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Statistics for the Social Sciences

Statistics for the Social Sciences. Psychology 340 Spring 2005. Hypothesis testing. Outline (for week). Review of: Basic probability Normal distribution Hypothesis testing framework Stating hypotheses General test statistic and test statistic distributions

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Statistics for the Social Sciences

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  1. Statistics for the Social Sciences Psychology 340 Spring 2005 Hypothesis testing

  2. Outline (for week) • Review of: • Basic probability • Normal distribution • Hypothesis testing framework • Stating hypotheses • General test statistic and test statistic distributions • When to reject or fail to reject

  3. Hypothesis testing • Example: Testing the effectiveness of a new memory treatment for patients with memory problems • Our pharmaceutical company develops a new drug treatment that is designed to help patients with impaired memories. • Before we market the drug we want to see if it works. • The drug is designed to work on all memory patients, but we can’t test them all (the population). • So we decide to use a sample and conduct the following experiment. • Based on the results from the sample we will make conclusions about the population.

  4. Memory treatment Memory Test Memory patients No Memory treatment Memory Test 5 error diff Hypothesis testing • Example: Testing the effectiveness of a new memory treatment for patients with memory problems 55 errors 60 errors • Is the 5 error difference: • A “real” difference due to the effect of the treatment • Or is it just sampling error?

  5. Testing Hypotheses • Hypothesis testing • Procedure for deciding whether the outcome of a study (results for a sample) support a particular theory (which is thought to apply to a population) • 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

  6. Basics of Probability • Probability • Expected relative frequency of a particular outcome • Outcome • The result of an experiment

  7. One outcome classified as heads 1 = = 0.5 2 Total of two outcomes Flipping a coin example What are the odds of getting a “heads”? n = 1 flip

  8. One 2 “heads” outcome = 0.25 Four total outcomes Flipping a coin example What are the odds of getting two “heads”? n = 2 Number of heads 2 1 1 0 # of outcomes = 2n This situation is known as the binomial

  9. Three “at least one heads” outcome = 0.75 Four total outcomes Flipping a coin example What are the odds of getting “at least one heads”? n = 2 Number of heads 2 1 1 0

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

  11. .4 .3 probability .2 .1 .125 .375 .375 .125 0 1 2 3 Number of heads Flipping a coin example Number of heads Distribution of possible outcomes (n = 3 flips) 3 2 2 1 2 1 1 0

  12. Flipping a coin example Can make predictions about likelihood of outcomes based on this distribution. Distribution of possible outcomes (n = 3 flips) .4 What’s the probability of flipping three heads in a row? .3 probability .2 .1 p = 0.125 .125 .375 .375 .125 0 1 2 3 Number of heads

  13. Flipping a coin example Can make predictions about likelihood of outcomes based on this distribution. Distribution of possible outcomes (n = 3 flips) .4 What’s the probability of flipping at least two heads in three tosses? .3 probability .2 .1 p = 0.375 + 0.125 = 0.50 .125 .375 .375 .125 0 1 2 3 Number of heads

  14. Flipping a coin example Can make predictions about likelihood of outcomes based on this distribution. Distribution of possible outcomes (n = 3 flips) .4 What’s the probability of flipping all heads or all tails in three tosses? .3 probability .2 .1 p = 0.125 + 0.125 = 0.25 .125 .375 .375 .125 0 1 2 3 Number of heads

  15. Hypothesis testing Can make predictions about likelihood of outcomes based on this distribution. Distribution of possible outcomes (of a particular sample size, n) • 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

  16. The Normal Distribution • The distribution of days before and after due date (bin width = 4 days). 14 -14 0 Days before and after due date

  17. The Normal Distribution • Normal distribution

  18. -2 -1 0 1 2 The Normal Distribution • Normal distribution is a commonly found distribution that is symmetrical and unimodal. • Not all unimodal, symmetrical curves are Normal, so be careful with your descriptions • It is defined by the following equation:

  19. The Unit Normal Table • The normal distribution is often transformed into z-scores. • Gives the precise proportion of scores (in z-scores) between the mean (Z score of 0) and any other Z score in a Normal distribution • Contains the proportions in the tail to the left of corresponding z-scores of a Normal distribution • This means that the table lists only positive Z scores

  20. 34.13% 2.28% 13.59% At z = +1: Using the Unit Normal Table 50%-34%-14% rule Similar to the 68%-95%-99% rule -2 -1 0 1 2 15.87% (13.59% and 2.28%) of the scores are to the right of the score 100%-15.87% = 84.13% to the left

  21. Using the Unit Normal Table • Steps for figuring the percentage above of below a particular raw or Z score: 1. Convert raw score to Z score (if necessary) 2. Draw normal curve, where the Z score falls on it, shade in the area for which you are finding the percentage 3. Make rough estimate of shaded area’s percentage (using 50%-34%-14% rule)

  22. Using the Unit Normal Table • Steps for figuring the percentage above of below a particular raw or Z score: 4.Find exact percentage using unit normal table 5. If needed, add or subtract 50% from this percentage 6. Check the exact percentage is within the range of the estimate from Step 3

  23. So 90.32% got your score or worse That’s 9.68% above this score SAT Example problems • The population parameters for the SAT are: m = 500, s = 100, and it is Normally distributed Suppose that you got a 630 on the SAT. What percent of the people who take the SAT get your score or worse? • From the table: • z(1.3) =.0968

  24. The Normal Distribution • You can go in the other direction too • Steps for figuring Z scores and raw scores from percentages: 1. Draw normal curve, shade in approximate area for the percentage (using the 50%-34%-14% rule) 2. Make rough estimate of the Z score where the shaded area starts 3. Find the exact Z score using the unit normal table 4. Check that your Z score is similar to the rough estimate from Step 2 5. If you want to find a raw score, change it from the Z score

  25. Inferential statistics • Hypothesis testing • 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 • A five step program • Step 1: State your hypotheses • Step 2: Set your decision criteria • Step 3: Collect your data • Step 4: Compute your test statistics • Step 5: Make a decision about your null hypothesis

  26. This is the one that you test Hypothesis testing • Hypothesis testing: a five step program • Step 1: State your hypotheses: as a research hypothesis and a null hypothesis about the populations • Null hypothesis (H0) • Research hypothesis (HA) • There are no differences between conditions (no effect of treatment) • Generally, not all groups are equal • You aren’t out to prove the alternative hypothesis • If you reject the null hypothesis, then you’re left with support for the alternative(s)(NOT proof!)

  27. In our memory example experiment: Testing Hypotheses • Hypothesis testing: a five step program • Step 1: State your hypotheses One -tailed • Our theory is that the treatment should improve memory (fewer errors). H0: mTreatment >mNo Treatment HA: mTreatment < mNo Treatment

  28. In our memory example experiment: no direction specified direction specified Testing Hypotheses • Hypothesis testing: a five step program • Step 1: State your hypotheses One -tailed Two -tailed • Our theory is that the treatment should improve memory (fewer errors). • Our theory is that the treatment has an effect on memory. H0: H0: mTreatment >mNo Treatment mTreatment = mNo Treatment HA: mTreatment < mNo Treatment HA: mTreatment ≠ mNo Treatment

  29. One-Tailed and Two-Tailed Hypothesis Tests • Directional hypotheses • One-tailed test • Nondirectional hypotheses • Two-tailed test

  30. Testing Hypotheses • Hypothesis testing: a five step program • Step 1: State your hypotheses • Step 2: Set your decision criteria • Your alpha () level will be your guide for when to reject or fail to reject the null hypothesis. • Based on the probability of making making an certain type of error

  31. Testing Hypotheses • Hypothesis testing: a five step program • Step 1: State your hypotheses • Step 2: Set your decision criteria • Step 3: Collect your data

  32. Testing Hypotheses • Hypothesis testing: a five step program • Step 1: State your hypotheses • Step 2: Set your decision criteria • Step 3: Collect your data • Step 4: Compute your test statistics • Descriptive statistics (means, standard deviations, etc.) • Inferential statistics (z-test, t-tests, ANOVAs, etc.)

  33. Testing Hypotheses • Hypothesis testing: a five step program • Step 1: State your hypotheses • Step 2: Set your decision criteria • Step 3: Collect your data • Step 4: Compute your test statistics • Step 5: Make a decision about your null hypothesis • Based on the outcomes of the statistical tests researchers will either: • Reject the null hypothesis • Fail to reject the null hypothesis • This could be correct conclusion or the incorrect conclusion

  34. Error types • Type I error (): concluding that there is a difference between groups (“an effect”) when there really isn’t. • Sometimes called “significance level” or “alpha level” • We try to minimize this (keep it low) • Type II error (): concluding that there isn’t an effect, when there really is. • Related to the Statistical Power of a test (1-)

  35. There really isn’t an effect There really is an effect Error types Real world (‘truth’) H0 is correct H0 is wrong Reject H0 Experimenter’s conclusions Fail to Reject H0

  36. I conclude that there is an effect I can’t detect an effect Error types Real world (‘truth’) H0 is correct H0 is wrong Reject H0 Experimenter’s conclusions Fail to Reject H0

  37. Error types Real world (‘truth’) H0 is correct H0 is wrong Type I error Reject H0 Experimenter’s conclusions Fail to Reject H0 Type II error

  38. One population Two populations the memory treatment sample are the same as those in the population of memory patients. they aren’t the same as those in the population of memory patients XA XA Performing your statistical test • What are we doing when we test the hypotheses? Real world (‘truth’) H0: is true (no treatment effect) H0: is false (is a treatment effect)

  39. 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

  40. Distribution of the test statistic “Generic” statistical test • The generic test statistic distribution (think of this as the distribution of sample means) • To reject the H0, you want a computed test statistics that is large • What’s large enough? • The alpha level gives us the decision criterion -level determines where these boundaries go

  41. Distribution of the test statistic If test statistic is here Reject H0 If test statistic is here Fail to reject H0 “Generic” statistical test • The generic test statistic distribution (think of this as the distribution of sample means) • To reject the H0, you want a computed test statistics that is large • What’s large enough? • The alpha level gives us the decision criterion

  42. Reject H0 Reject H0 a = 0.05 0.025 split up into the two tails 0.025 Fail to reject H0 Reject H0 Fail to reject H0 Fail to reject H0 “Generic” statistical test • The alpha level gives us the decision criterion Two -tailed One -tailed

  43. Reject H0 0.05 all of it in one tail a = 0.05 Fail to reject H0 “Generic” statistical test • The alpha level gives us the decision criterion Two -tailed One -tailed Reject H0 Reject H0 Fail to reject H0 Fail to reject H0

  44. Reject H0 a = 0.05 0.05 Fail to reject H0 “Generic” statistical test • The alpha level gives us the decision criterion Two -tailed One -tailed all of it in one tail Reject H0 Reject H0 Fail to reject H0 Fail to reject H0

  45. H0: the memory treatment sample are the same as those in the population of memory patients. Memory example experiment: • After the treatment they have an average score of = 55 memory errors. HA: they aren’t the same as those in 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

  46. 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?

  47. 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?

  48. 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

  49. 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

  50. 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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