True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.


A sampling distribution is normal only when the population is normal.

Verified step by step guidance

1. Understand the concept of a sampling distribution: A sampling distribution is the probability distribution of a statistic (e.g., the sample mean) obtained from a large number of samples drawn from a population.
2. Recall the Central Limit Theorem (CLT): The CLT states that the sampling distribution of the sample mean will be approximately normal if the sample size is sufficiently large (typically n ≥ 30), regardless of the population's distribution.
3. Identify the condition when the sampling distribution is normal: If the population itself is normal, then the sampling distribution of the sample mean will also be normal, regardless of the sample size.
4. Evaluate the given statement: The statement 'A sampling distribution is normal only when the population is normal' is false because the sampling distribution can also be approximately normal for non-normal populations if the sample size is large enough, as per the CLT.
5. Rewrite the statement as true: 'A sampling distribution is normal if the population is normal, or it is approximately normal for large sample sizes due to the Central Limit Theorem.'

Key Concepts

Sampling Distribution

A sampling distribution is the probability distribution of a statistic (like the sample mean) obtained from a large number of samples drawn from a specific population. It describes how the statistic varies from sample to sample and is crucial for understanding the behavior of estimators in statistics.

Central Limit Theorem

The Central Limit Theorem states that, regardless of the population's distribution, the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, typically when the sample size is 30 or more. This theorem is fundamental in inferential statistics, allowing for normal approximation in hypothesis testing.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. Many statistical methods assume normality, and understanding this concept is essential for interpreting results and making inferences about populations based on sample data.