Sampling Distributions and the Central Limit Theorem 5.4 SECTION 5.4 Sampling Distributions and the Central Limit Theorem 261 Sampling Distributions The Central LimitTheorem Probability and the Central LimitTheorem What You Should Learn How to find sampling distributions and verify their properties How to interpret the Central Limit Theorem How to apply the Central Limit Theorem to find the probability of a sample mean Sampling Distributions In previous sections, you studied the relationship between the mean of a population and values of a random variable. In this section, you will study the relationship between a population mean and the means of random samples taken from the population. A sampling distribution is the probability distribution of a sample statistic that is formed when random samples of size n are repeatedly taken from a population. If the sample statistic is the sample mean, then the distribution is the sampling distribution of sample means. Every sample statistic has a sampling distribution. DEFINITION Consider the Venn diagram below. The rectangle represents a large population, and each circle represents a random sample of size n. Because the sample entries can differ, the sample means can also differ. The mean of Random Sample 1 is x1; the mean of Random Sample 2 is x2; and so on. The sampling distribution of the sample means for samples of size n for this population consists of x1, x2, x3, and so on. If the samples are drawn with replacement, then an infinite number of samples can be drawn from the population. Population with Mean and Standard Deviation Random Sample 1, x 1 μ σ Random Sample 2, x 2 Random Sample 3, x 3 Random Sample 4, x 4 Random Sample 5, x 5 1. The mean of the sample means mx is equal to the population mean m. mx = m 2. The standard deviation of the sample means sx is equal to the population standard deviation s divided by the square root of the sample size n. sx = s2 n The standard deviation of the sampling distribution of the sample means is called the standard error of the mean. Properties of Sampling Distributions of Sample Means Study Tip Sample means can vary from one another and can also vary from the population mean. This type of variation is to be expected and is called sampling error. You will learn more about this topic in Section 6.1.
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