10.2 Hypothesis Testing with Two-Way Tables

Transcription

1 10.2 Hypothesis Testing with Two-Way Tables Part 2: more examples 3x3 Two way table 2x3 Two-way table (worksheet) 1

2 Example 2: n Is there an association between the type of school area and the students' choice of good grades, athletic ability, or popularity as most important? n Two categorical variables: School area (Rural, Suburban, Urban) Goals (Grades, Popularity, Sports) 2

3 n th, 5 th, and 6 th graders were asked, and their responses were recorded. The information is shown in the two-way table below: Goals Rural School area Suburban Urban

4 Hypothesis test for two-way tables n If there IS NOT a relationship, then the categorical variables do not impact each other." H 0 : the variables are independent (no " relationship exists) n If there IS a relationship, then the categorical variables DO impact each other." H a : there is a relationship between the two" variables" 4

5 Hypotheses in the context of the data n H 0 : The location of school has no bearing on what activities students consider to be most worth their time." n H a : The location of school does impact what activities students consider to be most worth their time." 5

6 Observed counts Rural As relative frequencies within each school type Rural Suburban Suburban Urban Urban Fewer Urban kids were surveyed

7 Are these three probability distributions similar enough to suggest that school type does not affect what kids think is most important? If so, accept H 0. As relative frequencies within each school type Rural Suburban If not, reject H Urban 7

8 n We need row totals and column totals. totals Rural Suburban Urban totals

9 n Let s calculate the frequencies (counts) we would have expected if the null were true (i.e. there the variables were independent). totals Rural??? 149 Suburban??? 151 Urban??? 35 totals

10 n We convert the row and column totals to relative frequencies totals Rural??? 149/335 Suburban??? 151/335 Urban??? 35/335 totals 168/335 98/335 69/ /335 10

11 n We convert the row and column totals to relative frequencies totals Rural??? Suburban??? Urban??? totals

12 n If the variables are independent (i.e. H 0 is true), then P(Rural and Grades) = P(Rural) x P(Grades) Relative frequency table = x = totals Rural 0.223?? Suburban??? Urban??? totals

13 n If the variables are independent (i.e. H 0 is true), then P(Rural and Poplr) = P(Rural) x P(Poplr) Relative frequency table = x = totals Rural ? Suburban??? Urban??? totals

14 n If the variables are independent (i.e. H 0 is true), then P(Rural and Sports) = P(Rural) x P(Sports) Relative frequency table = x = totals Rural Suburban??? Urban??? totals

15 n Filling-in each remaining cell by multiplying each respective row proportion by each respective column proportion gives Relative frequency table Rural Suburban Urban

16 n Convert the relative frequencies back to counts by multiplying by the total count of individuals (335 in this case) Rural Suburban Urban X 335 Rural Suburban Urban

17 n I can now compare the expected counts under H 0 true to the observed counts. observed counts Rural Suburb Urban expected counts Rural Suburb Urban n For each cell (there are 9 in this case), we will compare the observed and expected counts to create a test statistic for our hypothesis test. 17

18 n The Chi-Square Statistic: χ 2 = sum of all values (Observed - Expected)2 Expected observed counts Rural Suburb Urban expected counts Rural Suburb Urban

19 n The Chi-Square Statistic: χ 2 = (57-75) (50-43) 2 (5-7) = observed counts Rural Suburb Urban expected counts Rural Suburb Urban

20 n Making the decision: χ 2 = n Table 10.7 gives the critical values of χ 2 for two significance levels, 0.05 and Our test is significant at the 0.01 level because χ 2 = is larger than the 0.01 critical value of

21 n Conclusion: There is statistically significant evidence that the school type is related to which activity students find most important. 21