E-Cigarettes A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with 3.7% of them being regular users of e-cigarettes. Because e-cigarette use is relatively new, there is a need to obtain today’s usage rate. How many adults must be surveyed now if we want a confidence level of 95% and a margin of error of 1.5 percentage points?


c. Does the use of the result from the 2014 survey have much of an effect on the sample size?

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with determining the sample size required to estimate the proportion of adults who are regular users of e-cigarettes with a 95% confidence level and a margin of error of 1.5 percentage points. Additionally, we need to consider whether the 2014 survey result (3.7%) affects the sample size calculation.
2. Step 2: Recall the formula for sample size calculation for proportions: \( n = \frac{{Z^2 \cdot p \cdot (1-p)}}{{E^2}} \), where \( Z \) is the z-score corresponding to the confidence level, \( p \) is the estimated proportion, and \( E \) is the margin of error.
3. Step 3: Identify the values needed for the formula. For a 95% confidence level, the z-score \( Z \) is approximately 1.96. The margin of error \( E \) is given as 1.5 percentage points, which should be converted to a decimal (0.015). The proportion \( p \) can be taken from the 2014 survey result (3.7%, or 0.037), but we should also consider the worst-case scenario where \( p \) is 0.5 (maximum variability).
4. Step 4: Calculate the sample size using both \( p = 0.037 \) and \( p = 0.5 \). Substitute the values into the formula \( n = \frac{{Z^2 \cdot p \cdot (1-p)}}{{E^2}} \). Perform the calculation for each case to see how the choice of \( p \) affects the sample size.
5. Step 5: Compare the results. Discuss whether using the 2014 survey proportion (3.7%) significantly reduces the sample size compared to using \( p = 0.5 \). This will help determine if the prior survey data has a substantial effect on the sample size calculation.

Key Concepts

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed in a study to achieve a desired level of precision. It takes into account the population size, the expected proportion of the attribute of interest, the desired confidence level, and the margin of error. In this case, the calculation will help determine how many adults need to be surveyed to estimate the current e-cigarette usage rate accurately.

Confidence Level

The confidence level represents the degree of certainty that the true population parameter lies within the confidence interval calculated from the sample data. A 95% confidence level means that if the same survey were conducted multiple times, approximately 95% of the calculated intervals would contain the true population proportion. This concept is crucial for understanding the reliability of the survey results.

Margin of Error

The margin of error quantifies the amount of random sampling error in a survey's results. It indicates the range within which the true population parameter is expected to fall, based on the sample data. A smaller margin of error requires a larger sample size, which is important for ensuring that the survey results are precise and reflect the current usage rate of e-cigarettes accurately.