Answer: To determine the sample size needed to estimate the mean time to within 5 minutes with 96% confidence, we can use the formula for sample size calculation for a given level of confidence and margin of error. The formula is:
n = (Z^2 * σ^2) / E^2
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level (in this case, 96% confidence level, so Z = 1.96)
σ = standard deviation of the population (given as 16 minutes)
E = margin of error (desired precision, in this case, 5 minutes)
Substitute the values into the formula:
n = (1.96^2 * 16^2) / 5^2
n = (3.8416 * 256) / 25
n = 98.104 / 25
n ≈ 3.92416
Since the sample size should be a whole number (we can't have a fraction of a patient), the researcher should round up to the nearest integer.
Therefore, the researcher should take a sample size of at least 4 patients to estimate the mean relief time to within 5 minutes with 96% confidence.