1. AP Exam Prep: Essential Notes

2. Chapter 11: Inference for Distributions 11.1Inference for Means of a Population 11.2Comparing Two Means

3. Moving away from z … In chapter 10, when we knew σ, we calculated a z- score for a particular mean as follows: Now, we do not know σ, so we calculate a t-score, which provides somewhat of a “fudge-factor” because we do not know σ, but must estimate it from the sample : Standard error of the mean

4. One-sample t-procedures (p. 622) Confidence interval: Hypothesis test: In both cases, σ is unknown.

5. Matched Pairs t Procedures Matched pairs designs: subjects are matched in pairs and each treatment is given to one subject in the pair (randomly). One type of matched pairs design is to have a group of subjects serve as their own pair-mate. Each subject then gets both treatments (randomize the order). Apply one-sample t-procedures to the observed differences. Example 11.4, p. 629 Note H 0 Look at Figure 11.7, p. 631

6. Conditions for Inference about a Mean (p. 617) SRS Observations from the population have a normal distribution with mean µ and standard deviation σ. Symmetric and single-peaked essential.

7. Using t-procedures See Box, p. 636 SRS very important! n<15: do not use t-procedures if the data are clearly non-normal or if outliers are present. n at least 15: t-procedures can be used except in the presence of outliers or strong skewness. n at least 40: t-procedures can be used for even clearly skewed distributions. By CLT

8. 11.2 Comparing Two Means The goal of two-sample inference problems is to compare the responses of two treatments or to compare the characteristics of two populations. We must have a separate sample from each treatment or each population. Unlike the matched-pairs designs. A two-sample problem can arise from a randomized comparative experiment that randomly divides subjects into two groups and exposes each group to a different treatment.

9. Conditions for Significance Tests Comparing Two Means (p. 650) Two SRSs from distinct populations. Samples are independent (matching violates this assumption). We measure the same variable for each sample. Both populations are normally distributed. Means and standard deviations of both are unknown.

10. Two-sample t-test The appropriate t-statistic is as follows. The degrees of freedom calculation is complex; we will use our calculators to provide this for us (the df are usually not whole numbers for two-sample tests). =0 for the H 0 :µ 1 =µ 2

11. Two-sample confidence interval for µ 1 -µ 2 Draw an SRS of size n 1 from a normal population with unknown mean µ 1, and draw an independent SRS of size n 2 from a normal population with unknown mean µ 2. The confidence interval for µ 1 - µ 2 is given by the following: Again, we need the df for t*, but we will let the calculator do that for us.

12. Using t-procedures for two-sample analyses See Box, p. 636 SRS very important! n 1 +n 2 <15: do not use t-procedures if the data are clearly non-normal or if outliers are present. n 1 +n 2 at least 15: t-procedures can be used except in the presence of outliers or strong skewness. n 1 +n 2 at least 40: t-procedures can be used for even clearly skewed distributions. By CLT

13. Chapter 12: Inference for Proportions 12.1Inference for a Population Proportion 12.2Comparing Two Proportions

14. Conditions for Inference about a Proportion (p. 687) SRS N at least 10n For a significance test of H 0 :p=p 0 : The sample size n is so large that both np 0 and n(1-p 0 ) are at least 10. For a confidence interval: n is so large that both the count of successes, n*p-hat, and the count of failures, n(1 - p-hat), are at least 10.

15. Normal Sampling Distribution If these conditions are met, the distribution of p-hat is approximately normal, and we can use the z-statistic:

16. Inference for a Population Proportion Confidence Interval: Significance test of H 0 : p=p 0 :

17. Choosing a Sample Size (p. 695) Our guess p * can be from a pilot study, or we could use the most conservative guess of p * =0.5. Solve for n. Example 12.9, p. 696.

18. Conditions: Confidence Intervals for Comparing Two Proportions SRS from each population N>10n All of these are at least 5:

19. Calculating a Confidence Interval for Comparing Two Proportions (p. 704)

20. Significance Tests for Comparing Two Proportions The test statistic is: Where,

21. Conditions: Significance Test for Comparing Two Proportions SRS from each population N>10n All of these are at least 5:

22. Chapter 13: Chi-Square Procedures 13.1Test for Goodness of Fit 13.2Inference for Two-Way Tables

23. M&Ms Example Sometimes we want to examine the distribution of proportions in a single population. As opposed to comparing distributions from two populations, as in Chapter 12. Does the distribution of colors in your bags match up with expected values? We can use a chi-square goodness of fit test. Χ 2 We would not want to do multiple one-proportion z-tests. Why?

24. Performing a X 2 Test 1. H 0 : the color distribution of our M&Ms is as advertised: P brown =0.30, P yellow =P red =0.20, and P orange =P green =P blue =0.10 H a : the color distribution of our M&Ms is not as advertised. 2. Conditions: 1. All individual expected counts are at least 1. 2. No more than 20% of expected counts are less than 5. 3. Chi-square statistic:

25. Section 13.2 (Two-way tables)

26. Example 13.4, pp. 744-748 Is there a difference between proportion of successes? At left is a two-way table for use in studying this question. Explanatory Variable: Type of Treatment Response Variable: Proportion of no relapses Relapse? TreatmentNoYesTotal Desipramine141024 Lithium61824 Placebo42024 Total244872

27. Expected Counts and Conditions All expected counts are at least 1, no more than 20% less than 5.

28. Chapter 14: Inference about the Model

29. Confidence Intervals for the Regression Slope (p. 788) If we repeated our sampling and computed another model, would we expect a and b to be exactly the same? Of course not, given what we’ve learned about random variation and sampling error! We are interested in the true slope (β), which is unknowable, but we are able to estimate it. Confidence Interval for the slope β of the true regression line: Given in output from stats package.

30. Is β=0? H 0 : β=0 vs. H a : β ≠0 or β>0 or β<0 Perform a t-test: