Final answer:
To achieve a 95% confidence level with a margin of error of 2%, at least 2401 adults must be surveyed, using the conservative estimate of 50% for e-cigarette usage rate among the unknown current rate.
Step-by-step explanation:
The student has asked how many adults must be surveyed now if a confidence level of 95% and a margin of error of 2 percentage points are desired to estimate the current rate of e-cigarette usage among adults. To calculate this, we'll utilize the formula for determining sample size for a proportion, which is given by:
N = (Z^2 * p * (1 - p)) / E^2
where:
- N is the sample size,
- Z is the Z-score associated with the desired confidence level,
- p is the estimated proportion of the population attribute,
- E is the desired margin of error.
Since we desire a 95% confidence level, the Z-score (Z) would be approximately 1.96. Because we do not know the current rate of e-cigarette usage (p), we can use the most conservative estimate of 0.5 (50%) to ensure the largest sample size, which accommodates the highest variability. The margin of error (E) desired is 0.02 (2%).
Substituting the values we have:
N = (1.96^2 * 0.5 * (1 - 0.5)) / 0.02^2 = (3.8416 * 0.25) / 0.0004 = 0.9604 / 0.0004 = 2401
So, to achieve a 95% confidence level with a margin of error of 2%, it would be necessary to survey at least 2401 adults. Note that we should round up to ensure that the sample size is sufficient, so the final answer would be 2402 adults.