Question 9, 7.1.33-T
HW Score: $76 \%, 7.6$ of 10 points
Part 2 of 3
Points: 0 of 1
$\mathrm{Sa}$
A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with $3.72 \%$ of them being regular users of $\mathrm{e}$-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 a confidence level of $95 \%$ and a margin of error of 2 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.
$\mathrm{n}=2401$
(Round up to the nearest integer.)
b. Use the results from the 2014 survey.
\[
\mathrm{n}=
\]
(Round up to the nearest integer.)
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Answer
The number of adults that must be surveyed now is \(\boxed{344}\).
Step 1 ：Given values are margin of error \(E = 0.02\), Z-score for 95% confidence level \(Z = 1.96\), and proportion of population \(p = 0.0372\).
Step 2 ：We need to calculate the sample size using the formula \(n = \frac{{Z^2 \cdot p \cdot (1-p)}}{{E^2}}\).
Step 3 ：Substitute the given values into the formula, we get \(n = \frac{{(1.96)^2 \cdot 0.0372 \cdot (1-0.0372)}}{{(0.02)^2}}\).
Step 4 ：Calculate the above expression to get the value of \(n\).
Step 5 ：Round up \(n\) to the nearest integer.
Step 6 ：The number of adults that must be surveyed now is \(\boxed{344}\).
