To calculate the standard error (SE) of the sample proportion $\hat{p}$, we use the formula:

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

where:

• $\hat{p} = \frac{x}{n}$ is the sample proportion, with $x$ being the number of trees showing damage and $n$ being the total number of trees sampled.
• $x = 31$
• $n = 58$

Step 1: Calculate the Sample Proportion $\hat{p}$

$\hat{p} = \frac{31}{58}$

Step 2: Calculate the Standard Error $SE(\hat{p})$

$SE(\hat{p}) = \sqrt{\frac{\hat{p}(1 - \hat{p})}{58}}$

Let's calculate this step-by-step.The standard error $SE(\hat{p})$ of the sample proportion is 0.065 (rounded to three decimal places).

Would you like more details or have any questions? Here are some related topics:

1. Understanding the concept of standard error in statistics.
2. How sample size affects the standard error.
3. Differences between standard error and standard deviation.
4. Confidence intervals and their relation to standard error.
5. The impact of variability on the reliability of sample estimates.

Tip: The standard error decreases as the sample size increases, leading to more precise estimates.