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?


a. Assume that nothing is known about the rate of e-cigarette usage among adults.

Verified step by step guidance

1. Step 1: Identify the formula for determining the required sample size for estimating a population proportion. The formula is: n = (Z^2 * p * (1 - p)) / E^2, where n is the required sample size, Z is the z-score corresponding to the desired confidence level, p is the estimated population proportion, and E is the margin of error.
2. Step 2: Determine the values for the variables in the formula. Since the confidence level is 95%, the corresponding z-score (Z) is approximately 1.96. The margin of error (E) is given as 1.5%, which should be converted to a decimal: E = 0.015. Since nothing is known about the rate of e-cigarette usage, we use the most conservative estimate for p, which is 0.5 (this maximizes the product p * (1 - p)).
3. Step 3: Substitute the values into the formula. Replace Z with 1.96, p with 0.5, and E with 0.015 in the formula: n = (1.96^2 * 0.5 * (1 - 0.5)) / 0.015^2.
4. Step 4: Simplify the numerator of the formula. Calculate 1.96^2, then multiply it by 0.5 and (1 - 0.5). This will give you the value of the numerator.
5. Step 5: Simplify the denominator of the formula. Square the margin of error (0.015^2), then divide the numerator by the denominator to find the required sample size (n). Round up to the nearest whole number, as sample size must be an integer.

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 confidence level, margin of error, and variability in the population. In this case, we need to calculate how many adults should be surveyed to estimate the current e-cigarette usage rate with a specified confidence level and margin of error.

Confidence Level

The confidence level represents the degree of certainty that the population parameter lies within the confidence interval. A 95% confidence level means that if we were to take 100 different samples and compute a confidence interval for each sample, approximately 95 of those intervals would contain the true population parameter. This concept is crucial for understanding the reliability of the survey results.

Margin of Error

The margin of error indicates the range within which the true population parameter is expected to fall, based on the sample data. A margin of error of 1.5 percentage points means that the estimate of e-cigarette usage could be 1.5 percentage points higher or lower than the sample result. This concept is essential for assessing the precision of the survey findings and ensuring that the results are statistically valid.