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5.4SamDistCLT.notebook February 24, 2016 Properties of Sampling Distributions of Sampling Means: The Histogram represents the distribution of many sample means taken from the population The Density curve represents the distribution of the actual population The bigger the n, the closer the mean of the population is to the mean of the sampling distribution. 1 5.4SamDistCLT.notebook February 24, 2016 "The Central Limit Theorem" n=100 n=500 n=1500 The blue curve is the density curve of some sampling distribution of means. The red curve is the normal curve with mean and standard deviation that represents the population. As you increase n, the density curve looks more and more normal. Or put another way: it's shaped more like a classicBELL CURVE! And, is more like the actual population! 2 5.4SamDistCLT.notebook February 24, 2016 5.4 Sampling Distributions and the Central Limit Theorem The Central Limit Theorem Is the foundation on which the Inferential Branch of Statistics is built ≥ 30, are drawn from any population with mean =µ and standard deviation =σ , If samples of size n then the sampling distribution of the sample means approximates a normal distribution. The greater the sample size, the better the approximation, and the more sample means in the sampling distribution, the better as well. The Distribution of Sample Means will always be normal regardless of whether or not the population is normally distributed! 3 5.4SamDistCLT.notebook February 24, 2016 Properties of Sampling Distributions of Sampling Means: 1. 2. The mean of the sampling distribution will be the same value as the population mean The standard deviation of the sampling distribution will be equal to the population standard deviation divided by the square root of n Note: Another name for the standard deviation of the sampling distribution of the sample means is called the STANDARD ERROR OF THE MEAN 4 5.4SamDistCLT.notebook February 24, 2016 The mean of the sampling distribution will be the same value as the population mean The standard deviation of the sampling distribution will be equal to the population standard deviation divided by the square root of n standard deviation "standard error of the mean" 30 Normal Sample Size(n) approximation equal 5 5.4SamDistCLT.notebook February 24, 2016 6