Problem 21 - Basic Statistics for Business & Economics

Question

The estimate of the population proportion should be within plus or minus. \(05,\) with a \(95 \%\) level of confidence. The best estimate of the population proportion is .15. How large a sample is required?

Step-by-Step Solution

Answer

A sample size of 193 is required.

Step 1: Understanding the Problem

We need to find the sample size required to estimate a population proportion within a specified margin of error, using a specific confidence level. The given proportion (p̂) is 0.15, the margin of error (E) is 0.05, and the confidence level is 95%.

Step 2: Determine the Z-score for Confidence Level

For a 95% confidence level, the Z-score is typically 1.96. This value is derived from the standard normal distribution table and indicates the number of standard deviations needed to capture 95% of the data.

Step 3: Formula for Sample Size

The formula to calculate the sample size (n) for a proportion is: \[ n = \left( \frac{Z^2 \times p̂ \times (1-p̂)}{E^2} \right) \] where Z is the Z-score, p̂ is the estimated proportion, and E is the margin of error.

Step 4: Substitute Values into Formula

Substitute the values into the formula: \[ n = \left( \frac{1.96^2 \times 0.15 \times (1-0.15)}{0.05^2} \right) \]Calculate to find n.

Step 5: Calculate the Numerator

Calculate the expression inside the numerator: \[ 1.96^2 \times 0.15 \times 0.85 = 0.48069 \]

Step 6: Calculate the Denominator

Calculate the denominator: \[ 0.05^2 = 0.0025 \]

Step 7: Final Calculation for Sample Size

Divide the numerator by the denominator to find the sample size: \[ n = \frac{0.48069}{0.0025} = 192.276 \]Since the sample size must be a whole number, round up to 193.

Step 8: Conclusion

The required sample size is 193 to estimate the population proportion within a margin of error of 0.05 at a 95% confidence level.

Key Concepts

Population Proportion

Population proportion is an essential concept in statistics, referring to the ratio of members in a population with a certain characteristic. Imagine surveying a city where 15% of residents might be using bicycles as their primary mode of transportation. This 15% is referred to as the population proportion and is usually symbolized as \(p\). In exercises discussing sample size calculations, the population proportion often serves as the best estimate available for determining samples.

• Understand it as the assumed true ratio in the larger group we are interested in.
• It's a key factor in determining how accurately our sample reflects the population.

Determining the population proportion correctly is crucial because it affects the reliability of your survey or study outcomes. A correct population proportion helps ensure that the insights gathered from the sample accurately reflect the larger population's behavior or characteristics.

Margin of Error

The margin of error, depicted as \(E\), is the range that we expect our estimate to lie within, with a certain level of confidence. For example, if we find a margin of error of 5%, it means the estimate could be 5% more or less than the true population value. This is a key element of sample size calculations.

• It defines how close we want our estimate to be to the real population parameter.
• A smaller margin of error requires a larger sample size to accomplish more precision.

Selecting an appropriate margin of error is paramount in balancing the resources available for data collection with the desired accuracy of the results. Frequently in studies, a margin of error of \(0.05\) is used, signifying reasonable confidence and practicality without compromising too much on precision.

Confidence Level

The confidence level indicates how confident we can be in our estimation process. Expressed as a percentage, it tells us the probability that the population parameter falls within our interval estimate. For instance, a 95% confidence level implies we are 95% sure the actual population proportion falls within the range defined by the margin of error.

• Confidence level directly influences how trustworthy our intervals are.
• Common choices are usually 90%, 95%, or 99%, with higher levels offering more certainty but requiring larger samples.

Setting the confidence level depends on the context of the research. In most cases, a 95% confidence level is standard, striking a balance between certainty and feasibility, ensuring the findings are robust enough for decision-making while maintaining feasible study requirements.

Z-score

Z-score is a statistical metric that measures how many standard deviations a data point is from the mean. In the context of sample size calculation, Z-scores are critical for translating confidence levels into actual calculations. For example, a 95% confidence level is associated with a Z-score of 1.96. This comes from the standard normal distribution and recognizes how much deviation from the mean we consider acceptable to encapsulate our desired confidence level.

• Z-scores translate the abstract concept of confidence levels into actionable figures.
• Common Z-scores include 1.645 for 90%, 1.96 for 95%, and 2.576 for 99% confidence levels.

Understanding how to use and find the appropriate Z-score allows researchers to align their confidence level intentions with computational tasks. This alignment is key in ensuring the statistical strength and validity of inferred conclusions from a study.