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Figure 7.1 (p. 163) - A population of scores μ=5 Table 7.1 (p. 164) All possible random samples of n=2 Figure 7.2 The distribution of sample means is centered around the population mean The central limit theorem: For any population with a 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. 16 14 Pop Sample Mean (n=2) Frequency 12 10 Sample Mean (n=4) Sample Mean (n=8) 8 6 4 2 0 1 2 3 4 5 6 7 8 9 Hours of Sleep What is happening to s as n gets larger? 10 11 12 • The standard deviation of the distribution of sample means is called the standard error of M (σM). This approaches zero as sample size increases. Pop. N = 80 Samples 2 4 8 μ = 6.23 6.73 6.23 6.22 σ = 2.33 1.71 1.02 0.74 1.65 1.17 0.82 CL Theorem Predicts = The values predicted by the central limit theorem are the values that would have been attained if you had taken every possible sample of n=2, 4, and 8. Ours were still close. • A population has a size of N= 4. • What would be the approximate standard deviation of sample means for eighty samples of size (n=4)? • Law of large Numbers: As sample size increases, the sample means (M) will get closer to the population mean. • A population has a size of N= 4. • What would be the approximate standard deviation of sample means for eighty samples of size (n=1)? • Standard error (σM) will always be between 0 and the population standard deviation. – It will be close to zero when sample sizes are very large and equal to the population standard deviation (σ) when sample sizes are very small (n = 1). σM = σ/√ n • The distribution of sample means is used to tell us about the probability associates with a specific sample. – If a sample is drawn from a known population, how likely is it to have a particular mean? Figure 7.5 (p. 171) The population of scores on the SAT has a μ=500 and σ=100. What is the probability of drawing a sample of n=25 with a M > 540? What is σM? Formula for sample mean z should look familiar z = (M-μ)/(σM) • This is what it’s all about!! (almost) 1. You have developed the ultimate SAT prep course! 2. Select 25 High School students at random 3. Have them take your course 4. Have them take the SAT 5. Determine their mean scores and see if they are unusual. 6. Scientists generally consider unusual to be associated with p < 0.05 – This is the core of hypothesis testing • What kind of SAT scores can you expect from a random sample (n=25) 80% of the time? σM = 20 Μ = 500 Figure 7.6 (p. 173) Figure 7.7 (p. 174) Sampling error: Sample means will not perfectly match the population means, but most will be close Figure 7.8 (p. 175) Standard error: Gets smaller as sample sizes get larger Table 7.2 (p. 177) Reporting standard error with sample means Figure 7.9 (p. 177) Figure 7.10 (p. 177) Figure 7.11 (p. 179) Figure 7.12 (p. 180) A sample of 25 treated rats has a 95% chance of being between 392.16 and 407.84