Problem 58 - Basic Statistics for Business and Economics with Student

Question

You plan to conduct a survey to find what proportion of the workforce has two or more jobs. You decide on the 95 percent confidence level and state that the estimated proportion must be within 2 percent of the population proportion. A pilot survey reveals that 5 of the 50 sampled hold two or more jobs. How many in the workforce should be interviewed to meet your requirements?

Step-by-Step Solution

Answer

Survey approximately 864 people.

Step 1: Understand the Problem

We need to find the number of people to interview such that the sample proportion is within 2% of the population proportion with 95% confidence. We have a pilot survey where 5 out of 50 people hold two or more jobs, giving us an estimate for the proportion.

Step 2: Find the Sample Proportion

Calculate the sample proportion from the pilot survey. If 5 out of 50 hold two or more jobs, then the sample proportion \( \hat{p} = \frac{5}{50} = 0.1 \).

Step 3: Understand the Margin of Error Requirement

The margin of error (E) is given as 2%, or 0.02 in decimal form. We need to ensure this is the maximum potential error between our sample proportion and the true population proportion.

Step 4: Determine Z-Score for 95% Confidence

For a 95% confidence level, the Z-score is 1.96. This is the number of standard deviations away from the mean required for our desired confidence interval.

Step 5: Use the Sample Size Formula

The formula for estimating sample size \( n \) is given by \[ n = \left( \frac{Z^2 \times \hat{p} \times (1 - \hat{p})}{E^2} \right) \]. Substitute \( Z = 1.96 \), \( \hat{p} = 0.1 \), and \( E = 0.02 \) into the formula.

Step 6: Calculate the Sample Size

Using the values: \[ n = \left( \frac{1.96^2 \times 0.1 \times 0.9}{0.02^2} \right) \]. Simplify this calculation: \( n = \left( \frac{3.8416 \times 0.09}{0.0004} \right) \), which results in \( n \approx 864 \).

Step 7: Conclusion

To ensure that the sample proportion is within 2% of the population proportion with 95% confidence, approximately 864 people should be surveyed.

Key Concepts

Confidence Interval

When conducting a survey or an experiment, knowing how confident we are in our results is crucial. The confidence interval is a range of values, derived from your sample data, that is likely to contain the true value of an unknown population parameter. In this case, we are trying to estimate the proportion of the workforce with two jobs. The 95% confidence level signifies that if you were to take 100 different samples and compute a confidence interval for each sample, we would expect about 95 of those intervals to contain the actual population parameter. This helps to acknowledge the uncertainty inherent in sampling and provides a method to express how reliable our estimations are.

Margin of Error

The margin of error is a vital component that expresses the plus-or-minus figure reported in survey results. It represents the range within which the true population parameter is expected to lie. In this exercise, the margin of error is set to be 2%, denoted in decimal form as 0.02. This means that the calculated proportion of those who hold two or more jobs should not deviate by more than 2% from the actual population proportion. The confidence interval is calculated as the sample statistic plus or minus the margin of error. This makes margin of error critical when interpreting the range of values around your estimate, providing insights into how accurate the sample statistic is likely to be.

Pilot Survey

A pilot survey is a small-scale study conducted before the main research. It helps refine the design and methodology of the subsequent larger survey. In the context of this exercise, the pilot survey involved interviewing 50 individuals, out of which 5 people were found to hold two or more jobs. This initial data collection is used to provide an estimate of the population proportion, which can then be used in the calculation of the required sample size for the full survey. Using a pilot survey makes the larger survey more efficient as it provides an initial point estimate, helping in formulating a clearer understanding of the sample size and logistical needs for the main survey.

Proportion Estimation

Proportion estimation involves estimating the fraction or percentage of a population that possesses a particular attribute. Here, we aim to estimate the proportion of the workforce holding multiple jobs from the data collected. From the pilot survey, a proportion of 0.1, or 10%, was calculated. Proportion estimation is used not only for understanding characteristics of a population but also underscores the importance of sample size. The more accurate the initial pilot estimate, the more reliable the full sample size calculation. The sample size formula helps determine the number of participants needed to ensure that the estimate stays within an acceptable margin of error at a given confidence level. In this exercise, the estimated proportion plays a crucial role in calculating how large a sample is needed to meet the desired precision.