PSY 201 Professor Greg Francis Lecture 17 Sampling distribution of the mean Marvel at my predictive powers SAMPLING DISTRIBUTION suppose we have a population with a mean and a standard deviation the different X i sample means that are calculated will be related to each other because they all come from the same population which has a population mean of suppose we take a sample from the population and calculate a sample mean X 1 suppose we take a different sample from the population and calculate a sample mean X 2 suppose we take a different sample from the population and calculate a sample mean X 3 we can consider a distribution of the sample means same idea as distribution of sum of dice roles 0 03 Frequency Introduction to Statistics in Psychology 0 02 0 01 0 0 20 40 60 80 100 120 Sample Mean 2 3 DISTRIBUTION HOW TO GET EXAMPLE this distribution involves frequencies of means rather than frequencies of scores we said last time that if we take large enough sample sizes n we know the following about the sampling distribution of the mean let s create a sampling distribution for most of inferential statistics we do not deal with the frequency distribution of scores A sampling distribution is the distribution of values of the statistic under consideration from all possible samples of a given size currently the statistic is the sample mean X 4 1 The distribution is normal 2 The mean of the distribution is the same as the mean of the population two things 1 Write down your height in inches on the papers going around the room 2 Sample the height of 10 people close to you 3 The standard error of the mean is X n where is the standard deviation of the population 5 6 EXAMPLE EXAMPLE EXAMPLE I ll sit down and calculate the population mean and standard deviation OK I get with this information I can predict the frequency of sample means each of you calculated you calculate the sample mean X for the 10 scores you have Xi X 10 Xi N Xi 2 Xi 2 N N I predict that most of you calculated sample means close to X moreover I predict that the distribution of sample means is normal lets plot the sample means you calculated someone help me enter in scores in a spreadsheet 7 8 9 EXAMPLE EXAMPLE SAMPLE SIZE finally I could calculate the standard deviation of the sampling distribution as Let s calculate the standard deviation of the sampling distribution as If we have time do it all again but select new samples of size 5 X 2 X N and I predict that it will be very close to 10 10 X 2 X 2 N N I predicted that it will be very close to 10 X 11 12 EXAMPLE CONCLUSIONS NEXT TIME so we have two ways of finding the sampling distribution of the mean sampling distribution looks like a normal distribution hypothesis testing 1 gather lots of samples calculate means and standard deviations 2 calculate mean and standard deviation of the population use central limit theorem methods of calculating mean and standard deviation if and are known samples must be randomly selected using the sampling distribution in what looks to be reverse null hypothesis and Why I don t use herbal medicines the central limit theorem allows us to do inferential statistics without it what we are going to talk about next lecture would be virtually impossible 13 14 15
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