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A medical 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 16 minutes. How large a sample should she take to estimate the mean time to within 5 minutes with 96% confidence?

User Sarasgupta
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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.

User Jonathan Brown
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