Step 1 :Given that the confidence level is 95%, the corresponding z-score is approximately 1.96. The desired margin of error is 1.5 percentage points, or 0.015.
Step 2 :For part a, we don't know the proportion of e-cigarette users, so we'll use the worst-case scenario, which is p = 0.5. This maximizes the product p*(1-p). We can then use the formula for calculating the sample size for a proportion: \(n = \frac{Z^2 * p * (1-p)}{E^2}\). Substituting the given values, we find that the required sample size is \(n = \frac{(1.96)^2 * 0.5 * (1-0.5)}{(0.015)^2}\), which rounds up to 4269.
Step 3 :For part b, we'll use the proportion from the 2014 survey, which is 0.0368. Substituting this value into the formula, we find that the required sample size is \(n = \frac{(1.96)^2 * 0.0368 * (1-0.0368)}{(0.015)^2}\), which rounds up to 606.
Step 4 :For part c, we compare the sample sizes calculated in parts a and b. We see that using the result from the 2014 survey dramatically reduces the sample size.
Step 5 :Final Answer: For part a, the required sample size is \(\boxed{4269}\). For part b, the required sample size is \(\boxed{606}\). For part c, the answer is \(\boxed{\text{D. Yes, using the result from the 2014 survey dramatically reduces the sample size.}}\)