https://file.aiquickdraw.com/mathbot/file/1729460502389l4hp7gtk.jpg

The question you provided asks for the standard deviation of the sampling distribution of the sample proportion (\(\hat{p}\)), given the following information:

- Population proportion: \( p = 0.54 \)
- Sample size: \( n = 1380 \)
- Population size: \( N = 1,100,000 \)

The standard deviation of the sampling distribution for \(\hat{p}\) can be calculated using the formula for the standard error of the sample proportion:

\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n} \times \frac{N - n}{N - 1}}
\]

Where:
- \( p \) is the population proportion.
- \( n \) is the sample size.
- \( N \) is the population size.

Let's calculate this step-by-step:

1. Calculate \( p(1 - p) \):
   \[
   p(1 - p) = 0.54 \times (1 - 0.54) = 0.54 \times 0.46 = 0.2484
   \]

2. Divide by the sample size \( n \):
   \[
   \frac{p(1 - p)}{n} = \frac{0.2484}{1380} \approx 0.000180
   \]

3. Calculate the finite population correction factor:
   \[
   \frac{N - n}{N - 1} = \frac{1,100,000 - 1380}{1,100,000 - 1} \approx 0.9987
   \]

4. Multiply the results together:
   \[
   \sigma_{\hat{p}} = \sqrt{0.000180 \times 0.9987} \approx \sqrt{0.0001797} \approx 0.0134
   \]

So the standard deviation of the sampling distribution is approximately 0.013, rounded to three decimal places.

Now, let's analyze the choices:

- **A.** Approximately normal; \( \mu_{\hat{p}} = 0.54 \), \( \sigma_{\hat{p}} = 0.150 \)
- **B.** Exactly normal; \( \mu_{\hat{p}} = 0.54 \), \( \sigma_{\hat{p}} = 0.013 \)
- **C.** Approximately normal; \( \mu_{\hat{p}} = 0.54 \), \( \sigma_{\hat{p}} = 0.013 \)
- **D.** Exactly normal; \( \mu_{\hat{p}} = 0.54 \), \( \sigma_{\hat{p}} = 0.150 \)

Given that the sample size is large and \( n/N \) is small, the sampling distribution of \(\hat{p}\) is approximately normal. Therefore, **the correct answer is C**: "Approximately normal; \( \mu_{\hat{p}} = 0.54 \), \( \sigma_{\hat{p}} = 0.013 \)."

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

Here are 5 related questions to explore further:
1. What is the Central Limit Theorem, and why does it apply in this scenario?
2. How does the finite population correction factor impact the standard deviation?
3. What conditions must be met for the sampling distribution of \(\hat{p}\) to be approximately normal?
4. How would the answer change if the population size were infinite?
5. What is the difference between "exactly normal" and "approximately normal"?

**Tip:** Always check if a finite population correction factor is needed when sampling from a finite population!

A simple random sample of size n = 1380 is obtained from a population whose size is N = 1,100,000 and whose population proportion with a specified characteristic is p = 0.54. Describe the sampling distribution of p̂. Round the standard deviation of the sampling distribution to three decimal places.

Learn how to calculate the standard deviation of the sampling distribution for a population of 1,100,000 with a sample size of 1,380 and a population proportion of 0.54. This problem involves the use of the normal distribution and finite population correction factor, suitable for college-level statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation