psy 1950 chance, probability, and sampling september 24, 2008
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PSY 1950 Chance, Probability, and Sampling September 24, 2008. vs. Probability: Perspectives. Analytic: possible outcomes theoretical Relative frequency: past performance empirical Subjective: belief Psychological. Probability: Applications. Is data-generating process random? Yes - PowerPoint PPT PresentationTRANSCRIPT
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PSY 1950Chance, Probability, and Sampling
September 24, 2008
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vs
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Probability: Perspectives• Analytic: possible outcomes
– theoretical
• Relative frequency: past performance– empirical
• Subjective: belief– Psychological
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Probability: Applications• Is data-generating process random?– Yes
•Debunk or study psychological bias to see patterns – Chance is lumpy, brains are pattern-detectors
– e.g., bushy tiger vs. tigery bush– e.g., hot hand in basketball
•Debunk patterns vs. affirming randomness
– No•Pattern demands explanation
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Probability as Area
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Probability as Area
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NORMDIST(x, mean, standard_dev, cumulative) NORMSDIST(z) NORMINV(probability, mean, standard_dev) NORMSINV(probability)
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The Normal Distribution
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Why The Normal Distribution?
• Variables are often (or are often assumed to be) normally distributed in population– e.g., Quetelet’s (1835) measurements of heights– e.g., IQ scores
• Errors are often (or are often assumed to be) normally distributed– Sampling error (cf. terminology: normal, error)
• Assuming (approximate) normality allows inference
• Assuming normality enables parametric statistics– Normal distributions have “amazing” mathematical properties• Linear combinations of scores from two normally distributed variables are themselves normally distributed!
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Binomial Distribution• Two possible outcomes• Constant outcome probability• Trial-to-trial independence• If pn and qn 10:
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Binomial Distribution
http://www.socr.ucla.edu/htmls/SOCR_Distributions.html
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Example: Missing girls• 1049 males are born in the world for every 1000 females. From 2000-2005, there were approximately 17 million children born in China, approximately 7,730,000 of whom were female. What are the odds that this is by chance?
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Sampling• Overconfidence abhors uncertainty
– Law of small numbers– Correspondence bias– Overconfidence bias
• Bias– e.g., Who likes statistics?
•Characteristics of sample: representativeness
•Measurement of sample: response, non-response
– e.g., How many children in your family•Sampling unit: people vs. families
– Only family with children are represented– Families with multiples children are overrepresented
– How would you obtain a representative sample?
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Sampling Terms• Sampling error
– Variability of a statistic from sample to sample due to chance
• Sampling distribution– The distribution of a statistic over repeated sampling from a specified population
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X1 X2 X3 X4 X5
5 8 9 12 6
5 7 8 9 6
7 8 7 7 6
7 7 7 7 7
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Standard Error (of the Mean)
• Standard deviation of the distribution of sample means
• Estimate of how well, on the average, a sample mean estimates its population mean
• Expected error• Depends on sample size
– Law of large numbers
• Depends on population variability• Is not standard deviation of sample (s) or distribution ()
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Standard Error vs Standard Deviation
• Standard deviation– Descriptive statistic– Measure of dispersion– Standard distance between scores and their mean
– Does not depend on sample size
• Standard error (of the mean)– Inferential statistic*– Measure of precision– Standard distance between sample means and population mean
– Depends on sample size
• Standard error is type of standard deviation
• Equal when n = 1
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Central Limit Theorem• For any population with mean and standard deviation , the distribution of sample means for sample size n will:– have a mean of – have a standard deviation of /√n – will approach a normal distribution as n approaches infinity
• Valid for any population– Explains normal distribution of many psychological variables
• Distribution of sample means approaches normal distribution very quickly (n 30)
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Central Limit Theorem
Sampling distribution (CLT) experiment
http://www.socr.ucla.edu/htmls/SOCR_Experiments.html
• Examine mean, standard deviation, skewness, and kurtosis of sample mean
• Try different sample sizes• Try other sample statistics (e.g.,
variance)• Sampling from different (e.g.,
Poisson) distributions
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