Problem 56 - Basic Statistics for Business & Economics

Question

You plan to conduct a survey to find what proportion of the workforce has two or more jobs. You decide on the \(95 \%\) confidence level and a margin of error of \(2 \% .\) 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

You need to survey at least 865 people.

Step 1: Understand the Required Formula

To calculate the necessary sample size for a proportion, we use the formula:\[ n = \left( \frac{Z^2 \times p \times (1-p)}{E^2} \right) \]where \( Z \) is the critical value for the confidence level, \( p \) is the estimated proportion, and \( E \) is the margin of error.

Step 2: Determine Known Values

From the pilot survey, 5 out of 50 individuals hold two or more jobs. This gives the estimated proportion \( p = \frac{5}{50} = 0.1 \).For a \(95\%\) confidence level, the critical value \( Z \approx 1.96 \).The margin of error \( E \) is \( 0.02 \).

Step 3: Substitute Known Values into Formula

Substitute the known values into the sample size formula:\[ n = \left( \frac{(1.96)^2 \times 0.1 \times (1 - 0.1)}{(0.02)^2} \right) \]

Step 4: Perform the Calculations

Calculate each component inside the formula:1. \( Z^2 = (1.96)^2 = 3.8416 \)2. \( p(1-p) = 0.1 \times 0.9 = 0.09 \)3. \( E^2 = (0.02)^2 = 0.0004 \)Now substitute into the equation:\[ n = \left( \frac{3.8416 \times 0.09}{0.0004} \right) \approx 864.36\]

Step 5: Conclude with the Required Sample Size

Since the sample size must be a whole number, round up \(864.36\) to \(865\).Therefore, you should survey at least 865 people to meet your requirements of a \(95\%\) confidence level and a margin of error of \(2\%\).

Key Concepts

Confidence Level

When you hear the term "confidence level," think about how certain you want to be about your survey results. In our exercise, we have chosen a 95% confidence level, which is quite common. This means that if we were to repeat the survey many times, 95 out of 100 similar surveys would produce results that fall within the margin of error. In simple terms, it's your level of assurance that the survey reflects the true opinions of the entire workforce.
The confidence level affects the sample size. The higher the confidence level, the larger the sample needed. This is because a higher confidence level means you want to be more sure about your findings, which requires more data to minimize variability.

• 95% is a standard choice, offering a good balance between certainty and practical sample sizes.
• Higher confidence (like 99%) would require surveying more people.

Margin of Error

The margin of error represents how much you're allowed to be wrong. In our example, it's set at 2%. This 2% margin of error tells you how close the survey's results will be to the actual population's results. A margin of error lets us know the range within which the true value lies.
Let's say your survey finds that 10% of the workforce holds two jobs, but with a margin of error of 2%, you're saying the real value is likely between 8% and 12%.

• Smaller margins of error require larger sample sizes.
• It gives the maximum expected difference between the survey result and the actual population value.

This 2% margin is reasonable for many surveys but requiring a smaller margin (like 1%) would mean you need a much larger sample.

Proportion Estimation

Proportion estimation is about figuring out what fraction of your population has a particular characteristic. In our survey example, we're estimating what proportion of the workforce has two or more jobs. We found from the pilot survey data, 5 out of 50 people, which translates to 10%. This is the estimated proportion (denoted as \( p \)).
Keep in mind, the better your initial estimate, the more accurate your overall survey will be. If you don't have any prior data, you might assume it's a 50/50 split, but this is only an educated guess.

• The closer \( p \) is to 0.5, the larger the sample size needed for precision.
• Using previous data helps refine this estimate.

Critical Value

The critical value \( Z \) is associated with the confidence level. It acts like a multiplier that helps determine how straightforward or rigorous the estimates need to be. In our sample size calculation, a 95% confidence level corresponds to a critical value \( Z \) of approximately 1.96.
The critical value forms a part of the formula:

• The higher the confidence level, the larger the \( Z \).
• For 95%, use 1.96. For 99%, use approximately 2.576.

These values come from the standard normal distribution (think of the bell curve). They represent not just our desired accuracy, but how spread out or tight we wish our data to be within that margin of error. Understanding critical values is crucial in creating reliable statistical results.