Amedical researcher wants to investigate the amount of time it takes for patients' headache pain to be relieved after taking a new prescription painkiller. she plans to use statistical methods to estimate the mean of the population of relief times. she believes that the population is normally distributed with a standard deviation of 15 minutes. how large a sample should she take to make an estimate within 3 minutes of the mean time with 96% confidence?

We calculate the z-score, by evaluating the z value of 0.96/2 = 0.48
The z-score for 0.48 = 2.054
Next, we multiply the z-score by the standard deviation.
i.e. 2.054 x 15 = 30.81
Next, we divide by the margin of error. (from the question, we want to estimate to within 3 minutes of the mean, thus our margin of error is 3.)
i.e. 30.81 / 3 = 10.27
Finally, we square the outcome,
i.e. 10.27^2 = 105.47

Therefore, to estimate within 3 minutes of the mean with 96% confidence, the researcher should take a minimum of 106 samples.