SECTION 5.4 Sampling Distributions and the Central Limit Theorem 263 The Central Limit Theorem The Central Limit Theorem forms the foundation for the inferential branch of statistics. This theorem describes the relationship between the sampling distribution of sample means and the population from which the samples are taken. The Central Limit Theorem is an important tool that provides the information you will need to use sample statistics to make inferences about a population mean. 1. If random samples of size n, where n Ú 30, are drawn from any population with a mean m and a standard deviation s, then the sampling distribution of sample means approximates a normal distribution. The greater the sample size, the better the approximation. (See figures for “Any Population Distribution” below.) 2. If random samples of size n are drawn from a population that is normally distributed, then the sampling distribution of sample means is normally distributed for any sample size n. (See figures for “Normal Population Distribution” below.) In either case, the sampling distribution of sample means has a mean equal to the population mean. mx = m Mean of the sample means The sampling distribution of sample means has a variance equal to 1 n times the variance of the population and a standard deviation equal to the population standard deviation divided by the square root of n. s 2 x = s 2 n Variance of the sample means sx = s2 n Standard deviation of the sample means The Central Limit Theorem Recall that the standard deviation of the sampling distribution of the sample means, sx, is also called the standard error of the mean. 1. Any Population Distribution μ σ x Mean Standard deviation Distribution of Sample Means, n Ú 30 μ μ x x = Standard deviation of the sample means Mean σ σ x = n 2. Normal Population Distribution x μ Mean σ Standard deviation Distribution of Sample Means (any n) μ μ x x = Standard deviation of the sample means Mean σ σ x = n Study Tip The distribution of sample means has the same mean as the population. But its standard deviation is less than the standard deviation of the population. This tells you that the distribution of sample means has the same center as the population, but it is not as spread out. Moreover, the distribution of sample means becomes less and less spread out (tighter concentration about the mean) as the sample size n increases.
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