A company surveys employees. The company is too large to check each response, so samples of 50 are taken, and the tendency is that three-fourths of the employees are happy. For the mean of the sample distribution of sample proportions, answer the following questions. A) How does this affect the mean? B) How does this affect the standard Deviation? C) How does this affect the variance?

In this scenario, we are dealing with a **sampling distribution** of sample proportions, where the proportion of happy employees is \( p = \frac{3}{4} = 0.75 \) and the sample size is \( n = 50 \).

For the **sampling distribution of the sample proportion**, we can analyze the mean, standard deviation, and variance as follows:

### A) **How does this affect the mean?**
The **mean** of the sampling distribution of sample proportions is equal to the population proportion \( p \). This is because the sample proportions tend to center around the true population proportion in the long run.

\[
\mu_{\hat{p}} = p = 0.75
\]

**Conclusion**: The mean of the sampling distribution remains 0.75, as it directly reflects the population proportion of happy employees.

### B) **How does this affect the standard deviation?**
The **standard deviation** of the sampling distribution of sample proportions (often called the standard error of the proportion) is calculated as:

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

Substitute \( p = 0.75 \) and \( n = 50 \):

\[
\sigma_{\hat{p}} = \sqrt{\frac{0.75(1 - 0.75)}{50}} = \sqrt{\frac{0.75 \times 0.25}{50}} = \sqrt{\frac{0.1875}{50}} = \sqrt{0.00375} \approx 0.0612
\]

**Conclusion**: The standard deviation of the sampling distribution is approximately 0.0612.

### C) **How does this affect the variance?**
The **variance** of the sampling distribution of sample proportions is simply the square of the standard deviation:

\[
\text{Variance} = \sigma_{\hat{p}}^2 = \left(0.0612\right)^2 = 0.00375
\]

**Conclusion**: The variance of the sampling distribution is 0.00375.

---

### Summary:
- A) The **mean** of the sampling distribution of sample proportions is 0.75.
- B) The **standard deviation** of the sampling distribution is approximately 0.0612.
- C) The **variance** of the sampling distribution is 0.00375.

Do you want more details or have any questions? Here are five related questions to explore further:

1. How would increasing the sample size affect the standard deviation?
2. If the population proportion changes, how would the sampling distribution be affected?
3. How does the shape of the sampling distribution relate to the sample size?
4. Can you explain the concept of the **Central Limit Theorem** in this context?
5. What assumptions are made in calculating the sampling distribution?

**Tip**: As the sample size increases, the standard deviation of the sample proportion decreases, making estimates more precise!

Explore the statistical concepts behind sampling distribution of sample proportions. Learn how the mean remains 0.75, and calculate the standard deviation (approximately 0.0612) and variance (0.00375) for a sample size of 50 in a company survey. Perfect 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