Questions: A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with 3.68% of them being regular users of 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 90% and a margin of error of 1 percentage point are wanted? Complete parts (a) through (c) below. a. Assume that nothing is known about the rate of e-cigarette usage among adults. n=6766 (Round up to the nearest integer.) b. Use the results from the 2014 survey. n=960 (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. No, using the result from the 2014 survey only slightly reduces the sample size. B. Yes, using the result from the 2014 survey dramatically reduces the sample size. C. No, using the result from the 2014 survey does not change the sample size. D. Yes, using the result from the 2014 survey only slightly increases the sample size.

A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with 3.68% of them being regular users of 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 90% and a margin of error of 1 percentage point are wanted? Complete parts (a) through (c) below. a. Assume that nothing is known about the rate of e-cigarette usage among adults. n=6766 (Round up to the nearest integer.) b. Use the results from the 2014 survey. n=960 (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. No, using the result from the 2014 survey only slightly reduces the sample size. B. Yes, using the result from the 2014 survey dramatically reduces the sample size. C. No, using the result from the 2014 survey does not change the sample size. D. Yes, using the result from the 2014 survey only slightly increases the sample size.
Transcript text: A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with $3.68 \%$ of them being regular users of 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 $90 \%$ and a margin of error of 1 percentage point are wanted? Complete parts (a) through (c) below. a. Assume that nothing is known about the rate of e-cigarette usage among adults. \[ n=6766 \] (Round up to the nearest integer.) b. Use the results from the 2014 survey. \[ n=960 \] (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. No, using the result from the 2014 survey only slightly reduces the sample size. B. Yes, using the result from the 2014 survey dramatically reduces the sample size. C. No, using the result from the 2014 survey does not change the sample size. D. Yes, using the result from the 2014 survey only slightly increases the sample size.
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Solution

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Solution Steps

Step 1: Sample Size Calculation with Unknown Rate

When nothing is known about the rate of e-cigarette usage among adults, we assume the maximum variability by setting \( p = 0.5 \). The formula for the sample size \( n \) is given by:

\[ n = \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \]

Substituting the values:

  • \( Z = 1.645 \) (for a 90% confidence level)
  • \( p = 0.5 \)
  • \( E = 0.01 \)

We calculate:

\[ n = \frac{(1.645)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.01)^2} = 6765.0625 \]

Rounding up to the nearest integer, we find:

\[ n = 6766 \]

Step 2: Sample Size Calculation Using 2014 Survey Results

Using the results from the 2014 survey, where \( p = 0.0368 \), we apply the same formula:

\[ n = \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \]

Substituting the values:

  • \( Z = 1.645 \)
  • \( p = 0.0368 \)
  • \( E = 0.01 \)

We calculate:

\[ n = \frac{(1.645)^2 \cdot 0.0368 \cdot (1 - 0.0368)}{(0.01)^2} = 960.0001 \]

Rounding up to the nearest integer, we find:

\[ n = 960 \]

Step 3: Effect of Using 2014 Survey Results

Comparing the two sample sizes:

  • Sample size with unknown rate: \( n = 6766 \)
  • Sample size using 2014 survey results: \( n = 960 \)

Since \( 960 < 6766 \), we conclude that using the result from the 2014 survey dramatically reduces the sample size.

Final Answer

The answers to the sub-questions are:

  • a. \( n = 6766 \)
  • b. \( n = 960 \)
  • c. The answer is B.

Thus, the final boxed answer is:

\[ \boxed{6766, 960, B} \]

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