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Unformatted text preview: m GSU (Getting Smart Univ) and collect data on how many hours they study for week. Each time you may get a different sample, so the sample mean you get each time is different. Drawing many samples, you’ll get a distribution for the mean study hours. 2. Let X be the outcome of flipping a coin. Heads = 1, tails = 0, so X can take on 0 or 1 with some probability ≠ 1. Assume that the coin is fair, then prob (X = 0) = prob (X = 1) = 0.5, so by definition, X is a r.v. Flip the coin 10 times, then you get a sample of 10 observations. Suppose you get 6 heads, and 4 tails. The sample mean X = 0.6 This sample mean is a r.v.. The sample mean might have come to be 0.4, or 0.7, or 0.01, etc… Intuition tells us that each value comes with a different 9 probability. For example, there is a low chance (less than 1/210) for it to be zero, and a very high chance for it to be 0.6 (about .205), so X has a distribution. The one specific mean that you obtain from your sample is one observation of the r.v. X . Suppose you do 500 experiments, and each time you flip the coin 10 times, you will have 500 observations on X . An estimate is one realization of an estimator, i.e., a specific value of the estimator based on a particular sample. Example: If you calculate sample mean, sample variance etc. based on a particular sample you have, these values are estimates. Sampling distribution of an estimator: The prob. distribution for an estimator is the estimator’s sampling distribution Comments: • Lecture 1 briefly describes how to estimate Y = α + β X + u . • Important‐‐those estimators (a rule about how to estimate an unknown parameter, oftentimes it is an explicit formula) are all r.v.’s, and hence have distributions. This is the foundation for doing statistical inference or hypothesis testing after estimation (discussed later in our course). , in a simple linear model The Normal and Related Distributions If a r.v., X has a normal dis...
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