CHAPTER 11 SECTION 2 Inference for Relationships

CHAPTER 11 SECTION 2  Essential Question: Describe how a chi-square test helps determine a sample data is consistent.  Learning Target: Students will be able to check the random, large sample size, and independent conditions before performing a chi-square test; use a chi-square test for homogeneity to determine whether the distribution of a categorical variable differs for several populations or treatments; interpret computer output for a chi-square test based on a two-way table; examine individual components of the chi-square statistic as part of a follow-up analysis; show that the two-sample z test for comparing two proportions and the chi-square test for a 2- by-2 two-way table give equivalent results; use a chi-square test of association/independence to determine whether there is convincing evidence of an association between two categorical variables; distinguish between the three types of chi-square tests.

INFERENCE FOR RELATIONSHIPS  Two-sample z compares two proportions… What if we wanted to compare more than two samples??  What if we wanted to compare distributions of a single categorical variable across several populations or treatments?? This is where a new test comes into play. But first, we need to present the data in a two-way table  Let’s take a look at example on pp

COMPARING DISTRIBUTIONS OF A CATEGORICAL VARIABLE  Remember explanatory and response variables. This example using music and wine, the music influences sales (explanatory variable) and wine is what is affected (response variable).  Null hypothesis would illustrate that there is no difference between the sell of wine and the music played. Whereas, the alternate hypothesis would indicate that there is a difference between the sell of wine and the music played.

COMPARING DISTRIBUTIONS OF A CATEGORICAL VARIABLE  When we do many individual tests or construct many confidence intervals, the individual P-values and confidence levels don’t tell us how confident we can be in all the inferences taken together.  The problem of how to do many comparisons at once with an overall measure of confidence in all our conclusions is common in statistics. This is the problem of multiple comparisons.

COMPARING DISTRIBUTIONS OF A CATEGORICAL VARIABLE  Statistical methods for dealing with multiple comparisons usually have two parts: 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 of the parameters differ and to estimate how large the differences are.

COMPARING DISTRIBUTIONS OF A CATEGORICAL VARIABLE  The overall test uses the familiar chi-square statistic and distributions. But in this new setting, the test will be used to compare the distribution of a categorical variable for several populations or treatments.  The follow-up analysis can be quite elaborate. We will concentrate on the overall test and use data analysis to describe the nature of the differences.

EXPECTED COUNTS AND THE CHI- SQUARE STATISTIC  To perform a test – this is what you indicate in the statement part of your four step process… Null – there is no difference in the distribution of a categorical variable for several populations or treatments Alternate – there is a difference in the distribution of a categorical variable for several populations or treatments Let’s take a look at example on pp. 700

FINDING EXPECTED COUNTS  General formula for the expected count in any cell of a two-way table. (row total – column total) / table total this indicates the expected count

EXPECTED COUNTS AND THE CHI- SQUARE STATISTIC  Plan: remember with chi-square you have to satisfy the following: The large sample size condition that all expected counts are at least 5. The random condition is met The independency is met: remember with independency, you need to evaluate the x 2 statistic (refer back to section 1 for formula) Take a look at example on pp. 702

THE CHI-SQUARE TEST FOR HOMOGENEITY  New procedure is known as chi-square test for homogeneity. Take a look at yellow box on pp. 703 – 704  Now take a look at example on pp. 704 (you should start seeing repetition from section 1 except one additional part when finding the degrees of freedom)

CHAPTER 11 SECTION2  The rest of this section has multiple of examples to look over to measure your understanding of this section. I would take a look at a couple of them for further understanding. Once you have done so, now take a look at the following Check your understanding…  Pp. 698, 703, 705, 708, 713, 718