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?

b. Use the results from the 2014 survey.

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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.

Step 2: Determine the values for the variables in the formula. From the problem, the confidence level is 95%, so the z-score (Z) is approximately 1.96. The estimated population proportion (p) is 3.7% or 0.037, and the margin of error (E) is 1.5 percentage points or 0.015.

Step 3: Substitute the values into the formula. Replace Z with 1.96, p with 0.037, and E with 0.015 in the formula: n = (1.96^2 * 0.037 * (1 - 0.037)) / 0.015^2.

Step 4: Simplify the numerator of the formula. Calculate Z^2 (1.96^2), p * (1 - p) (0.037 * (1 - 0.037)), and multiply these values together.

Step 5: Simplify the denominator of the formula. Calculate E^2 (0.015^2), and 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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed to ensure that a survey or experiment can yield reliable results. It takes into account the desired confidence level, margin of error, and the estimated proportion of the population exhibiting a certain characteristic. In this case, the calculation will help determine how many adults need to be surveyed to accurately estimate the current e-cigarette usage rate.

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 proportion of e-cigarette users. This concept is crucial for understanding the reliability of the survey results.

Margin of Error

Margin of error quantifies the uncertainty associated with survey results, indicating the range within which the true population parameter is expected to fall. A margin of error of 1.5 percentage points means that the estimated proportion of e-cigarette users could be 1.5% higher or lower than the reported figure. This concept is essential for assessing the precision of the survey findings and guiding the sample size determination.

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