1. Thursday, September 19, 2013 Introduction to t-tests & one-sample t-test

2. Exam I Exams are graded, but waiting on one make-up test. Summary results and individual grades will be announced on Tuesday. General comment: Modal grade is in the B range. If you found Exam I very difficult, you should see me asap to clarify material that you don’t understand & discuss strategies you can use to learn the material.

3. A word about homework assignments To help the TAs in their efforts to provide feedback to you in a timely manner, please – write legibly (homework may be typed) – staple your papers – remove frayed edges – show your work – circle your answer To improve your own grade, please – Complete all parts of all questions

4. NEW TOPIC (CHAPTER 9) t-tests Starting with the one-sample t-test

5. Statistical analysis follows design The one-sample z-test can be used when: –1 sample –One score per subject –Population mean (μ) and standard deviation (  )are known

6. Statistical analysis follows design The one-sample t-test can be used when: –1 sample –One score per subject –Population mean (μ) is known –but standard deviation (  ) is NOT known

7. Testing Hypotheses –Step 1: State your hypotheses –Step 2: Set your decision criteria –Step 3: Collect your data & compute your test statistics –Step 4: Make a decision about your null hypothesis Hypothesis testing: a four step program

8. Testing Hypotheses –Step 1: State your hypotheses –Step 2: Set your decision criteria –Step 3: Collect your data & compute your test statistics Compute your estimated standard error Compute your t-statistic Compute your degrees of freedom –Step 4: Make a decision about your null hypothesis Hypothesis testing: a four step program

9. Performing your statistical test What are we doing when we test the hypotheses? – Consider a variation of our memory experiment example Population of memory patients MemoryTest μ is known Memory treatment Memory patients Memory Test M Compare these two means Conclusions: 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 H0:H0: HA:HA:

10. Performing your statistical test What are we doing when we test the hypotheses? Real world (‘truth’) H 0 : is false (is a treatment effect) Two populations MAMA they aren’t the same as those in the population of memory patients H 0 : is true (no treatment effect) One population MAMA the memory treatment sample are the same as those in the population of memory patients.

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

12. Performing your statistical test Test statistic One sample z One sample t identical

13. Performing your statistical test Test statistic Diff. Expected by chance Standard error One sample z One sample t different If we don’t know this, we need to estimate it

14. Performing your statistical test Test statistic Diff. Expected by chance Standard error Estimated standard error One sample z One sample t different If we don’t know this, we need to estimate it Degrees of freedom

15. One sample t-test The t-statistic distribution (a transformation of the distribution of sample means transformed) – Varies in shape according to the degrees of freedom New table: the t-tablet-table

16. One sample t-test – To reject the H 0, you want a computed test statistic (  t  ) that is large The alpha level gives us the decision criterion If test statistic is here Fail to reject H 0 Distribution of the t-statistic If test statistic is here Reject H 0 New table: the t-tablet-table The t-statistic distribution (a transformation of the distribution of sample means transformed)

17. One sample t-test New table: the t-table One tailed - or - Two-tailed α levels Critical values of t t crit Degrees of freedom df

18. One sample t-test What is the t crit for a two-tailed hypothesis test with a sample size of n = 6 and an α -level of 0.05? Distribution of the t-statistic α= 0.05 Two-tailed n = 6 df = n - 1 = 5 t crit = + 2.571

19. One sample t-test Distribution of the t-statistic  = 0.05 One-tailed n = 6 df = n - 1 = 5 t crit = +2.015 What is the t crit for a one-tailed hypothesis test with a sample size of n = 6 and an α-level of 0.05?

20. One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of M = 55, s = 8 memory errors. Step 1: State your hypotheses H0:H0: Treatment sample same as (or worse than) those in the population of memory patients. HA:HA: Treatment sample better than memory patient population μ Treatment > μ pop = 60 μ Treatment < μ pop = 60

21. One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. Step 2: Set your decision criteria H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60 α = 0.05 One -tailed How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of M = 55, s = 8 memory errors.

22. One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. Step 2: Set your decision criteria H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60 α = 0.05 One -tailed How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of M = 55, s = 8 memory errors.

23. One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. α = 0.05 One -tailed Step 3: Collect your data & compute your test statistic H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of M = 55, s = 8 memory errors.

24. One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. α = 0.05 One -tailed Step 3: Collect your data & compute your test statistic = -2.5 H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of M = 55, s = 8 memory errors.

25. One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. α = 0.05 One -tailed Step 3: Collect your data & compute your test statistic t = -2.5 H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of M = 55, s = 8 memory errors.

26. One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. α = 0.05 One -tailed Step 4: Make a decision about your null hypothesis H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of M = 55, s = 8 memory errors. t crit = -1.753

27. One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. α = 0.05 One -tailed Step 4: Make a decision about your null hypothesis H 0 : μ Treatment > μ pop = 60 H A : μ Treatment < μ pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60? After the treatment they have an average score of M = 55, s = 8 memory errors. -1.753 = t crit t obs =-2.5 - Reject H 0

28. Practice Problem A new over-the-counter cold medication includes a warning label stating that it “may cause drowsiness.” A researcher would like to evaluate this effect. Under regular circumstances, the distribution of reaction times is normal with μ=200. A sample of n=9 participants is given the medication, and, one hour later, reaction time is measured for each individual. The average reaction time for this sample is M=206 with SS=648. Use the 4-step hypotheses-testing program to evaluate the effect of the medication on reaction time.

29. Using spss to conduct t-tests One-sample t-test: Analyze =>Compare Means =>One sample t-test. Select the variable you want to analyze, and type in the expected mean based on your null hypothesis.

30. Practice problem Use SPSS to conduct a one-sample t-test to see if the students in this class are taller than the average college student (μ= 66.5 inches). Save your output file and submit on-line.

31. Homework #4 (due 9/24/13) Chapter 9: 1, 8, 9, 16, 17