The standard deviation for a binomial distribution, when dealing with a proportion $p$, can be calculated using the formula:

$\sigma = \sqrt{\frac{p(1 - p)}{n}}$

Given:

• $p = 0.58$
• $n = 410$

Let's calculate:

1. Calculate $1 - p$: $1 - p = 1 - 0.58 = 0.42$

2. Compute the variance $\frac{p(1 - p)}{n}$: $\frac{0.58 \times 0.42}{410} = \frac{0.2436}{410} \approx 0.00059415$

3. Take the square root to find the standard deviation: $\sigma = \sqrt{0.00059415} \approx 0.0244$

4. Convert to a percentage and round to 1 decimal place: $0.0244 \times 100 \approx 2.4\%$

Answer:

The standard deviation is approximately 2.4%.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. How would changing $p$ to a higher or lower value affect the standard deviation?
2. What would be the standard deviation if the sample size $n$ were doubled?
3. How is the binomial distribution’s standard deviation different from that of a normal distribution?
4. Can you explain why we use $p(1 - p)$ in the formula?
5. How would you interpret the result of a 2.4% standard deviation in a real-world context?

Tip: The smaller the sample size, the larger the standard deviation, indicating greater variability in proportion estimates.