© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   1 Chapter 7 THE DISTRIBUTION OF SAMPLE MEANS

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   2 Imagine an urn filled with balls…  Two-third of the balls are one color, and the remaining one-third are a second color  One individual select 5 balls from the urn and finds that 4 are red and 1 is white  Another individual select 20 balls and finds that 12 are red and 8 are white  Which of these two individuals should feel more confident that the urn contains two third red balls and one-third white balls than the opposite?

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   3 The CORRECT ANSWER  The larger sample gives a much stronger justification for concluding that the balls in the urn predominantly red  With a small number, you risk obtaining an unrepresentative sample  The larger sample is much more likely to provide an accurate representation of the population  This is an example of the law of large number which states that large samples will be representative of the population from which they are selected

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   4 OVERVIEW  Whenever a score is selected from a population, you should be able to compute a z-score  And, if the population is normal, you should be able to determine the probability value for obtaining any individual score  In a normal distribution, a z-score of correspond to an extreme score out in the tail of the distribution, and a score at least large has a probability of only p =.0228

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   5 OVERVIEW  In this chapter we will extend the concepts of z-scores and probability to cover situation with larger samples  We will introduce a procedure for transforming a sample mean into a z-score

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   6  Two separate samples probably will be different even though they are taken from the same population  The sample will have different individual, different scores, different means, and so on  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

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   7 COMBINATION  Consider a population that consist of 5 scores: 3, 4, 5, 6, and 7  Mean population = ?  Construct the distribution of sample means for n = 1, n = 2, n = 3, n = 4, n = 5 nCr = n! r! (n-r)!

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   8 SAMPLING DISTRIBUTION  … is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population 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

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   9 The STANDARD ERROR OF MEAN  The value we will be working with is the standard deviation for the distribution of sample means, and it called the σ M  Remember the sampling error  There typically will be some error between the sample and the population  The σ M measures exactly how much difference should be expected on average between sample mean M and the population mean μ

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   10 The MAGNITUDE of THE σ M  Determined by two factors: ○ The size of the sample, and ○ The standard deviation of the population from which the sample is selected

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   11  A population of scores is normal with μ = 100 and σ = 15 ○ Describe the distribution of sample means for samples size n = 25 and n =100  Under what circumstances will the distribution of samples means be a normal shaped distribution? LEARNING CHECK

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   12 PROBABILITY AND THE DISTRIBUTION OF SAMPLE MEANS  The primary use of the standard distribution of sample means is to find the probability associated with any specific sample  Because the distribution of sample means present the entire set of all possible Ms, we can use proportions of this distribution to determine probabilities

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   13 EXAMPLE  The population of scores on the SAT forms a normal distribution with μ = 500 and σ = 100. If you take a random sample of n = 16 students, what is the probability that sample mean will be greater that M = 540? σ M = σ √n√n = 25 z = M - μ σMσM = 1.6 z = 1.6  Area C  p =.0548

© aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS   14  The population of scores on the SAT forms a normal distribution with μ = 500 and σ = 100. We are going to determine the exact range of values that is expected for sample mean 95% of the time for sample of n = 25 students See Example 7.3 on Gravetter’s book page 207 LEARNING CHECK