level, you can use a Z-table or a statistical calculator. The Z-score for a 95% confidence level is approximately 1.96.
Once you have the Z-score, substitute the values into the formula:
n = ((1.96 * σ) / (0.1 * σ))^2
The population standard deviation cancels out, simplifying the formula to:
n = (1.96 / 0.1)^2
Simplify further:
n = 19.6^2
Calculate:
n ≈ 384
Hence, the minimum sample size needed to be 95% confident that the sample standard deviation is within 10% of the population standard deviation is approximately 384.