Statistical Sciences 1024A/B Chapter 15: Chapter 15 - Sampling Distributions

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Statistical Sciences
Statistical Sciences 1024A/B
Lori Murray

February 28, 2017 Chapter 15: Sampling Distribution 15.1 Parameters and Statistics A parameter is a number that describes some aspect of the population. Parameters are usually unknown! A statistic is a number that is computer from the sample and is often used to estimate the unknown parameter Example: is a population parameter, x is a statistic, x is an estimate of The population parameter is called the true population mean The statistic is called the sample mean 15.2 Statistical Estimation and the Law of Large Numbers The population mean is rarely exactly equal to the population mean. However, if we keep taking large and large samples, the statistic is guaranteed to get closer and closer to the true population parameter. Eventually we will estimate the population mean very accurately This is called the Law of Large Numbers: as the number of randomlydrawn observations (n) in a sample increases, the mean of the sample gets closer and closer to the population mean. 15.3 Sampling Distributions Variability: the sample mean changes sample by sample with each SRS of size n Stimulation sampling distribution 1. Take a random sample of size n from a population 2. Calculate the sample average x of the n observations 3. Repeat the procedure m times 4. Make a histogram of the m sample averages. Examine the shape, centre and spread. 5. This shows the approximate sampling distribution of the sample average x The population distribution of a variable is the distribution of values of the variable among all the individuals in the population (when x is height) The sampling distribution of a statistic is the distribution of values taken by the statistic in al possible samples of the same size from the same population (when x is the average hight) 15.4 The Sampling Distribution of X The sampling distribution of a statistic is the distribution of all possible values taken by the statistic when all possible samples of a fixed size n are taken from the population. It is a theoretical idea we cannot actually build it The sampling distribution of a statistic is the probability distribution of that statistic Note: When sampling randomly from a given population, The law of large numbers describes what happens when the sample size n is gradually increased
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