Under what conditions is the sampling distribution of x̄ normal?

Verified step by step guidance

1. Understand that the sampling distribution of the sample mean \(\bar{x}\) describes the distribution of sample means obtained from all possible samples of a given size from a population.
2. Recall the Central Limit Theorem (CLT), which states that the sampling distribution of \(\bar{x}\) approaches a normal distribution as the sample size \(n\) becomes large, regardless of the population's distribution shape.
3. Identify the key conditions for the sampling distribution of \(\bar{x}\) to be approximately normal: either the population itself is normally distributed, or the sample size \(n\) is sufficiently large (commonly \(n \geq 30\) is used as a rule of thumb).
4. Note that if the population distribution is normal, then the sampling distribution of \(\bar{x}\) is exactly normal for any sample size \(n\).
5. Summarize that the sampling distribution of \(\bar{x}\) is normal if either the population is normal or the sample size is large enough for the Central Limit Theorem to apply.

Key Concepts

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean (x̄) is the probability distribution of all possible sample means from a population. It describes how the sample mean varies from sample to sample and is fundamental for making inferences about the population mean.

Central Limit Theorem (CLT)

The Central Limit Theorem states that, for a sufficiently large sample size, the sampling distribution of the sample mean approaches a normal distribution regardless of the population's shape. This allows normal approximation even when the population is not normal.

Conditions for Normality of Sampling Distribution

The sampling distribution of x̄ is normal if the population is normal or if the sample size is large (usually n ≥ 30). For small samples from non-normal populations, the distribution may not be normal, so normality depends on population shape and sample size.