The statistic that best estimates $\mu$ (the population mean) is the sample mean, denoted as $\bar{x}$. This is because the sample mean is an unbiased estimator of the population mean, meaning that the expected value of the sample mean is equal to the population mean.

In a random sample of size $n$ from a population, the sample mean $\bar{x}$ is calculated as:

$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$

Where $x_i$ represents each individual observation in the sample.

Key points:

• The sample mean $\bar{x}$ is used because it's an unbiased estimator of $\mu$.
• It becomes more accurate as the sample size $n$ increases, due to the Law of Large Numbers.
• The sample mean minimizes the Mean Squared Error (MSE) when estimating the population mean.

Would you like further details on the properties of estimators or unbiased estimation?

Here are 5 follow-up questions related to the concept:

1. Why is the sample mean considered an unbiased estimator?
2. How does the sample size affect the reliability of the sample mean as an estimator of $\mu$?
3. What is the difference between the sample mean and the population mean?
4. What other statistics might be used to estimate the population mean in non-standard distributions?
5. How does the Central Limit Theorem support the use of the sample mean in estimating $\mu$?

Tip: The accuracy of the sample mean improves as the sample size increases, especially when the population has a large variance.