Distributions of Sample Means

z-scores for Samples  What do I mean by a “z-score” for a sample? This score would describe how a specific sample is related to all other possible samples selected from a population This score would describe how a specific sample is related to all other possible samples selected from a population  How do we account for the difference between a given sample and the population as a whole? Sampling error is the discrepancy, or amount of error, between a sample statistic and its corresponding population parameter Sampling error is the discrepancy, or amount of error, between a sample statistic and its corresponding population parameter

What is the Distribution of Sample Means?  If we collected every possible sample from a population and created a frequency histogram from those samples, what would it look like? It starts to look normal It starts to look normal  The distribution of sample means is the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population.

How Does a Distribution of Samples Differ From Other Distributions We Have Looked At?  A sampling distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population Thus, the distribution of sample means is a sampling distribution of M Thus, the distribution of sample means is a sampling distribution of M

What Does the Sampling Distribution of M Look Like?  Where will the sample means collect? Around the population mean Around the population mean  What will the shape of this distribution be? Again, the shape should be normal Again, the shape should be normal  What happens to the accuracy of the statistic (with respect to the parameter) as we increase the n in each sample? The statistic is more representative of the parameter The statistic is more representative of the parameter

Central Limit Theorem  For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will have a mean of μ and a standard deviation of σ/√n and will approach a normal distribution as n approaches infinity. Actually once n hits 30 the distribution is approximately normal Actually once n hits 30 the distribution is approximately normal

What Would You Expect The Mean of the Distribution of Sample Means To Be?  The mean of the distribution of sample means is equal to μ (the population mean) and is called the expected value of M.

What Are The Two Parameters That Describe a Normal Distribution?  For the distribution of sample means… μ (the mean) μ (the mean) This we expect to be roughly equal to the population meanThis we expect to be roughly equal to the population mean σ (the standard deviation) σ (the standard deviation) This we expect to equal the standard error of MThis we expect to equal the standard error of M σ m = σ/√n = √σ 2 /√n = √(σ 2 /n)σ m = σ/√n = √σ 2 /√n = √(σ 2 /n) This is the standard distance between M and μThis is the standard distance between M and μ What happens when n gets very large?What happens when n gets very large?

The Law of Large Numbers  The larger the sample size (n), the more probable it is that the sample mean will be close to the population mean  What happens when n = 1? σ m = σ σ m = σ

SAT Example  Mu = 500  σ = 100  If we select 25 students randomly, what is the S.E.? σ m = 20 σ m = 20

So, What Is The Probability of Having an M > 540?  2.28% for n = 25 (randomly chosen students)  How do we calculate a z-score for sample means? Essentially the same as we did before, except with S.E. instead of σ Essentially the same as we did before, except with S.E. instead of σ z = (M – μ) / σ m z = (M – μ) / σ m