Problem 64 - Basic Statistics for Business and Economics with Student

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 percent of visitors are campers. How large a sample would you take to estimate at a 95 percent confidence level the population proportion with an allowable error of 2 percent?

Step-by-Step Solution

Answer

A sample size of 2187 is needed.

Step 1: Identify the Problem

We need to find the sample size required to estimate the population proportion of visitors who plan to camp, given that we want a 95% confidence level and an allowable error of 2%.

Step 2: Determine Known Values

The known values are the confidence level (95%), the estimated population proportion (35%, or 0.35), and the allowable error (2%, or 0.02).

Step 3: Find the Z-Score for the Confidence Level

For a 95% confidence level, the Z-score is approximately 1.96, which is found using a standard Z-table.

Step 4: Use the Sample Size Formula for Proportion

The formula used is: \[ n = \left( \frac{Z^2 \cdot p \cdot (1-p)}{E^2} \right) \]Where:- \( n \) is the required sample size,- \( Z \) is the Z-score (1.96), - \( p \) is the estimated proportion (0.35),- \( E \) is the allowable error (0.02).

Step 5: Calculate the Sample Size

Substitute the known values into the formula:\[ n = \left( \frac{1.96^2 \cdot 0.35 \cdot (1-0.35)}{0.02^2} \right) \]Calculate the value:\[ n = \left( \frac{1.96^2 \cdot 0.35 \cdot 0.65}{0.0004} \right) \]Perform the calculations:\[ n = \left( \frac{3.8416 \cdot 0.2275}{0.0004} \right) \]\[ n = \left( \frac{0.87468}{0.0004} \right) \approx 2186.7 \]Since the sample size must be a whole number, we round up to 2187.

Key Concepts

Population Proportion

The population proportion is a fundamental concept in statistics, referring to the fraction of the overall population that exhibits a particular trait or characteristic. In the context of the exercise, the population proportion is presented as 35%, or 0.35 in decimal form. This indicates that 35% of visitors are expected to camp during their stay. It is vital for statistical calculations because it helps determine sample size requirements by giving us an initial estimate of the characteristic we want to measure across the whole population.

The population proportion is often denoted by the symbol \( p \). It is used in the sample size formula for proportions, where it represents our best guess of the true proportion within the entire audience we are investigating. Understanding this element is crucial; by making an informed estimate, we can better design surveys or studies to gather accurate data.

• If \( p \) is not known, assumptions or historical data might be used as an estimate.
• This estimate influences the margin of error and confidence the study achieves.

In practical applications, adjustments may be necessary if the preliminary estimates are adjusted based on new information or newer studies.

Confidence Level

The confidence level is a measure of how certain we are that our sample truly represents the population. In this exercise, a 95% confidence level is chosen, which is a commonly used standard in statistical research. It signifies that if the study were conducted 100 times, the result would lie within the specified margin of error for 95 of those times, assuming the method is reliable.

This percentage reflects our assurance and provides a sense of security regarding the validity of the sample's reflection of the broader population. The confidence level directly affects the sample size needed – higher confidence levels often necessitate larger samples because they require more evidence to achieve the same degree of certainty in the estimates.

• A 95% confidence level corresponds to a Z-score of approximately 1.96, which is derived from the standard normal distribution.
• Researchers choose a confidence level based on the context of the study and the level of precision required.

Deciding on the appropriate confidence level is crucial as it impacts both the reliability and the scalability of the study outcomes.

Allowable Error

Allowable error, also known as the margin of error, is the permissible range of error for the population proportion estimate. For this exercise, the allowable error is 2%, or 0.02 in decimal form. It dictates how much the sample proportion could differ from the actual population proportion while still considering the study results reliable.

The allowable error represents the precision or accuracy needed in the survey. When designing a study, researchers select this margin to balance between precision and practicality. A smaller allowable error results in a more precise result but requires larger sample sizes, which can increase the cost and complexity of the survey.

• Allowable error is denoted by \( E \) in the sample size formula for proportions.
• Reducing \( E \) necessitates a larger sample size, thus increasing resource requirements.

This concept ensures that while the results are not exact, they fall within an acceptable and reasonable range, offering valuable insights while limiting uncertainty.

Z-score

In the realm of statistics, a Z-score is a numerical measurement that describes a value's relationship to the mean of a group of values. It is an essential part of calculating confidence intervals and sample sizes. In this particular exercise, a Z-score of 1.96 is derived from the chosen confidence level of 95% and is used in the formula to determine the necessary sample size.

The Z-score links the sample mean and the population mean by accounting for variability within the sample. It helps translate the confidence level into a precise numerical value in the sample size equation. To find the correct Z-score for any given confidence level, statisticians refer to a Z-table or standard normal distribution.

• Z-scores are not only used for sample size calculations but also for hypothesis testing and determining statistical significance.
• The choice of Z-score depends on the confidence level selected for the study.

Ultimately, the Z-score is crucial for achieving a balance between practical sample sizes and robust confidence, ensuring the study's accuracy and effectiveness.