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Probability and the Sampling Distribution Quantitative Methods in HPELS 440:210

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Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

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Introduction Recall: Any raw score can be converted to a Z-score Provides location relative to µ and Assuming NORMAL distribution: Proportion relative to Z-score can be determined Z-score relative to proportion can be determined Previous examples have looked at single data points Reality most research collects SAMPLES of multiple data points Next step convert sample mean into a Z- score Why? Answer probability questions

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Introduction Two potential problems with samples: 1. Sampling error Difference between sample and parameter 2. Variation between samples Difference between samples from same taken from same population How do these two problems relate?

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Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

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Distribution of Sample Means Distribution of sample means = sampling distribution is the distribution that would occur if: Infinite samples were taken from same population The µ of each sample were plotted on a FDG Properties: Normally distributed µ M = the “mean of the means” M = the “SD of the means” Figure 7.1, p 202

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Distribution of Sample Means Sampling error and Variation of Samples Assume you took an infinite number of samples from a population What would you expect to happen? Example 7.1, p 203

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Assume a population consists of 4 scores (2, 4, 6, 8) Collect an infinite number of samples (n=2)

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Total possible outcomes: 16 p(2) = 1/16 = 6.25%p(3) = 2/16 = 12.5% p(4) = 3/16 = 18.75%p(5) = 4/16 = 25% p(6) = 3/16 = 18.75%p(7) = 2/16 = 12.5% p(8) = 1/16 = 6.25%

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Central Limit Theorem For any population with µ and , the sampling distribution for any sample size (n) will have a mean of µ M and a standard deviation of M, and will approach a normal distribution as the sample size (n) approaches infinity If it is NORMAL, it is PREDICTABLE!

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Central Limit Theorem The CLT describes ANY sampling distribution in regards to: 1. Shape 2. Central Tendency 3. Variability

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Central Limit Theorem: Shape All sampling distributions tend to be normal Sampling distributions are normal when: The population is normal or, Sample size (n) is large (>30)

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Central Limit Theorem: Central Tendency The average value of all possible sample means is EXACTLY EQUAL to the true population mean µ M = µ If all possible samples cannot be collected? µ M approaches µ as the number of samples approaches infinity

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µ = 2+4+6+8 / 4 µ = 5 µ M = 2+3+3+4+4+4+5+5+5+5+6+6+6+7+7+8 / 16 µ M = 80 / 16 = 5

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Central Limit Theorem: Variability The standard deviation of all sample means is denoted as M M = /√n Also known as the STANDARD ERROR of the MEAN (SEM)

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SEM Measures how well statistic estimates the parameter The amount of sampling error between M and µ that is reasonable to expect by chance Central Limit Theorem: Variability

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SEM decreases when: Population decreases Sample size increases Other properties: When n=1, M = (Table 7.2, p 209) As SEM decreases the sampling distribution “tightens” (Figure 7.7, p 215) M = /√n

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Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

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Probability Sampling Distribution Recall: A sampling distribution is NORMAL and represents ALL POSSIBLE sampling outcomes Therefore PROBABILITY QUESTIONS can be answered about the sample relative to the population

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Probability Sampling Distribution Example 7.2, p 209 Assume the following about SAT scores: µ = 500 = 100 n = 25 Population normal What is the probability that the sample mean will be greater than 540? Process: 1. Draw a sketch 2. Calculate SEM 3. Calculate Z-score 4. Locate probability in normal table

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Step 1: Draw a sketch Step 2: Calculate SEM SEM = M = /√n SEM = 100/√25 SEM = 20 Step 3: Calculate Z-score Z = 540 – 500 / 20 Z = 40 / 20 Z = 2.0 Step 4: Probability Column C p(Z = 2.0) = 0.0228

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Agenda Introduction Distribution of Sample Means Probability and the Distribution of Sample Means Inferential Statistics

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Looking Ahead to Inferential Statistics Review: Single raw score Z-score probability Body or tail Sample mean Z-score probability Body or tail What’s next? Comparison of means experimental method

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Textbook Assignment Problems: 13, 17, 25 In your words, explain the concept of a sampling distribution In your words, explain the concept of the Central Limit Theorum

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