Question

which of the following conditions must be met in order to make a statistical inference about a population based on a sample if the sample does not come from a normally distributed population?
○ $\mu \geq 30$
○ $\bar{x} \geq 30$
○ $n \geq 30$
○ $n \geq 30$

Explanation:

Step1: Recall Central Limit Theorem

The Central Limit Theorem (CLT) states that for a sample from a non - normally distributed population, when the sample size \(n\) is large enough (usually \(n\geq30\)), the sampling distribution of the sample mean \(\bar{x}\) is approximately normally distributed. This allows us to make statistical inferences about the population even when the population is not normally distributed.

• The symbol \(\mu\) represents the population mean, and the condition for \(\mu\) being greater than or equal to 30 has no relation to making inferences from a non - normal population.
• \(\bar{x}\) is the sample mean, and the size of the sample mean itself (whether \(\bar{x}\geq30\)) is not the relevant condition for inference from a non - normal population.
• \(N\) usually represents the population size, and the population size being greater than or equal to 30 is not the key condition for making inferences from a sample of a non - normal population.

Step2: Identify the correct option

Based on the Central Limit Theorem, when the sample size \(n\geq30\), we can approximate the sampling distribution of the sample mean as normal and make statistical inferences about the population even if the population is not normally distributed. So the correct condition is \(n\geq30\).

Answer:

\(n\geq30\) (the option with \(n\geq30\))