1. Class Seven Turn In: Chapter 18: 32, 34, 36 Chapter 19: 26, 34, 44 Quiz 3 For Class Eight: Chapter 20: 18, 20, 24 Chapter 22: 34, 36 Read Chapters 23 & 24 Complete Final Exam

2. Objectives for Class Seven Compute confidence intervals and perform significance tests for proportions for two sample problems. Construct and interpret two way tables of counts of categorical variables. Discuss ways of representing and analyzing categorical data.

3. Conditions for Two Sample Proportions when the samples are large, the distribution is approximately Normal –large sample confidence interval: use this interval only when the populations are at least 10 times as large as the samples and the counts of successes and failures are each 10 or more in both samples –plus four confidence interval: use this interval whenever both samples have 5 or more observations –significance test: use this test when the populations are at least 10 times as large as the samples and the counts of successes and failures are each 5 or more in both samples the mean of is p 1 – p 2 i.e. the difference between sample proportions is an unbiased estimator of the difference between population proportions. the standard deviation of the difference is:

4. Large Sample Confidence Interval an approximate confidence interval for p 1 – p 2 is:

5. Plus Four Confidence Interval a more accurate interval can be found using the plus four method where we add one success and one failure to each sample.

6. Significance Tests for Comparing Proportions the null hypothesis says that there is no difference in the two populations: H 0 : p 1 = p 2. The alternative hypothesis state what kind of difference we expect.

7. Two Way Tables two way tables compare categorical variables. –each row should be totaled –each column should be totaled –the table should be totaled –each cell can be represented as either a percentage of its row total or its column total or the total of the table –percentages for each row should add to 100% (given rounding error) –percentages for each column should total 100% (given rounding error)

8. Multiple Comparisons separate tests do not yield an overall conclusion with a single p value the represents the probability that two distributions differ by chance multiple comparisons entail: –an overall test to see if there is good evidence of any difference among the parameters that we want to compare –a detailed follow-up analysis to decide which parameters differ and to estimate how large the differences are

9. Expected Counts if the observed counts are far from the expected counts, that is evidence against the null hypothesis that there is no relationship between the two categorical variables

10. Chi-Square Test the statistical test that tells us whether the observed difference between two categorical variables are statistically significant compares the observed and expected counts the chi square statistic is a measure of how far the observed counts in a two way table are from the expected counts, given by the formula –large values of X 2 are evidence against H 0 because they say that the observed counts are far from what we would expect if H 0 were true –the chi square test is one sided –small values of X 2 are not evidence against H 0

11. When the Chi Square Test is Significant compare appropriate percents :which cells have the largest or smallest conditional distributions? compare observed and expected cell counts: which cells have more or fewer observations than we would expect if H 0 were true? look at the terms in the chi square statistic: which cells contribute the most to the value of X 2 ?

12. Chi Square Distributions take only positive values are skewed right specific chi square distributions are specified by their degrees of freedom found by taking (# of rows – 1)(# of columns – 1) –use critical values from the chi square distribution (Table E) for the degrees of freedom –the p value is the area to the right of X 2 under the density curve of this chi square distribution the mean of any chi square distribution is equal to the degrees of freedom

13. Uses of the Chi Square Test Use the chi square test to test the hypothesis H 0 : that there is no relationship between two categorical variables when you have a two way table from one of these situations: –independent SRSs from each of several populations, with each individual classified according to one categorical variable (the other variable says which sample the individual comes from). –a single SRS with each individual classified according to both of two categorical variables You can safely use the chi square test with critical values form the chi square distribution when no more than 20% of the expected counts are less than 5 and all individual counts are 1 or greater.

14. Chi Square and z Tests If the rows of a table are r groups and the columns are successes and failures then the p values coming from the chi square test with r – 1 degrees of freedom are comparing two proportions just as we did in Chapter 19. –the two tests (chi square and z) always agree –X 2 statistic is the z statistic squared –p values for the X 2 and the z statistic are equal

15. The Chi Square Test for Goodness of Fit A categorical variable has k possible outcomes with probabilities p 1, p 2, p 3,..., p k. p i is the probability of the i th outcome we have n independent observations from this categorical variable To test H 0 : p 1 = p 10, p 2 = p 20,..., p k = p k0 us the chi square statistic the p value is the area to the right of X 2 under the density curve of the chi square distribution with k – 1 degrees of freedom

16. Objectives for Class Seven Compute confidence intervals and perform significance tests for proportions for two sample problems. Construct and interpret two way tables of counts of categorical variables. Discuss ways of representing and analyzing categorical data.

17. Next Week Class Eight To Be Completed Before Class Eight: Chapter 20: 18, 20, 24 Chapter 22: 34, 36 Read Chapters 23 & 24 Complete Final Exam