The __________ of the sample mean, x̄, is the probability distribution of all possible values of the random variable x̄ computed from a sample of size n from a population with mean μ and standard deviation σ.

Verified step by step guidance

1. Identify the key concept described: it refers to the distribution of the sample mean \( \bar{x} \) when taking all possible samples of size \( n \) from a population with mean \( \mu \) and standard deviation \( \sigma \).
2. Recall that this concept is known as the sampling distribution of the sample mean, which is a fundamental idea in inferential statistics.
3. Understand that the sampling distribution of \( \bar{x} \) has its own mean and standard deviation, where the mean is equal to the population mean \( \mu \), and the standard deviation (called the standard error) is \( \frac{\sigma}{\sqrt{n}} \).
4. Recognize that this distribution describes the variability of the sample mean \( \bar{x} \) across different samples, and it is crucial for constructing confidence intervals and hypothesis tests.
5. Summarize that the blank in the problem should be filled with 'sampling distribution' to complete the definition accurately.

Key Concepts

Sampling Distribution

The sampling distribution is the probability distribution of a statistic, such as the sample mean, calculated from all possible samples of a specific size drawn from a population. It describes how the sample mean varies from sample to sample.

Sample Mean (x̄)

The sample mean is the average value of observations in a sample. It serves as an estimate of the population mean (μ) and is a random variable because it varies depending on the sample selected.

Population Parameters (μ and σ)

Population parameters include the mean (μ) and standard deviation (σ), which describe the entire population's characteristics. These parameters influence the shape and spread of the sampling distribution of the sample mean.