This Wine Stinks. How sensitive are the untrained noses of students? Exercise 16.27 (page 381) gives the lowest levels of dimethyl sulfide (DMS) that 10 students could detect. You want to estimate the mean DMS odor threshold among all students, and you would be satisfied to estimate the mean to within ±0.1 with 99% confidence. The standard deviation of the odor threshold for untrained noses is known to be σ = 7 micrograms per liter of wine. How large an MIS of untrained students do you need?
To estimate the mean DMS odor threshold among all students with the desired level of precision and confidence, you can use the formula for sample size calculation for estimating a mean. The formula is:
n = (Z * σ / E)^2
where:
n = sample size
Z = Z-value for the desired confidence level (in this case, for 99% confidence level, which corresponds to a Z-value of approximately 2.576)
σ = standard deviation of the odor threshold (given as 7 micrograms per liter of wine)
E = desired margin of error (in this case, ±0.1 micrograms per liter of wine)
Plugging in these values into the formula:
n = (2.576 * 7 / 0.1)^2
n = (18.032 / 0.1)^2
n = 180.32^2
n ≈ 32593.5424
Therefore, you would need a minimum sample size of approximately 32594 untrained students in order to estimate the mean DMS odor threshold among all students to within ±0.1 with 99% confidence.