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An introduction to sampling distributions, including the central limit theorem and the law of large numbers. It covers the concept of population and sample mean, standard deviation, and the relationship between sample size and standard error. The document also includes sample problems and examples to illustrate the concepts.
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Sampling Distributions
What is a sampling distribution? Population μ, σ Sample 1 M1, s 1 Sample 5 M5, s 5 Sample 2 M2, s 2 Sample 3 M3, s 3 Sample 4 M4, s 4 Sample 7 M 7 , s 7 Sample 6 M6, s 6 Population μ, σ Sample M, s Treatment
All the possible samples, n = 2 from the population of scores 2, 4, 6, 8 μ = 5, σ = 2.
The Central Limits Theorem
mean equal to the population mean.
The Law of Large numbers As the sample size ( n ) increases, the more probable it is that the sample mean ( M )
Application of the Law of Large Numbers - The Obama Effect K. Schmidt & Nosek (2010) IAT (implicit association test) Over time (2.5 year period) Large sample ( n = 479,405)
Fig. 1. Data points indicate the mean daily IAT effects from September 28, 2006 to May 11, 2009 with 7-days moving average in black. Light-gray line indicates a moving average of the number of daily news articles containing “Obama” in the Lexis–Nexis database. K. Schmidt & Nosek (2010) Fig. 2. Mean daily IAT effects by daily sample size with lines representing ±2 standard errors of the grand mean by sample size. K. Schmidt & Nosek (2010) Grand M =.
σ (^) M = σ √ n
Population μ=3400, σ= Smoking Sample M = 3200 Sample problems Cereal Boxes: Machine designed to fill boxes with μ = 32 oz, σ = 2 oz Sample: n = 16, what is the probability that the sample mean would be less than 31 if the machine were working properly? Standard error and reporting results.
Standard error and reporting results.
