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 error 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 = □
(round up to the nearest integer.)

Explanation:

Step1: Recall the formula for sample size

The formula for sample size \(n\) when estimating a proportion is \(n=\frac{z^{2}\cdot p(1 - p)}{E^{2}}\). When nothing is known about the proportion \(p\), we use \(p = 0.5\) (this gives the maximum value of \(p(1 - p)\)). For a \(99\%\) confidence level, the critical value \(z\) is \(z = 2.576\) (from standard normal distribution tables), and the margin of error \(E=0.025\) (since \(2.5\%\) is \(0.025\)).

Step2: Calculate \(p(1 - p)\)

Substitute \(p = 0.5\) into \(p(1 - p)\). We get \(p(1 - p)=0.5\times(1 - 0.5)=0.25\).

Step3: Substitute values into the formula

Substitute \(z = 2.576\), \(p(1 - p)=0.25\), and \(E = 0.025\) into the formula \(n=\frac{z^{2}\cdot p(1 - p)}{E^{2}}\).

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Step4: Round up

Since \(n\) represents the number of samples and it must be an integer, we round up \(2654.3104\) to the next whole number.

Answer:

\(2655\)