The sampling error in this context is a measure of how much the sample proportion might differ from the true population proportion. We can calculate the sampling error using the standard error of the proportion formula:

$\text{Standard Error (SE)} = \sqrt{\frac{p(1 - p)}{n}}$

where:

• $p = 0.30$ (the assumed true proportion of first-time customers),
• $n = 100$ (the sample size).

Step-by-Step Calculation

1. Substitute the values into the formula: $\text{SE} = \sqrt{\frac{0.30 \times (1 - 0.30)}{100}}$

2. Simplify the terms inside the square root: $\text{SE} = \sqrt{\frac{0.30 \times 0.70}{100}} = \sqrt{\frac{0.21}{100}}$

3. Continue simplifying: $\text{SE} = \sqrt{0.0021} \approx 0.0458$

Answer

The sampling error (standard error of the proportion) is approximately:

$0.0458$

Would you like to see further details on this calculation or have any questions?

Here are five related questions to consider:

1. What would happen to the sampling error if the sample size increased?
2. How does changing the true proportion $p$ affect the sampling error?
3. What if we weren't sure about $p$ being exactly 0.30; how would we estimate the error?
4. Can this method apply if the distribution of first-time customers isn’t exactly 30%?
5. How does sampling error relate to the confidence level in estimations?

Tip: Increasing the sample size decreases the sampling error, giving a more precise estimate of the population proportion.