# Stat400 05.3-5.4 lecture notes - Stat 400 section 5.3-5.4...

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Stat 400, section 5.3-5.4 Sampling Distributions & the Central Limit Theorem notes by Tim Pilachowski If you haven’t done it yet, go to the Stat 400 page and download the handout 5.4 supplement Central Limit Theorem. The homework (both practice and hand-in) homework for section 5.4 will be from that supplement. From the previous Lecture, 5.3a Populations and Samples: Random variables X1, X2, …,Xnform a (simple) random sample of size nif they meet two (important) requirements: 1. The Xi’s are independent random variables. 2. Every Xihas the same probability distribution. Data is collected from the sample, i.e., the random variables X1, X2, …,Xneach receive values x1, x2, …,xn.These values are used to calculate sample statistics. The sample statistics we’ll be most interested in are: 1. The sample total T0= X1 +X2 ++Xn. 2. The sample mean 01TXn*=. (The calculated sample mean is symbolized by x.) 3. The sample variance ()=--*=niinXXS12112. (The calculated sample variance is symbolized by s2.) Note that T0, Xand S2are themselves random variables. Sections 5.3 and 5.4 focus on what the probability distributions of these random variables look like, and what they can tell us about the overarching population’s distribution. Theory Probability models exist in a theoretical world where everything is known. If you constructed every possible sample of a specified size nfrom a given population (Example 1 in the supplement), or were able to toss a coin an infinite number of times (Example 2 in the supplement), you would create what statisticians call a sampling distribution. In the Examples below, we’ll use a hypothetical population Ψconsisting of the numbers 10, 20, 30, 40 and 50. The parameter and statistic we’ll consider first is the mean. Example A-1: Calculate the mean and standard deviation of a population Ψwhich consists of elements from the set {10, 20, 30, 40, 50} with probabilities given in the table below.