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THE DISTRIBUTION OF SAMPLE MEANS Chapter 7 THE DISTRIBUTION OF SAMPLE MEANS © aSup-2007 1 THE DISTRIBUTION OF SAMPLE MEANS 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 2 THE DISTRIBUTION OF SAMPLE MEANS 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 3 THE DISTRIBUTION OF SAMPLE MEANS 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 +2.00 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 4 THE DISTRIBUTION OF SAMPLE MEANS 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 5 THE DISTRIBUTION OF SAMPLE MEANS THE DISTRIBUTION OF SAMPLE MEANS 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 6 THE DISTRIBUTION OF SAMPLE MEANS COMBINATION n! nCr = r! (n-r)! 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 © aSup-2007 7 THE DISTRIBUTION OF SAMPLE MEANS 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 8 THE DISTRIBUTION OF SAMPLE MEANS 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 σM Remember the sampling error There typically will be some error between the sample and the population The σM measures exactly how much difference should be expected on average between sample mean M and the population mean μ © aSup-2007 9 THE DISTRIBUTION OF SAMPLE MEANS The MAGNITUDE of THE σM Determined by two factors: ○ The size of the sample, and ○ The standard deviation of the population from which the sample is selected M © aSup-2007 n 10 THE DISTRIBUTION OF SAMPLE MEANS LEARNING CHECK 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? © aSup-2007 11 THE DISTRIBUTION OF SAMPLE MEANS 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 12 THE DISTRIBUTION OF SAMPLE MEANS 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? σM = σ √n = 25 M-μ z= σ M = 1.6 z = 1.6 Area C p = .0548 © aSup-2007 13 THE DISTRIBUTION OF SAMPLE MEANS LEARNING CHECK 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 © aSup-2007 14