Determine whether the statement is true or false. If it is...

Determine whether the statement is true or false. If it is false, rewrite it as a true statement.
If the size of a sample is at least $30,$ then you can use $z$ -scores to determine the probability that a sample mean falls in a given interval of the sampling distribution.

Key Concepts

Central Limit Theorem

The Central Limit Theorem (CLT) states that the distribution of the sample mean will approach a normal distribution as the sample size increases, typically considered to be 30 or more, regardless of the shape of the underlying population, provided the population has a finite variance. This theorem is the foundation for making approximate inferences about the population mean using normally distributed models when dealing with sufficiently large samples.

Sampling Distribution of the Sample Mean

This concept refers to the probability distribution of all possible sample means from a given population. The sampling distribution of the sample mean is key to understanding how sample means behave and is used in hypothesis testing and the construction of confidence intervals. The distribution becomes approximately normal for large samples due to the Central Limit Theorem.

z-scores vs. t-scores for Inference

A z-score is a type of standard score that measures how many standard deviations an element is from the mean, and it is typically used when the population standard deviation is known. However, when the population standard deviation is unknown and must be estimated from the sample, particularly in smaller samples, the t-distribution (and t-scores) are more appropriate. Even for large samples, if the population variance is not known, statisticians generally prefer t-scores to account for additional uncertainty unless the sample size is very large.

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