To calculate the standard deviation of a proportion, we use the formula for the standard deviation of a sample proportion, $\sigma_p$:

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

Given:

• $p = 0.41$
• $n = 225$

1. First, we calculate $1 - p$: $1 - p = 1 - 0.41 = 0.59$

2. Substitute these values into the formula: $\sigma_p = \sqrt{\frac{0.41 \times 0.59}{225}}$

3. Calculate the value inside the square root: $0.41 \times 0.59 = 0.2419$ $\frac{0.2419}{225} \approx 0.0010751$

4. Take the square root: $\sigma_p \approx \sqrt{0.0010751} \approx 0.0328$

5. Convert to a percentage and round to 1 decimal place: $\sigma_p \approx 3.3\%$

Answer:

The standard deviation is approximately 3.3%.

Would you like more details, or do you have any questions?

Here are five related questions to consider:

1. How would the standard deviation change if the sample size, $n$, increased?
2. What is the significance of a high vs. low standard deviation in probability?
3. How would you interpret the result in terms of data spread?
4. How would changing $p$ affect the standard deviation?
5. What other formulas are commonly used in calculating sampling variability?

Tip: In probability, a larger sample size generally decreases the standard deviation, making the sample proportion a more reliable estimate of the population proportion.