To find the standard deviation of the sample proportion, we use the formula for the standard deviation of a proportion, which is also known as the standard error of the proportion:

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

where:
- \(p = 0.62\) is the population proportion (62% of U.S. adults drink coffee daily),
- \(n = 225\) is the sample size.

Substitute the values into the formula:

\[
\sigma_{\hat{p}} = \sqrt{\frac{0.62(1 - 0.62)}{225}}
\]

Now calculate the values:

\[
\sigma_{\hat{p}} = \sqrt{\frac{0.62 \cdot 0.38}{225}}
\]

\[
\sigma_{\hat{p}} = \sqrt{\frac{0.2356}{225}}
\]

\[
\sigma_{\hat{p}} \approx \sqrt{0.0010462}
\]

\[
\sigma_{\hat{p}} \approx 0.0323
\]

So, the standard deviation (standard error) of the sample proportion is approximately 0.0323.

Do you have any questions or need further clarification?

Here are five follow-up questions you might consider:
1. How do you calculate the confidence interval for a proportion?
2. What is the interpretation of the standard error in this context?
3. Why is the sample size important when determining the standard error?
4. What effect does a larger sample size have on the standard error?
5. How can you use the standard error to assess the reliability of the sample estimate?

**Tip:** When calculating standard deviations or standard errors for proportions, always ensure that the sample size and proportion make the standard error calculation valid, especially when applying the normal approximation.

Learn how to calculate the standard deviation (standard error) of a sample proportion using the formula. This example involves determining the standard deviation of a proportion where 62% of U.S. adults drink coffee daily, with a sample size of 225. Understand the significance of sample size and proportion in standard error calculations.

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