A market researcher for a provider of music player accessories wants to know the proportion of customers who own cars to assess the market for a new car charger. A survey of 500 customers indicates that 74​% own cars.
​a) What is the estimated standard deviation of the sampling distribution of the​ proportion?
​b) How large would the estimated standard deviation have been if he had surveyed only 125 customers​ (assuming the proportion is about the​ same)?
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Part 1
​a) The estimated standard deviation of the sampling distribution of the proportion is 
  
enter your response here. 
​(Round to four decimal places as​ needed.)

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.

A market researcher for a provider of music player accessories wants to know the proportion of customers who own cars to assess the market for a new car charger. A survey of 500 customers indicates that 74​% own cars.
​a) What is the estimated standard deviation of the sampling distribution of the​ proportion?
​b) How large would the estimated standard deviation have been if he had surveyed only 125 customers​ (assuming the proportion is about the​ same)?

Learn how to calculate the estimated standard deviation of the sampling distribution for a proportion based on a survey of 500 customers, where 74% own cars. The guide also explores how the standard deviation changes when the sample size is reduced to 125 customers. Suitable for college-level statistics students.

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