μ_\bar{x}

Definition

The mean of the sampling distribution of the sample mean, $\bar{x}$. It represents the expected value or central tendency of the sampling distribution, which is a crucial concept in statistical inference and the application of the Central Limit Theorem.

5 Must Know Facts For Your Next Test

1. The mean of the sampling distribution of the sample mean, $\mu_{\bar{x}}$, is equal to the population mean, $\mu$, regardless of the sample size or the shape of the population distribution.
2. The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, with a mean equal to the population mean, $\mu$.
3. The standard error of the sample mean, $\sigma_{\bar{x}}$, is the standard deviation of the sampling distribution of the sample mean, and it decreases as the sample size increases.
4. The standard error of the sample mean, $\sigma_{\bar{x}}$, is calculated as $\sigma_{\bar{x}} = \sigma / \sqrt{n}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.
5. The Central Limit Theorem and the properties of the sampling distribution of the sample mean are essential for making statistical inferences, such as constructing confidence intervals and conducting hypothesis tests.

Review Questions

• Explain the relationship between the population mean, $\mu$, and the mean of the sampling distribution of the sample mean, $\mu_{\bar{x}}$.
  • The mean of the sampling distribution of the sample mean, $\mu_{\bar{x}}$, is equal to the population mean, $\mu$, regardless of the sample size or the shape of the population distribution. This is a fundamental property of the sampling distribution of the sample mean, which is a crucial concept in statistical inference and the application of the Central Limit Theorem. The fact that $\mu_{\bar{x}} = \mu$ means that the sample mean, $\bar{x}$, is an unbiased estimator of the population mean, $\mu$, and that the expected value of the sample mean is equal to the population mean.
• Describe how the Central Limit Theorem relates to the sampling distribution of the sample mean and the standard error of the sample mean.
  • The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean, $\bar{x}$, approaches a normal distribution, regardless of the shape of the population distribution. This has important implications for the properties of the sampling distribution of the sample mean. Specifically, the Central Limit Theorem guarantees that the mean of the sampling distribution, $\mu_{\bar{x}}$, is equal to the population mean, $\mu$, and that the standard deviation of the sampling distribution, known as the standard error of the sample mean ($\sigma_{\bar{x}}$), decreases as the sample size increases ($\sigma_{\bar{x}} = \sigma / \sqrt{n}$). These properties are essential for making statistical inferences, such as constructing confidence intervals and conducting hypothesis tests.
• Explain how the properties of the sampling distribution of the sample mean, including $\mu_{\bar{x}}$ and $\sigma_{\bar{x}}$, can be used to draw conclusions about the population mean, $\mu$.
  • The properties of the sampling distribution of the sample mean, $\bar{x}$, are crucial for making statistical inferences about the population mean, $\mu$. Since the mean of the sampling distribution, $\mu_{\bar{x}}$, is equal to the population mean, $\mu$, the sample mean, $\bar{x}$, can be used as an unbiased estimator of the population mean. Additionally, the standard error of the sample mean, $\sigma_{\bar{x}}$, provides a measure of the variability or precision of the sample mean as an estimate of the population mean. By understanding the properties of the sampling distribution, particularly the Central Limit Theorem, researchers can construct confidence intervals and conduct hypothesis tests to draw conclusions about the population mean based on sample data.

Related terms

Sampling Distribution:

The probability distribution of a statistic, such as the sample mean, obtained by repeatedly drawing samples from a population and calculating the statistic for each sample.

Central Limit Theorem:

A fundamental theorem in probability and statistics that states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.

Standard Error:

The standard deviation of the sampling distribution of a statistic, such as the sample mean, which provides a measure of the variability or precision of the statistic as an estimate of a population parameter.