1. Reasoning in Psychology Using Statistics Psychology 138 2015

2. Reasoning in Psychology Using Statistics Announcements Quiz 5 Due Friday March 20 th

3. Reasoning in Psychology Using Statistics Exam(s) 2 Lab Ex2, mean = 69.3/75 = 92.4%Lecture Ex2, mean = 56.9/75 = 75.9% Combined Ex2, mean = 126.3/150 = 84.2%

4. Reasoning in Psychology Using Statistics Inferential statistics Estimation –Using sample statistics to estimate the population parameters Population Sample Inferential statistics used to generalize back Sampling to make data collection manageable Hypothesis testing –Testing claims about populations (and the effect of variables) based on data collected from samples

5. Reasoning in Psychology Using Statistics Hypothesis testing Example: Testing the effectiveness of a new memory treatment for patients with memory problems Memory treatment No Memory treatment Memory patients Memory Test Memory Test 60 errors 64 errors 4 error diff Is the 4 error difference: –a “ real ” difference due to the effect of the treatment –or is it just sampling error?

6. Reasoning in Psychology Using Statistics Testing Hypotheses –Step 1: State your hypotheses –Step 2: Set your decision criteria –Step 3: Collect your data from your sample –Step 4: Compute your test statistics –Step 5: Make a decision about your null hypothesis Hypothesis testing: a five step program

7. Reasoning in Psychology Using Statistics Testing Hypotheses –Step 1: State your hypotheses –Step 2: Set your decision criteria –Step 3: Collect your data from your sample –Step 4: Compute your test statistics –Step 5: Make a decision about your null hypothesis Hypothesis testing: a five step program Today ’ s focus

8. Reasoning in Psychology Using Statistics Null hypothesis (H 0 ) Alternative hypothesis (H A ) Testing Hypotheses There are no differences between conditions (no effect of treatment) Not all conditions are equal This is the one that you test Hypothesis testing: a five step program –You are NOT out to prove the alternative hypothesis If you reject the null hypothesis, then you are left with support for the alternative(s) (NOT proof!) –Step 1: State your hypotheses Note: This is general form, more complexity is coming up

9. Reasoning in Psychology Using Statistics In our memory example experiment: Testing Hypotheses μ Treatment > μ No Treatment μ Treatment < μ No Treatment H0:H0: HA:HA: – Our theory is that the treatment should improve memory (fewer errors). –Step 1: State your hypotheses Hypothesis testing: a five step program One -tailed About populations

10. Reasoning in Psychology Using Statistics In our memory example experiment: Testing Hypotheses μ Treatment > μ No Treatment μ Treatment < μ No Treatment H0:H0: HA:HA: – Our theory is that the treatment should improve memory (fewer errors). –Step 1: State your hypotheses Hypothesis testing: a five step program μ Treatment = μ No Treatment μ Treatment ≠ μ No Treatment H0:H0: HA:HA: – Our theory is that the treatment has an effect on memory. One -tailedTwo -tailed no direction specified direction specified

11. Reasoning in Psychology Using Statistics Testing Hypotheses –Step 1: State your hypotheses –Step 2: Set your decision criteria Hypothesis testing: a five step program Your alpha (α) level will be your guide for when to reject or fail to reject the null hypothesis (see step 5) –Based on the probability of making making an certain type of error »Type I error »Type II error This step is basically deciding how big a difference is big enough to feel safe in concluding that there is an effect. This decision is made before the data is collected.

12. Reasoning in Psychology Using Statistics Error types Real world (‘truth’) H 0 is correct H 0 is wrong Experimenter’s conclusions Reject H 0 Fail to Reject H 0 There really is not an effect There really is an effect

13. Reasoning in Psychology Using Statistics Error types Real world (‘truth’) H 0 is correct H 0 is wrong Experimenter’s conclusions Reject H 0 Fail to Reject H 0 I conclude that there is an effect I cannot detect an effect

14. Reasoning in Psychology Using Statistics Error types Real world (‘truth’) H 0 is correct H 0 is wrong Experimenter’s conclusions Reject H 0 Fail to Reject H 0 Type I error Type II error Concluding that there is a difference between groups ( “ an effect ” ) when there really isn’t Concluding that there isn ’ t an effect, when there really is

15. Reasoning in Psychology Using Statistics Error types: Courtroom analogy Real world (‘truth’) Defendant is innocent Jury’s decision Find guilty Type I error Type II error Defendant is guilty Find not guilty Guilty person gets out of jail!!! Innocent person goes to jail!!!

16. Reasoning in Psychology Using Statistics Error types Summary of error types –Type I error (α): concluding that there is a difference between groups ( “ an effect ” ) when there really is not. Sometimes called “ significance level ” or “ alpha level ” We try to minimize this (keep it low) by picking a low level of alpha –Psychology: 0.05 and 0.01 most common –Explicit acknowledgement of the uncertainty of our conclusion about the effect, our conclusions are based on likelihood (probability) –Type II error (β): concluding that there is not an effect, when there really is. Related to the Statistical Power of a test (1-β) –How likely are you able to detect a difference if it is there

17. Reasoning in Psychology Using Statistics Testing Hypotheses –Step 1: State your hypotheses –Step 2: Set your decision criteria –Step 3: Collect your data from your sample Hypothesis testing: a five step program Memory treatment No Memory treatment Memory patients Memory Test Memory Test 60 errors 64 errors 4 error diff Is the 4 error difference: –a “ real ” difference due to the effect of the treatment –or is it just sampling error? How do we estimate our sampling error?

18. Reasoning in Psychology Using Statistics Inferential Statistics Consider two bags of marbles (populations) –We can estimate how likely a particular sample is Bag 1 50 black marbles 50 white marbles Bag 2 10 black marbles 90 white marbles sample p(4black) = 0.0625p(4black) = 0.0001 = 0.5 * 0.5 * 0.5 * 0.5 = 0.1 * 0.1 * 0.1 * 0.1 Roughly 6 out of 100 samples Roughly 1 out of 10,000 samples

19. Reasoning in Psychology Using Statistics Distribution of sample means A simpler case –Population: –All possible samples of size n = 2 2468 Assumption: sampling with replacement

20. Reasoning in Psychology Using Statistics Distribution of sample means A simpler case –Population: –All possible samples of size n = 2 2468 2 4 62 2 82 2 44 4 6 8 28 8 8 8 8464 6 6 6 6 4 6 8 242 mean 2 3 4 5 3 4 5 6 4 5 6 7 5 6 7 8 There are 16 of them

21. Reasoning in Psychology Using Statistics Distribution of sample means 2 4 6 8 2 4 6 8 2 46 2 6 2 6 46 4 6 8 28 8 8 8 4 4 4 6 8 2 2 mean 2 3 4 5 3 4 5 6 4 5 6 7 5 6 7 8 means 2345678 5 2 3 4 1 In long run, the random selection of tiles leads to a predictable pattern The distribution of sample means

22. Reasoning in Psychology Using Statistics Distribution of sample means means 2345678 5 2 3 4 1 Xfp 8 7 6 5 4 3 2 1 = 1/16 0.0625

23. Reasoning in Psychology Using Statistics Distribution of sample means means 2345678 5 2 3 4 1 Xfp 8 7 6 5 4 3 2 1 2 0.0625 = 2/16 0.1250

24. Reasoning in Psychology Using Statistics Distribution of sample means means 2345678 5 2 3 4 1 Xfp 8 7 6 5 4 3 2 1 2 3 0.0625 0.1250 = 3/16 0.1875

25. Reasoning in Psychology Using Statistics Distribution of sample means means 2345678 5 2 3 4 1 Xfp 8 7 6 5 4 3 2 1 2 3 4 0.0625 0.1250 0.1875 = 4/16 0.2500

26. Reasoning in Psychology Using Statistics Distribution of sample means means 2345678 5 2 3 4 1 Xfp 8 7 6 5 4 3 2 1 2 3 4 3 0.0625 0.1250 0.1875 0.2500 = 3/16 0.1875

27. Reasoning in Psychology Using Statistics Distribution of sample means means 2345678 5 2 3 4 1 Xfp 8 7 6 5 4 3 2 1 2 3 4 3 2 0.0625 0.1250 0.1875 0.2500 0.1875 = 2/16 0.1250

28. Reasoning in Psychology Using Statistics Distribution of sample means means 2345678 5 2 3 4 1 Xfp 8 7 6 5 4 3 2 1 2 3 4 3 2 1 0.0625 0.1250 0.1875 0.2500 0.1875 0.1250 = 1/16 0.0625

29. Reasoning in Psychology Using Statistics Distribution of sample means means 2345678 5 2 3 4 1 Xfp 810.0625 720.1250 630.1875 540.2500 430.1875 320.1250 210.0625 Sample problem: –What is the probability of getting a sample (n = 2) with a mean of 6 or more? P(X > 6) =.1875 +.1250 +.0625 = 0.375 Same as before, except now we are asking about sample means rather than single scores Using the distribution of sample means Finding out how likely is a particular sample

30. Reasoning in Psychology Using Statistics Distribution of sample means Distribution of sample means is a “ virtual ” distribution between the sample and population PopulationDistribution of sample meansSample There is a different one these for each sample size

31. Reasoning in Psychology Using Statistics Properties of the distribution of sample means Shape –If population is Normal, then the dist of sample means will be Normal Population Distribution of sample means n > 30 –If the sample size is large (n > 30), the DSM will be approximately Normal ( regardless of shape of the population)

32. Reasoning in Psychology Using Statistics –The mean of the dist of sample means is equal to the mean of the population PopulationDistribution of sample means same numeric value different conceptual values Center Properties of the distribution of sample means

33. Reasoning in Psychology Using Statistics Center –The mean of the dist of sample means is equal to the mean of the population –Consider our earlier example 2468 Population μ = 2 + 4 + 6 + 8 4 = 5 Distribution of sample means means 2345678 5 2 3 4 1 2+3+4+5+3+4+5+6+4+5+6+7+5+6+7+8 16 = = 5 Properties of the distribution of sample means

34. Reasoning in Psychology Using Statistics Spread Standard deviation of the population Sample size Properties of the distribution of sample means –The stand. Dev. of the distrib. of sample mean depends on 2 things

35. Reasoning in Psychology Using Statistics Spread Standard deviation of the population μ X 1 X 2 X 3 μ X 1 X 2 X 3 –The smaller the population variability, the closer the sample means are to the population mean Properties of the distribution of sample means

36. Reasoning in Psychology Using Statistics Spread Sample size μ n = 1 X Properties of the distribution of sample means

37. Reasoning in Psychology Using Statistics Spread Sample size μ n = 10 X Properties of the distribution of sample means

38. Reasoning in Psychology Using Statistics Spread Sample size μ n = 100 X The larger the sample size the smaller the spread Properties of the distribution of sample means

39. Reasoning in Psychology Using Statistics 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 Properties of the distribution of sample means - The smaller the population variability, … the smaller the spread - The larger the sample size the smaller the spread

40. Reasoning in Psychology Using Statistics Sample s X Population  μ Distribution of sample means The standard error is the average amount that you would 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 (it is our estimate of the sampling error) Keep your distributions straight by taking care with your notation Properties of the distribution of sample means

41. Reasoning in Psychology Using Statistics Properties of the distribution of sample means All three of these properties are combined to form the Central Limit Theorem –For any population with mean μ and standard deviation , the distribution of sample means for sample size n will approach a normal distribution with a mean of μ and a standard deviation of as n approaches infinity (good approximation if n > 30).

42. Reasoning in Psychology Using Statistics Hypothesis testing Example: Testing the effectiveness of a new memory treatment for patients with memory problems Memory treatment No Memory treatment Memory patients Memory Test Memory Test 60 errors 64 errors 4 error diff Is the 4 error difference: –a “ real ” difference due to the effect of the treatment –or is it just sampling error? So if our standard error (estimated sampling error) is Small (e.g., 1 error), then a 4 error difference is unlikely due to chance, so we will probably conclude there is a treatment effect Large (e.g., 4 errors), then a 4 error difference may well be due to chance, so we will probably NOT conclude there is a treatment effect

43. Reasoning in Psychology Using Statistics In lab In Labs –Make hypotheses (both null and alternative) –Get a feel for distributions of sample means Next time –Finishing up steps of hypothesis testing Step 1: State your hypotheses Step 2: Set your decision criteria Step 3: Collect your data from your sample Step 4: Compute your test statistics Step 5: Make a decision about your null hypothesis Wed ’ s focus

44. Reasoning in Psychology Using Statistics In lab In Labs –Make hypotheses (both null and alternative) –Get a feel for distributions of sample means Next time –Finish 5 steps of hypothesis testing Questions?