1. Today’s Agenda Review of ANOVA Module 9 Review for Exam 2 Please log in with your UMID and your participation will be graded by the number of questions you answer (no matter whether correct or not)

2. Second Midterm Exam 27 th Mar. 6pm-7pm30 1324 EH Material: Ch. 9-13: Inference for the big 5 parameters HW 4-9 and all lab material Bring: PICTURE ID, Pen/pencils, erasers, and a calculator. Formula cards with tables will be provided with the exam. Alternate, randomized seating will be used in the exam rooms. No cell phones, beepers, palm pilots allowed in the exam rooms. All bags/backpacks will be placed against the exam room wall, so the less extra things you bring, the better.

3. Comments on p-value When using tables from yellow card… Z-statistic: the probability from z-table (of yellow card) corresponds to the left direction T-statistic: the probability from t-table (of yellow card) corresponds to the tail direction, or to the left direction when t 0 Then, one-minus or divided by two when needed.

4. When to tell whether there is a difference of two population means Using hypothesis test with H0: μ1=μ2 Ha: μ1≠μ2 and check whether p-value<α Constructing CI and check whether 0 is included in CI When the significant level α and confidence level of CI are consistent, e.g., 5% and 95%, two methods always give the same conclusion When Ha is μ1>μ2 or μ1<μ2, the two methods do not always give the same conclusion. In other words, when the question asks whether the mean of population 1 is larger (smaller) than the mean of population 2, only hypothesis test can be used.

5. ANOVA ANOVA (Analysis Of Variance): basically an extension of two independent samples pooled t test Hypothesis: H 0 : all the population means are equal, or μ 0 = μ 1 = … = μ k H a : at least one population mean is different Assumptions: 1) Normality of each population 2) The k groups have equal population variances 3) k independent random samples

6. ANOVA Table MSE is a good (unbiased) estimator of the common population variance σ 2 MSG is a good estimator for the variance σ 2 only when the null hypothesis for ANOVA F- Test is true F-statistic is a ratio of two estimators

7. ANOVA F statistic has F(k-1,N-k) distribution When F is large (p-value is smaller than α), reject H 0 (equal population means) P-value is the area to the right of the observed F-test statistic value When H 0 is rejected, use Tukey’s procedure to tell which population mean(s) appear to be different Tukey’s procedure produces confidence intervals for the differences in each pair of population means. We need to check whether 0 is in the interval or not

8. Module 9: ANOVA

9. What is the appropriate null hypothesis? H o : μ 1 -μ 2 -μ 3 = 0 H o : μ 1 =μ 2 =μ 3 = 0 H o : μ 1 =μ 2 =μ 3 Any of the above would be appropriate.

10. What is the appropriate alternative hypothesis? H a : μ 1 ≠ μ 2 ≠ μ 3 H a : at least one is different H a : at least one μ i is different Any of the above would be appropriate.

11. Which provides an estimate of the common population standard deviation for the response? Mean square groups Mean square error None of the above

12. What is the distribution of the test statistic if there really is no difference between the popul. average GPA for the three social classes? N(0,1) t(n-1) t(n 1 +n 2 -2) F(k-1,N-k)

13. Yes or No Based on our decision, it would be appropriate to use Tukey’s multiple comparison procedure.

14. Select the pair(s) for which the popul. means are significantly different. Choose all that apply... A)Lower class, Middle class B)Lower class, Upper class C)Middle class, Upper class

15. Exam 2 Review

16. Inference We’ve talked about two broad inference procedures? –Do you know what “inference” means? –Do you know what the two general procedures are? –Do you know when it is appropriate to use each one?

17. Suppose we want to estimate the magnitude of an effect. Which procedure would be best to use? Confidence interval Hypothesis test

18. Confidence Intervals (CI) Can you interpret the confidence interval? Can you interpret the confidence level? Do you know how to use a CI to make a decision?

19. Hypothesis Tests (HT) What are the 5 steps to conducting a HT? Can you interpret the test statistic? Can you interpret the p-value?

20. Suppose we want to make a “maybe yes” or “maybe no” type decision. Which procedure would be best to use? Confidence interval Hypothesis test

21. The “Big 5” Do you know what this refers to? Do you know the distinction between the following symbols? (Hint: 2 are parameters, 2 are statistics) p μ What other 3 parameters complete the “Big 5”?

22. Order the steps for conducting a hypothesis test... 1)Assumptions and test statistic 2)Conclusion 3)Find the p-value assuming H o true 4)State the hypotheses 5)Decision

23. Which of the following tools could be used to make a decision for a two-sided hypothesis test? A)Confidence interval B)p-value C)Either of the above

24. A 95% CI for the popul. mean is (35, 51). Which of the following are correct interpretation(s) of the confidence level? Choose all that apply... A)The popul. mean will be in the interval (35, 51) 95% of the time. B)We are 95% confident that the popul. mean is in the interval (35,51). C)If the procedure were repeated many times, we would expect 95% of the resulting CIs to contain the popul. mean.

25. Which of the following are correct interpretation(s) of the p-value? Choose all that apply... The probability of observing a test statistic as extreme or more extreme (in the direction of H a ) than what we found, assuming H o is true. The probability the null hypothesis is true.

26. Which of the following inference scenarios will NOT be on Exam 2? Choose all that apply... Single population proportion p Single population mean  Population mean difference  d Difference in 2 population means  1 -  2 Difference in 2 popul. proportions p 1 - p 2 ANOVA  1,…,  k