A simple random sample will be obtained from a normally distributed population. Find the minimum sample size needed to be 99% confident that the sample standard deviation s is within 5% of the standard deviation. Is such a sample size practical in most cases?
The minimum sample size needed is...
To find the minimum sample size needed to be 99% confident that the sample standard deviation is within 5% of the population standard deviation, you can use the formula:
n = [(Z * σ) / (E)]^2
Where:
n = required sample size
Z = Z-score corresponding to the desired level of confidence (in this case, 99% confidence level)
σ = population standard deviation
E = desired margin of error (in this case, 5% of the standard deviation)
Assuming we don't have the population standard deviation (σ), we can use a conservative estimate for σ based on previous studies or use a pilot sample to estimate σ.
However, if we assume σ = 1, the formula becomes:
n = [(Z * 1) / (0.05)]^2 = (Z / 0.05)^2
Since we want a 99% confidence level, the Z-score corresponding to this level is approximately 2.576. Substituting this value into the formula:
n = [(2.576 / 0.05)^2 = 664.467
Rounding up to the nearest whole number, the minimum sample size needed would be 665.
As for whether such a sample size is practical in most cases, it depends on various factors such as the availability of resources (time, budget, personnel) and the nature of the population being studied. In some cases, obtaining a sample size of 665 or larger may be feasible, especially with larger populations. However, in certain situations with limited resources or small populations, it might not be practical or necessary to achieve such a high sample size.