https://file.aiquickdraw.com/mathbot/file/1727577294585lsjnc8h9.jpg

The quiz in the image is asking about the distribution of sample proportions, and we are given the following information:

- Sample size \( n = 568 \)
- Population proportion \( p = 0.51 \)

### Part 1: Mean of the Distribution of Sample Proportions

The mean of the distribution of sample proportions is the same as the population proportion:

\[
\mu_{\hat{p}} = p = 0.51
\]

### Part 2: Standard Deviation of the Distribution of Sample Proportions

The formula for the standard deviation of the sample proportions is given by:

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

Substituting the values:

\[
\sigma_{\hat{p}} = \sqrt{\frac{0.51(1 - 0.51)}{568}} = \sqrt{\frac{0.51 \times 0.49}{568}}
\]

Let me calculate that.The standard deviation of the distribution of sample proportions is approximately \( \sigma_{\hat{p}} = 0.02 \).

### Summary:
- **Mean of the distribution of sample proportions**: \( \mu_{\hat{p}} = 0.51 \)
- **Standard deviation of the distribution of sample proportions**: \( \sigma_{\hat{p}} = 0.02 \)

Would you like further explanation or have any questions?

Here are 5 related questions to deepen your understanding:
1. How does increasing the sample size affect the standard deviation of the sample proportions?
2. What happens to the distribution if the population proportion \( p \) is close to 0 or 1?
3. Can the standard deviation of sample proportions ever be zero?
4. How is the sample proportion different from the population proportion?
5. What is the significance of the Central Limit Theorem in the context of sample proportions?

**Tip**: The larger the sample size, the smaller the standard deviation, making the sample proportion more representative of the population proportion.

You intend to draw a random sample of size n = 568 from a population whose parameter is p = 0.51. What is the mean of the distribution of sample proportions? What is the standard deviation of the distribution of sample proportions?

Learn how to calculate the mean and standard deviation of the distribution of sample proportions for a population parameter p = 0.51 and a sample size n = 568. This problem includes statistical concepts, key formulas, and the Central Limit Theorem, suitable for advanced high school or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation