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Statistics for the Social Sciences

Statistics for the Social Sciences. Psychology 340 Fall 2006. Sampling distribution. Outline. Review 138 stuff: What are sample distributions Central limit theorem Standard error (and estimates of) Test statistic distributions as transformations. How do we determine this?.

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Statistics for the Social Sciences

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  1. Statistics for the Social Sciences Psychology 340 Fall 2006 Sampling distribution

  2. Outline • Review 138 stuff: • What are sample distributions • Central limit theorem • Standard error (and estimates of) • Test statistic distributions as transformations

  3. How do we determine this? Testing Hypotheses • From last time: • Core logic of hypothesis testing • Considers the probability that the result of a study could have come about if the experimental procedure had no effect • If this probability is low, scenario of no effect is rejected and the theory behind the experimental procedure is supported Based on standard error or an estimate of the standard error

  4. Flipping a coin example Number of heads HHH 3 HHT 2 HTH 2 HTT 1 2 THH THT 1 TTH 1 TTT 0 = 23 = 8total outcomes 2n

  5. .4 .3 probability .2 .1 .125 .375 .375 .125 0 1 2 3 Number of heads Flipping a coin example Number of heads 3 Distribution of possible outcomes (n = 3 flips) 2 2 1 2 1 1 0

  6. Hypothesis testing Distribution of Sample Means Distribution of possible outcomes (of a particular sample size, n) Can make predictions about likelihood of outcomes based on this distribution. • In hypothesis testing, we compare our observed samples with the distribution of possible samples (transformed into standardized distributions) • This distribution of possible outcomes is often Normally Distributed

  7. Hypothesis testing Distribution of Sample Means • Mean of a group of scores • Comparison distribution is distribution of means Distribution of possible outcomes (of a particular sample size, n)

  8. Population Sample Distribution of sample means • Distribution of sample means is a “theoretical” distribution between the sample and population • Mean of a group of scores • Comparison distribution is distribution of means Distribution of sample means

  9. 2 4 6 8 Distribution of sample means • A simple case • Population: • All possible samples of size n = 2 Assumption: sampling with replacement

  10. 2 4 6 8 mean mean mean 4 6 5 8 2 5 4 8 6 8 4 6 2 6 6 2 4 8 6 7 2 8 6 4 5 8 8 8 4 2 6 6 6 4 4 6 8 7 Distribution of sample means • A simpler case • Population: • All possible samples of size n = 2 There are 16 of them 2 2 2 2 4 3 4 5 3 4

  11. 5 4 3 2 1 2 3 4 5 6 7 8 means Distribution of sample means In long run, the random selection of tiles leads to a predictable pattern mean mean mean 2 2 2 4 6 5 8 2 5 2 4 3 4 8 6 8 4 6 2 6 4 6 2 4 8 6 7 2 8 5 6 4 5 8 8 8 4 2 3 6 6 6 4 4 4 6 8 7

  12. 5 4 3 2 1 2 3 4 5 6 7 8 means P(X > 6) = Distribution of sample means • Sample problem: • What’s the probability of getting a sample with a mean of 6 or more? .1875 + .1250 + .0625 = 0.375 • Same as before, except now we’re asking about sample means rather than single scores

  13. N > 30 Properties of the distribution of sample means • Shape • If population is Normal, then the dist of sample means will be Normal • If the sample size is large (n > 30), regardless of shape of the population Distribution of sample means Population

  14. Distribution of sample means same numeric value different conceptual values Properties of the distribution of sample means • The mean of the dist of sample means is equal to the mean of the population • Center Population

  15. 5 4 3 2 1 2 3 4 5 6 7 8 means Population Distribution of sample means 2 4 6 8 2 + 4 + 6 + 8 4 2+3+4+5+3+4+5+6+4+5+6+7+5+6+7+8 = 16 Properties of the distribution of sample means • Center • The mean of the dist of sample means is equal to the mean of the population • Consider our earlier example m = = 5 = 5

  16. Properties of the distribution of sample means • Spread • The standard deviation of the distribution of sample mean depends on two things • Standard deviation of the population • Sample size

  17. 3 X X X X X 2 2 1 1 X 3 m m Properties of the distribution of sample means • Spread • Standard deviation of the population • The smaller the population variability, the closer the sample means are to the population mean

  18. X m Properties of the distribution of sample means • Spread • Sample size n = 1

  19. X m Properties of the distribution of sample means • Spread • Sample size n = 10

  20. X m Properties of the distribution of sample means • Spread • Sample size n = 100 The larger the sample size the smaller the spread

  21. Properties of the distribution of sample means • Spread • Standard deviation of the population • Sample size • Putting them together we get the standard deviation of the distribution of sample means • Commonly called the standard error

  22. Standard error • The standard error is the average amount that you’d expect a sample (of size n) to deviate from the population mean • In other words, it is an estimate of the error that you’d expect by chance (or by sampling)

  23. X Population Distribution of sample means Sample s s m Distribution of sample means • Keep your distributions straight by taking care with your notation

  24. For any population with mean  and standard deviation , the distribution of sample means for sample size n will approach a normal distribution with a mean of  and a standard deviation of as n approaches infinity • (good approximation if n > 30). Properties of the distribution of sample means • All three of these properties are combined to form the Central Limit Theorem

  25. Could be difference between a sample and a population, or between different samples Based on standard error or an estimate of the standard error Performing your statistical test • What are we doing when we test the hypotheses? • Computing a test statistic: Generic test

  26. Hypothesis Testing With a Distribution of Means • It is the comparison distribution when a sample has more than one individual • Find a Z score of your sample’s mean on a distribution of means

  27. H0: the memory treatment sample are the same (or worse) as the population of memory patients. Memory example experiment: • After the treatment they have an average score of = 55 memory errors. HA: Their memory is better than the population of memory patients “Generic” statistical test An example: One sample z-test • Step 1: State your hypotheses • We give a n = 16 memory patients a memory improvement treatment. mTreatment >mpop > 60 • How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, m = 60, s = 8? mTreatment < mpop < 60

  28. Memory example experiment: • After the treatment they have an average score of = 55 memory errors. One -tailed “Generic” statistical test An example: One sample z-test H0: mTreatment >mpop > 60 HA: mTreatment < mpop < 60 • We give a n = 16 memory patients a memory improvement treatment. • Step 2: Set your decision criteria a = 0.05 • How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, m = 60, s = 8?

  29. Memory example experiment: • After the treatment they have an average score of = 55 memory errors. “Generic” statistical test An example: One sample z-test H0: mTreatment >mpop > 60 HA: mTreatment < mpop < 60 • We give a n = 16 memory patients a memory improvement treatment. a = 0.05 One -tailed • Step 3: Collect your data • How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, m = 60, s = 8?

  30. Memory example experiment: • After the treatment they have an average score of = 55 memory errors. “Generic” statistical test An example: One sample z-test H0: mTreatment >mpop > 60 HA: mTreatment < mpop < 60 • We give a n = 16 memory patients a memory improvement treatment. a = 0.05 One -tailed • Step 4: Compute your test statistics • How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, m = 60, s = 8? = -2.5

  31. Memory example experiment: • After the treatment they have an average score of = 55 memory errors. 5% “Generic” statistical test An example: One sample z-test H0: mTreatment >mpop > 60 HA: mTreatment < mpop < 60 • We give a n = 16 memory patients a memory improvement treatment. a = 0.05 One -tailed • Step 5: Make a decision about your null hypothesis • How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, m = 60, s = 8? Reject H0

  32. Memory example experiment: • After the treatment they have an average score of = 55 memory errors. “Generic” statistical test An example: One sample z-test H0: mTreatment >mpop > 60 HA: mTreatment < mpop < 60 • We give a n = 16 memory patients a memory improvement treatment. a = 0.05 One -tailed • Step 5: Make a decision about your null hypothesis - Reject H0 • How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, m = 60, s = 8? - Support for our HA, the evidence suggests that the treatment decreases the number of memory errors

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