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!