If the true population standard deviation was known to be 3.0, then approximately what minimum sample size would you need in a simple random sample if you want a 95% confidence interval for the mean to have a margin of error of .5 or less?
To determine the minimum sample size required for a given margin of error and confidence level, we can use the formula:
n = ((Z * σ) / E)^2
where:
n is the minimum sample size needed
Z is the z-score corresponding to the desired confidence level
σ is the population standard deviation
E is the desired margin of error
In this case, we want a 95% confidence level, which corresponds to a z-score of approximately 1.96 (you can find this value in a standard normal distribution table or use a calculator). The population standard deviation (σ) is given as 3.0, and the desired margin of error (E) is 0.5.
Substituting these values into the formula, we can calculate the minimum sample size:
n = ((1.96 * 3.0) / 0.5)^2
n = 11.76^2
n ≈ 138.34
Therefore, you would need a minimum sample size of approximately 139 in a simple random sample to obtain a 95% confidence interval for the mean with a margin of error of 0.5 or less.