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.
n = 2655
(round up to the nearest integer.)
b. use the results from the 2014 survey.
n = 374
(round up to the nearest integer.)
c. does the use of the result from the 2014 survey have much of an effect on the sample size?
a. yes, using the result from the 2014 survey dramatically reduces the sample size.
b. no, using the result from the 2014 survey only slightly reduces the sample size.
c. yes, using the result from the 2014 survey only slightly increases the sample size.
d. no, using the result from the 2014 survey does not change the sample size.

Explanation:

Step1: Recall the formula for sample size \(n=\frac{z^{2}\cdot p(1 - p)}{E^{2}}\)

For a \(99\%\) confidence level, the critical value \(z\) is \(z = 2.576\) (from standard normal distribution tables). The margin of error \(E=0.025\).

Step2: Case when no prior estimate (\(p = 0.5\))

When \(p=0.5\), then \(n=\frac{(2.576)^{2}\times0.5\times(1 - 0.5)}{(0.025)^{2}}\)

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Step3: Case when using prior estimate (\(p = 0.0365\))

\(n=\frac{(2.576)^{2}\times0.0365\times(1 - 0.0365)}{(0.025)^{2}}\)

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Step4: Analyze the effect of using the 2014 - survey result

Since \(374\) (using prior - estimate) is much smaller than \(2655\) (no prior - estimate), using the 2014 - survey result (prior - estimate) dramatically reduces the sample size.

Answer:

A. Yes, using the result from the 2014 survey dramatically reduces the sample size.