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 = 2655
(round up to the nearest integer.)
b. use the results from the 2014 survey.
n =
(round up to the nearest integer.)

Explanation:

Step1: Recall the sample size formula for proportion

When nothing is known about the proportion \(p\), we use \(p = 0.5\) (this gives the maximum value of \(p(1 - p)\)). The formula for sample size \(n\) is \(n=\frac{z^{2}_{\alpha/2}p(1 - p)}{E^{2}}\). For a \(99\%\) confidence level, \(\alpha=1 - 0.99 = 0.01\), and \(\alpha/2=0.005\). From the standard normal table, \(z_{\alpha/2}=z_{0.005} = 2.576\), and \(E = 0.025\) (since the margin of error is \(2.5\) percentage points, \(E=0.025\)). Substitute \(p = 0.5\) into the formula:

$$n=\frac{(2.576)^{2}\times0.5\times(1 - 0.5)}{(0.025)^{2}}$$

Step2: Calculate the numerator and denominator

First, calculate \((2.576)^{2}=6.635776\), \(0.5\times(1 - 0.5)=0.25\), and \((0.025)^{2}=0.000625\). Then the numerator is \(6.635776\times0.25 = 1.658944\).

$$n=\frac{1.658944}{0.000625}$$

Step3: Compute the value of \(n\)

\(n=\frac{1.658944}{0.000625}=2654.3104\). Since we round up to the nearest integer, \(n = 2655\).

Step4: For part (b)

We know from the 2014 survey that \(p=0.0365\) (since \(3.65\%=0.0365\)), \(1 - p=1 - 0.0365 = 0.9635\), \(z_{\alpha/2}=2.576\), \(E = 0.025\). Substitute into the formula \(n=\frac{z^{2}_{\alpha/2}p(1 - p)}{E^{2}}\):

$$n=\frac{(2.576)^{2}\times0.0365\times0.9635}{(0.025)^{2}}$$

Calculate \((2.576)^{2}=6.635776\), \(0.0365\times0.9635 = 0.03516775\). The numerator is \(6.635776\times0.03516775\approx0.2334\).

$$n=\frac{0.2334}{0.000625}=373.44$$

Rounding up to the nearest integer, \(n = 374\).

Answer:

a. \(n = 2655\)
b. \(n = 374\)