
The president of Doerman Distributors, Inc., believes that 28% of the firm's orders come from first-time customers. A random sample of 100 orders will be used to estimate the proportion of first-time customers. Assume that the president is correct and p = 0.28. What is the sampling error of p for this study? If required, round your answer to four decimal places.

To calculate the **sampling error** for the proportion \( p \) in this study, we use the formula for the **standard error of a sample proportion**:

\[
SE = \sqrt{\frac{p(1 - p)}{n}}
\]

Where:
- \( p = 0.28 \) is the proportion of first-time customers (as given by the president),
- \( n = 100 \) is the sample size.

### Step-by-Step Calculation:
1. \( p = 0.28 \)
2. \( 1 - p = 1 - 0.28 = 0.72 \)
3. \( n = 100 \)

Now, substitute these values into the formula:

\[
SE = \sqrt{\frac{0.28 \times 0.72}{100}} = \sqrt{\frac{0.2016}{100}} = \sqrt{0.002016}
\]

\[
SE = 0.0449
\]

So, the **standard error** is approximately **0.0449**.

### Sampling Error:
The **sampling error** is essentially the **standard error** for the sample proportion \( p \). Therefore, the sampling error for this study is **0.0449**.

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Would you like further clarification on this, or do you have any questions on other related topics?

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Here are some related questions to explore:

1. How would the standard error change if the sample size increased to 400?
2. What happens to the sampling error if the proportion of first-time customers is much higher or lower than 0.28?
3. How does the formula for standard error differ when estimating the mean instead of the proportion?
4. How can we compute a confidence interval for the proportion of first-time customers using the standard error?
5. What would be the margin of error at a 95% confidence level for this study?

**Tip:** When calculating standard errors for proportions, always check if the sample size is large enough to apply the normal approximation. Generally, both \( np \) and \( n(1-p) \) should be at least 10.

The president of Doerman Distributors, Inc., believes that 28% of the firm's orders come from first-time customers. A random sample of 100 orders will be used to estimate the proportion of first-time customers. Assume that the president is correct and p = 0.28. What is the sampling error of p for this study? If required, round your answer to four decimal places.

Learn how to calculate the sampling error for estimating the proportion of first-time customers in a sample of 100 orders. This problem involves standard error for sample proportions and is suitable for students in Grades 10-12 studying statistics.

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