Problem 62 - Basic Statistics for Business & Economics

Question

The Tennessee Tourism Institute (TTI) plans to sample information center visitors entering the state to learn the fraction of visitors who plan to camp in the state. Current estimates are that \(35 \%\) of visitors are campers. How many visitors would you sample to estimate the population proportion of campers with a \(95 \%\) confidence level and an allowable error of \(2 \% ?\)

Step-by-Step Solution

Answer

Sample 1083 visitors.

Step 1: Understand the Problem

The task is to determine the sample size needed to estimate the population proportion of campers among visitors, with a certain confidence level and margin of error. We know the current estimated proportion of campers is \( p = 0.35 \), and we're aiming for a 95% confidence level with a margin of error of \( E = 0.02 \).

Step 2: Determine the Necessary Formula

To find the required sample size \( n \), we use the formula \( n = \left(\frac{Z^2 \, p \, (1-p)}{E^2}\right) \). Here, \( Z \) is the z-value corresponding to the desired confidence level.

Step 3: Find the Z-value for Confidence Level

For a 95% confidence level, the Z-value is approximately \( 1.96 \). This value corresponds to the probability of the sample proportion being within 2% of the population proportion.

Step 4: Substitute Values into the Formula

Substitute \( Z = 1.96 \), \( p = 0.35 \), and \( E = 0.02 \) into the formula: \[ n = \left(\frac{1.96^2 \, \times 0.35 \, \times (1 - 0.35)}{0.02^2}\right) \]

Step 5: Calculate Numerator and Denominator

Calculate the numerator: \( 1.96^2 \times 0.35 \times 0.65 = 0.4333 \approx 0.4331 \). Calculate the denominator: \( 0.02^2 = 0.0004 \).

Step 6: Compute the Sample Size

Divide the numerator by the denominator: \[ n = \frac{0.4331}{0.0004} \approx 1082.75 \] Since sample size must be an integer, round up to 1083.

Key Concepts

Confidence Interval

When conducting a survey or study, we often want to understand what can be said about the whole population based on a smaller sample. A confidence interval provides an estimated range that is likely to contain the true value of a population parameter, such as a proportion, within a certain level of confidence.
For instance, if we say we have a 95% confidence interval, we are indicating that we are 95% confident that the interval contains the true population proportion.
This does not mean that there is a 95% probability that the interval contains the parameter, but rather that if we were to take 100 different samples and compute a confidence interval for each, we would expect about 95 of those intervals to contain the true population parameter.

• Confidence levels are usually expressed as a percentage, such as 90%, 95%, or 99%.
• The width of the confidence interval depends on the sample size and the variability in the data.
• Greater confidence levels result in wider intervals.

Population Proportion

The population proportion refers to the fraction of the entire population that holds a specific characteristic. In the context of the TTI example, it is the percentage of information center visitors who plan to camp in Tennessee.
Current surveys might provide an estimate, such as the known 35% proportion of campers, but determining the true population proportion requires sampling and analysis.
The formula used to calculate the sample size relies on an initial estimated proportion, often denoted by the symbol \( p \). This estimate is crucial as:

• It gives us a starting point for calculations regarding the sample size needed for accuracy.
• It impacts the margin of error—larger deviations in estimated \( p \) could result in larger or smaller required samples.
• The proportion also affects the confidence interval for our estimates.

By determining the right sample size, we ensure that our study accurately reflects the population and helps make data-driven decisions.

Margin of Error

The margin of error is an important measure in statistics that quantifies the amount of random sampling error in a survey's results. It represents the radius of the confidence interval for a given confidence level.
In simpler terms, the margin of error indicates how close your sample result is expected to be to the true population parameter. For the TTI case, a margin of error of 2% means we are looking to have 95% confidence that our sample proportion is within 2% of the true population proportion.

• Smaller margins result in more precise estimates but may require larger sample sizes.
• This figure directly influences how wide the confidence interval will be.
• A commonly accepted margin of error for proportion estimates is between 1% to 5%, depending on the study's needs and resources.

The margin of error ensures that the results have the desired level of precision and reliability.

Z-value

The Z-value in statistics is a measure that tells how many standard deviations an element is from the mean. When calculating sample sizes and constructing confidence intervals, Z-values correspond to the desired confidence level.
For example, using the standard normal distribution:

• A 95% confidence level has a Z-value of approximately 1.96.
• A 99% confidence level requires a Z-value of approximately 2.576.
• The Z-value is key in determining the sample size required for achieving certain statistical proprieties.

In the case of the TTI study, a 95% confidence level is chosen, giving us a Z-value of 1.96. This Z-value is plugged into the sample size formula to ensure the sample accurately represents the population within the desired confidence level and margin of error.
Understanding Z-values allows you to apply the correct standards when calculating these intervals and sample sizes.