M&Ms Two-way Tables

1. M&Ms Two-way Tables Ellen Gundlach STAT 301 Course Coordinator Purdue University

2. M&Ms Color Distribution % according to their website

3. Skittles Color Distribution % according to their hotline

4. My M&Ms data in counts

5. My M&Ms data: joint %(divide counts by total = 76)

6. My M&Ms data: marginal %s for color (add down the columns)

7. My M&Ms data: marginal %s for flavor (add across the rows)

8. My M&Ms data: joint and marginal %s

9. Conditional distribution of flavor for color • We know the color of our M&M already, but now how is flavor distributed for this color?

10. Conditional distribution example • We know we have a red M&M, so what is the probability it is a plain M&M?

11. Conditional distribution of color for flavor • We know the flavor of our M&M already, but now how is color distributed for this color?

12. Conditional distribution example • We know we have a peanut M&M, so what is the probability it is green?

13. Conditional distributions in general Conditional distribution of X for Y (we know Y for sure already, but we want to know the probability or % of having X be true as well):

14. Bar graphs for conditional distribution of color for both flavors

15. Chi-squared hypothesis test H0: There is no association between color distribution and flavor for M&Ms. Ha: There is association between color distribution and flavor for M&Ms. Use an  = 0.01 for this story.

16. Full-class M&Ms data in counts(large sample size necessary for test)

17. Chi-squared test SPSS results

18. Chi-squared test conclusions • Test statistic = 14.396 and P-value = 0.013 • Since P-value is > our  of 0.01, we do not reject H0. • We do not have enough evidence to say there is association between color distribution and flavor for M&Ms.

19. Skittles vs. M&Ms • Now we will compare the proportion of yellow candies for Skittles and for M&Ms. • The previous two-way table with plain and peanut M&Ms was of size 2 x 6. • This table will be of size 2x2 because we only care about whether a candy is yellow or non-yellow.

20. Full-class M&Ms and Skittles data in counts(large sample size necessary for test)

21. Chi-squared hypothesis test H0: There is no association between color distribution and flavor for these candies. Ha: There is association between color distribution and flavor for these candies. Use an  = 0.01 for this story.

22. Chi-squared test SPSS results

23. Chi-squared test conclusions • Test statistic = 11.839 and P-value = 0.001 • Since P-value is < our  of 0.01, we reject H0. • We have evidence that there is association between color distribution and flavor for these candies.

24. Another way to do this test Since this is a 2x2 table, and if we are only interested in a 2-sided () hypothesis test, we can use the 2-sample proportions test here.

25. 2-sample proportion test hypotheses H0: pM&Ms = pSkittles Ha: pM&Ms  pSkittles

26. Defining the proportions

27. Test statistic

28. Results from the proportion test • Sample proportions: • Test statistic Z = -3.44 • P-value = 2(0.0003) = 0.0006 • Since P-value < our  of 0.01, we reject H0.

29. Conclusion to the proportion test • We have evidence the proportion of yellow M&Ms is not the same as the proportion of yellow Skittles. • In other words, the type of candy makes a difference to the color distribution.

30. How do our results from the 2 tests compare? • The X2 test statistic = 11.839, which is actually the (Z test statistic = -3.44)2. • If you take into account the rounding, the P-values for both tests are  0.001. • We rejected H0 in both tests.

31. When do you use which test? • Chi-squared tests are best for: two-sided hypothesis tests only 2x2 or bigger tables • Proportion (Z) tests are best for: one- or two-sided hypothesis tests only 2x2 tables