Problem 30 - Basic Statistics for Business and Economics with Student

Question

Past surveys reveal that 30 percent of tourists going to Las Vegas to gamble during a weekend spend more than \(\$ 1,000 .\) Management wants to update this percentage. a. The new study is to use the 90 percent confidence level. The estimate is to be within 1 percent of the population proportion. What is the necessary sample size? b. Management said that 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

a) Sample size is 5683. b) Increase margin of error to reduce sample size to 1421.

Step 1: Identify Initial Parameters

We are given an initial proportion \( p = 0.30 \), a confidence level of 90%, and a margin of error \( E = 0.01 \). The Z-score for a 90% confidence level is \( 1.645 \).

Step 2: Calculate Required Sample Size

To find the sample size \( n \), use the formula:\[ n = \left( \frac{Z^2 \cdot p \cdot (1-p)}{E^2} \right)\]Substituting, \( n = \left( \frac{1.645^2 \cdot 0.30 \cdot 0.70}{0.01^2} \right) \).

Step 3: Solve for Sample Size

Calculate the intermediate value: \( 1.645^2 = 2.706025 \). Then substitute:\[ n = \left( \frac{2.706025 \cdot 0.21}{0.0001} \right)\]This simplifies to \( n \approx \frac{0.56826525}{0.0001} = 5682.6525 \). Rounding up gives \( n = 5683 \).

Step 4: Suggest Reducing Sample Size

To reduce the sample size, either decrease the confidence level or increase the margin of error \( E \). Both adjustments can significantly reduce the required sample size.

Step 5: Recalculate with Increased Margin of Error

Suppose we increase \( E \) to 2% (\( E = 0.02 \)) while keeping the confidence level at 90%:\[ n = \left( \frac{1.645^2 \cdot 0.30 \cdot 0.70}{0.02^2} \right)\]Calculate this as:\[ n = \frac{2.706025 \cdot 0.21}{0.0004} = 1420.6625\]This rounds up to \( n = 1421 \).

Key Concepts

Confidence Level

In statistical terms, the confidence level is a crucial aspect when performing a sample size calculation. It refers to how confident we are that the sample truly reflects the population. In simpler words, how sure we can be that the sample data collected represents the whole group's behavior or characteristics. Typically, a confidence level of 90%, like in the given problem, suggests that if we were to repeat the survey multiple times, we would expect the results to reflect the true population value in 90% of the cases.

A higher confidence level gives more reliable results. However, it also requires collecting a larger sample size, thus increasing the efforts and costs involved in the survey. That's why adjusting the confidence level is a common strategy to manage the sample size. But, beware that lowering it too much could make the survey results less trustworthy. It's all about finding that perfect balance between practicality and precision.

In the sample problem, with a 90% confidence level, they opted for moderate certainty, balancing between having a reasonable sample size and ensuring that the results are trustworthy most of the time.

Margin of Error

The margin of error represents how much we expect our estimate to differ from the true population proportion. In the exercise, they required a very small margin of error of 1%, meaning they want the estimate to be very precise. This tiny margin ensures that the study's result almost accurately matches the actual population's spending behavior.

To calculate the sample size precisely, the margin of error must play a key role. The formula directly uses the margin of error to gauge how large the sample should be. A smaller margin of error implies greater precision, but this comes at the cost of needing a larger sample. Conversely, increasing the margin of error makes the required sample smaller, lowering costs and efforts.

Example Adjustment:

• In our problem, managing the margin of error from 1% to 2%, decreased the sample from 5683 to 1421. This shows how crucial the margin of error is when calculating sample sizes.

Careful consideration of the margin of error ensures that we neither overspend on extra samples nor compromise on our study's accuracy.

Population Proportion

Population proportion is a term that represents the fraction or percentage of the total population exhibiting a particular trait. In the problem, it was stated at 30% (0.30), showing the portion of tourists spending more than $1,000 over a weekend. Understanding this concept is essential in sample size calculation, as the formula takes into account the proportion to estimate efficiently.

Since this proportion determines the variability in your population, knowing it helps in forming predictions about how much the survey data will deviate from the actual population value. The formula for calculating sample size uses the expression \(p imes (1-p)\) where \p\ is the population proportion, indicating how the proportions encompass as much variation in the data as possible.

Key Points:

• Before any calculation, ensure the population proportion is realistically estimated through prior studies or reliable sources like past surveys.
• An accurate population proportion leads to more reliable predictions of sample size.

Accurately understanding and implementing population proportion ensures that the survey results stay true to representing the actual group's behavior, making research effective and efficient.