We can solve this problem using the formula for the standard deviation of the sampling distribution of a proportion. The standard deviation is given by:

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

where:

• $\hat{p}$ is the sample proportion (74% or 0.74 in decimal form),
• $n$ is the sample size (500 in part a, and 125 in part b).

Part (a)

For part (a), we use the given values:

• $p = 0.74$
• $n = 500$

Substitute these values into the formula:

$\sigma_{\hat{p}} = \sqrt{\frac{0.74(1 - 0.74)}{500}} = \sqrt{\frac{0.74(0.26)}{500}} = \sqrt{\frac{0.1924}{500}} \approx \sqrt{0.0003848} \approx 0.0196$

Thus, the estimated standard deviation of the sampling distribution is approximately 0.0196.

Part (b)

If the survey size were 125 customers, we follow the same procedure with $n = 125$:

$\sigma_{\hat{p}} = \sqrt{\frac{0.74(1 - 0.74)}{125}} = \sqrt{\frac{0.1924}{125}} \approx \sqrt{0.0015392} \approx 0.0392$

Thus, the estimated standard deviation of the sampling distribution with 125 customers would be approximately 0.0392.

Would you like more details or have any questions?

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Here are 5 related questions:

1. What is the relationship between sample size and the standard deviation of the sampling distribution?
2. How does the confidence interval for a proportion change with a smaller sample size?
3. How does increasing the sample size affect the precision of the estimated proportion?
4. What happens to the standard deviation of a sample proportion if the proportion itself is closer to 0.5?
5. Why is it important for the researcher to know the standard deviation of the sampling distribution?

Tip: A larger sample size decreases the standard deviation of the sampling distribution, leading to more precise estimates of population parameters.