1. Use a desired maximum margin of error to calculate a sample size.
2. Use a desired maximum interval width to calculate a sample size.

Other than level of confidence, there is one other thing we can control in the confidence interval: the sample size \(n\).

• We can specify the confidence level and the maximum acceptable interval width and use these to determine sample size.
• We know that \[\text{interval width} \ge 2z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\]
• Letting interval width equal \(w\), we can solve for \(n\): \[ n \ge \left(2z_{\alpha/2}\frac{\sigma}{w}\right)^2\]

Alternately, we may specify a maximum margin of error \(m\) instead: \[ n \ge \left(z_{\alpha/2}\frac{\sigma}{m}\right)^2\]

• Once we’ve done a sample size calculation, we need a whole number for \(n\).
• Since \(n \ge\) something, we will always round up.

Suppose we want a 95% confidence interval for the mean of a normally distributed population with standard deviation \(\sigma=10\). It is important for our margin of error to be no more than 2. What sample size do we need?

• As desired width/margin of error decreases, \(n\) will increase.
• As \(\sigma\) increases, \(n\) will also increase. (More population variability will necessitate a larger sample size.)
• As confidence level increases, \(n\) will also increase.

Prior experience with SAT scores in the CSU system suggests that SAT scores are well-approximated by a normal distribution with standard deviation known to be 50.

Find the sample size required for a 98% confidence interval with maximum margin of error 10.