Question

a new york times article reported that a survey conducted in 2014 included 36,000 adults, with 3.65% of them being regular users of e - cigarettes. because e - cigarette use is relatively new, there is a need to obtain todays usage rate. how many adults must be surveyed now if a confidence level of 99% and a margin of err of 2.5 percentage points are wanted? complete parts (a) through (c) below.
a. assume that nothing is known about the rate of e - cigarette usage among adults. ( n = 2655 ) (round up to the nearest integer.)
b. use the results from the 2014 survey. ( n=square ) (round up to the nearest integer.)

Explanation:

Step1: Recall the sample size formula for proportion

The formula for sample size \(n\) when estimating a proportion is \(n=\frac{z^{2}\cdot p(1 - p)}{E^{2}}\). For a \(99\%\) confidence level, the critical value \(z\) (from standard normal distribution tables) is \(z = 2.576\). When nothing is known about the proportion \(p\), we use \(p=0.5\) (this gives the maximum value of \(p(1 - p)\)). The margin of error \(E = 0.025\).

Step2: Substitute values into the formula

Substitute \(z = 2.576\), \(p = 0.5\), and \(E=0.025\) into the formula:

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Step3: Round up the value

Since we cannot have a fraction of a sample, and we need to meet the margin of error requirement, we round up \(2654.3104\) to \(n = 2655\).

Answer:

\(n = 2655\)