module 20 distribution of sample means one of 12 how sample means vary in random samples so in this unit we work with quantitative variables so the statistics and parameters will be means instead of proportions We Begin this module with a discussion of the sample distribution of sample means our goal is to understand how sample means vary when we select random samples right from a population with a known mean we did the same type of thinking previously with sample proportions to understand the distribution of sample proportions ultimately we will develop a probability model based on the distribution of sample means we use a probability model with an actual sample mean to test a claim about a population mean or to estimate a population mean so example birth weights the World Health Organization who monitors many variables to assess a population's overall health one of these variables is birth weight birth weight is a quantitative variable we can calculate the mean birth weight for a population suppose that babies in a town had a mean birth weight of 3,500 G in 2005 this year a random sample of nine babies has a mean weight of 3,400 G the 3500 is a parameter from a population we use the Greek letter mu to represent it so mu equals 3,500 G the 3,400 is a statistic from a sample so we write xar equal 3,400 G obviously this sample weighs less on average than the population of babies in the town a decrease in the town's mean birth weight could indicate a decline in overall health of the town but does this sample give strong evidence that the town's mean birth weight is less than 3,500 gram this year to answer this question we need to understand how much the means from random samples vary would a sample be likely or unlikely to have a mean birth weight of 3,400 gr if the mean weight of all babies is 3,500 G so we outline this investigation in the following diagram so as before the logic of inference is the same begin with the population with mean of 3,500 G so here's a population right there okay and take random samples of nine babies at a time so here's one random sample and here's the mean for that sample here's a different random sample of nine babies different random sample of nine babies Etc right if a sample mean of 3,400 right is likely to occur when sampling from a population with mean of 3,500 then this sample could have come from a population with a mean of 3,500 the evidence from the sample there is not strong enough to reject the idea that the mean is equal to 3,500 and this is Illustrated here where you know this is the actual sample that we got and the sample from the population has a mean so those nine babies have a mean uh birth weight of 3,5 400 excuse me and the question is does this fit with the other samples that are um collected or are sampled from the population of babies in the town so if a sample mean of 3,400 is unlikely when sampling from a samp from a population with a mean of 3,500 then the sample provides evidence that the mean weight for all babies in the population is less than 3,500 likely or unlikely it depends on how much the sample means vary we need to investigate the sampling distribution of sample means