Questions: A travel agent wants to estimate, with 98% confidence, the proportion of vacationers who use an online service or the Internet to make reservations for lodging. The estimate must be accurate within 4% of the population proportion. a) How many vacationers must be surveyed? b) If past industry data suggest that 10% of vacationers use an online service or the Internet to make such reservations, how many vacationers much be surveyed?

A travel agent wants to estimate, with 98% confidence, the proportion of vacationers who use an online service or the Internet to make reservations for lodging. The estimate must be accurate within 4% of the population proportion. a) How many vacationers must be surveyed? b) If past industry data suggest that 10% of vacationers use an online service or the Internet to make such reservations, how many vacationers much be surveyed?
Transcript text: 15. A travel agent wants to estimate, with $98 \%$ confidence, the proportion of vacationers who use an online service or the Internet to make reservations for lodging. The estimate must be accurate within $4 \%$ of the population proportion. a) How many vacationers must be surveyed? b) If past industry data suggest that $10 \%$ of vacationers use an online service or the Internet to make such reservations, how many vacationers much be surveyed?
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Solution

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Solution Steps

Step 1: Determine Sample Size Without Prior Estimate

To estimate the proportion of vacationers who use an online service for lodging reservations without any prior estimate, we use the maximum variability assumption, setting \( p = 0.5 \). The formula for the required sample size \( n \) is given by:

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

Substituting the values:

  • \( Z = 2.33 \) (Z-score for 98% confidence)
  • \( p = 0.5 \)
  • \( E = 0.04 \)

Calculating:

\[ n = \frac{(2.33)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.04)^2} = 848 \]

Thus, the required sample size without a prior estimate is \( n = 848 \).

Step 2: Determine Sample Size With Prior Estimate

Using past industry data that suggests \( p = 0.1 \), we apply the same formula for the required sample size:

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

Substituting the values:

  • \( Z = 2.33 \)
  • \( p = 0.1 \)
  • \( E = 0.04 \)

Calculating:

\[ n = \frac{(2.33)^2 \cdot 0.1 \cdot (1 - 0.1)}{(0.04)^2} = 305 \]

Thus, the required sample size with the prior estimate is \( n = 305 \).

Final Answer

The required sample sizes are:

  • Without prior estimate: \\(\boxed{n = 848}\\)
  • With prior estimate: \\(\boxed{n = 305}\\)
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