Problem 26 - Basic Statistics for Business & Economics

Question

Past surveys reveal that 30% of tourists going to Las Vegas to gamble spend more than \(\$ 1,000 .\) The Visitor's Bureau of Las Vegas wants to update this percentage. a. How many tourists should be randomly selected to estimate the population proportion with a 90% confidence level and a 1% margin of error? b. The Bureau feels the sample size determined above is too large. What can be done to reduce the sample? Based on your suggestion, recalculate the sample size.

Step-by-Step Solution

Answer

Sample size is 5682. Increase error margin to 2% to reduce sample to 1421.

Step 1: Understand the formula for sample size

For estimating a population proportion, the sample size can be calculated using the formula:\[ n = \left( \frac{Z^2 \cdot p \cdot (1-p)}{E^2} \right) \]where \(n\) is the sample size, \(Z\) is the z-score corresponding to the desired confidence level, \(p\) is the estimated proportion, and \(E\) is the margin of error.

Step 2: Plug in the values for the initial sample size calculation

Given, \(p = 0.30\), \(E = 0.01\), and a 90% confidence level, which corresponds to a \(Z\) value of approximately 1.645. Substitute these values into the formula:\[ n = \left( \frac{1.645^2 \times 0.30 \times (1 - 0.30)}{0.01^2} \right)\]This calculates to:\[ n \approx \left( \frac{2.706025 \times 0.21}{0.0001} \right) \approx 5682\]

Step 3: Interpret how to reduce the sample size

To reduce the sample size, you can either increase the margin of error or decrease the confidence level. Increasing \(E\), the margin of error, would make the estimated range wider but with fewer participants required.

Step 4: Recalculate sample size with adjusted margin of error

Suppose we increase the margin of error to \(E = 0.02\). Recalculate:\[ n = \left( \frac{1.645^2 \times 0.30 \times (1 - 0.30)}{0.02^2} \right)\]This calculates to:\[ n \approx \left( \frac{2.706025 \times 0.21}{0.0004} \right) \approx 1421\]

Step 5: Decision and conclusion

The revised sample size with a margin of error of 2% is 1421, significantly smaller than the original 5682 with a 1% error margin.

Key Concepts

Population Proportion

The population proportion, often denoted as \( p \), is a statistical term that represents the fraction of individuals in a population that exhibit a particular characteristic. In the context of the problem, the population proportion is the percentage of tourists who spend more than \(1,000 while gambling in Las Vegas. The previously known population proportion is given as 30%, or \( p = 0.30 \). When estimating this proportion for a larger population, it is utilized in the sample size formula to help determine how many individuals need to be observed to achieve accurate results.
Understanding this concept helps in planning and implementing surveys, ensuring that the selected sample is representative of the entire population. The correct estimate of \( p \) improves the reliability of the conclusions drawn from the sample data.
When applying the sample size formula, \( p \) is multiplied by \( 1-p \) because it accounts for both the success (people spending more than \)1,000) and the failure (people spending less than $1,000) in the population.

Confidence Level

The confidence level in statistics indicates the probability that a parameter will fall between a pair of values based on your sample data. In simple terms, it tells you how certain you can be about your results. In our example, a 90% confidence level means that if the sample study were repeated numerous times, 90% of those samples would contain the true population proportion.
Choosing a confidence level involves a trade-off. Higher confidence levels mean less risk of error, but they often require a larger sample size. In practical scenarios, 90%, 95%, and 99% are common choices for confidence levels. For the Visitor's Bureau of Las Vegas, the 90% level provides a good balance between certainty and cost-effectiveness.
Confidence levels are crucial in statistical analysis, especially for businesses or policymakers, as they allow decision-makers to weigh the reliability of their data-driven decisions.

Margin of Error

The margin of error, denoted as \( E \), expresses the range within which we expect the true value of the population parameter to lie. In this context, it defines how close the estimated population proportion is to the actual population proportion.
For example, a 1% margin of error means that the calculated sample proportion will likely be within 1% of the actual population proportion. Reducing the margin of error requires a larger sample size, as seen in the solution where increasing \( E \) from 1% to 2% significantly reduced the required sample size from 5682 to 1421.
A smaller margin of error gives more precise results but at the cost of requiring more resources. Thus, it's essential to balance the need for precision with practical considerations like budget and time.

Z-Score

The z-score is a statistical measure that correlates with the confidence level and is used in the formula for determining sample size. It indicates how many standard deviations an element is from the mean of the distribution.
In the sample size formula, the z-score corresponds to the selected confidence level. For the 90% confidence level in this exercise, the z-score is approximately 1.645. This means that the population mean is expected to be within 1.645 standard deviations of the sample mean, 90% of the time.
Understanding z-scores is vital because they directly impact the sample size calculation. A higher z-score, associated with a higher confidence level, will increase the required sample size, ensuring that the results are statistically robust. Z-scores are a cornerstone of inferential statistics, crucial for making accurate predictions and analyses.