1. Chi-squared test or c2 test

2. Chi-square test Used to test the counts of categorical data
Three types
Goodness of fit (univariate)
Independence (bivariate)
Homogeneity (univariate with two samples)

3. c2 distribution –
df=3
df=5
df=10

4. c2 distribution Different df have different curves Skewed right
As df increases, curve shifts toward right & becomes more like a normal curve

5. c2 assumptions SRS – reasonably random sample
Have counts of categorical data & we expect each category to happen at least once
Sample size – to ensure that the sample size is large enough we should expect at least five in each category.
***Be sure to list expected counts!!
Combine these
together:
All expected
counts are at
least 5.

6. c2 formula

7. df = number of categories - 1
c2 Goodness of fit test
Based on df –
df = number of categories - 1
Uses univariate data
Want to see how well the observed counts “fit” what we expect the counts to be
Use c2cdf function on the calculator to find p-values

8. Hypotheses – written in words
H0: proportions are equal
Ha: at least one proportion is not the same
Be sure to write in context!

9. I would expect CEOs to be equally born under all signs.
Does your zodiac sign determine how successful you will be? Fortune magazine collected the zodiac signs of 256 heads of the largest 400 companies. Is there sufficient evidence to claim that successful people are more likely to be born under some signs than others?
Aries 23 Libra 18 Leo 20
Taurus 20 Scorpio 21 Virgo 19
Gemini 18 Sagittarius 19 Aquarius 24
Cancer 23 Capricorn 22 Pisces 29
How many would you expect in each sign if there were no difference between them?
How many degrees of freedom?
I would expect CEOs to be equally born under all signs.
So 256/12 =
Since there are 12 signs –
df = 12 – 1 = 11

11. Because we do NOT have counts of the type of nuts.
A company says its premium mixture of nuts contains 10% Brazil nuts, 20% cashews, 20% almonds, 10% hazelnuts and 40% peanuts. You buy a large can and separate the nuts. Upon weighing them, you find there are 112 g Brazil nuts, 183 g of cashews, 207 g of almonds, 71 g or hazelnuts, and 446 g of peanuts. You wonder whether your mix is significantly different from what the company advertises?
Why is the chi-square goodness-of-fit test NOT appropriate here?
What might you do instead of weighing the nuts in order to use chi-square?
Because we do NOT have counts of the type of nuts.
We could count the number of each type of nut and then perform a c2 test.

12. Since there are 4 categories,
Offspring of certain fruit flies may have yellow or ebony bodies and normal wings or short wings. Genetic theory predicts that these traits will appear in the ratio 9:3:3:1 (yellow & normal, yellow & short, ebony & normal, ebony & short) A researcher checks 100 such flies and finds the distribution of traits to be 59, 20, 11, and 10, respectively. What are the expected counts? df?
Are the results consistent with the theoretical distribution predicted by the genetic model? (see next page)
Since there are 4 categories,
df = 4 – 1 = 3
Expected counts:
Y & N = 56.25
Y & S = 18.75
E & N = 18.75
E & S = 6.25
We expect 9/16 of the 100 flies to have yellow and normal wings. (Y & N)

13. Assumptions:
Have a random sample of fruit flies
All expected counts are greater than 5.
Expected counts:
Y & N = 56.25, Y & S = 18.75, E & N = 18.75, E & S = 6.25
H0: The proportions of fruit flies are the same as the theoretical model.
Ha: At least one of the proportions of fruit flies is not the same as the theoretical model.
P-value = c2cdf(5.671, 10^99, 3) = a = .05
Since p-value > a, I fail to reject H0. There is not sufficient evidence to suggest that the distribution of fruit flies is not the same as the theoretical model.

14. c2 test for independence
Used with categorical, bivariate data from ONE sample
Used to see if the two categorical variables are associated (dependent) or not associated (independent)

15. Assumptions & formula remain the same!

16. Hypotheses – written in words
H0: two variables are independent
Ha: two variables are dependent
Be sure to write in context!

17. A beef distributor wishes to determine whether there is a relationship between geographic region and cut of meat preferred. If there is no relationship, we will say that beef preference is independent of geographic region. Suppose that, in a random sample of 500 customers, 300 are from the North and 200 from the South. Also, 150 prefer cut A, 275 prefer cut B, and 75 prefer cut C.

18. If beef preference is independent of geographic region, how would we expect this table to be filled in?
North
South
Total
Cut A
150
Cut B
275
Cut C 75
300
200
500 90 60
165
110 45 30

19. Expected Counts
Assuming H0 is true,

20. Degrees of freedom
Or cover up one row & one column & count the number of cells remaining!

21. Suppose that in the actual sample of 500 consumers the observed numbers were as follows:
 Is there sufficient evidence to suggest that geographic regions and beef preference are not independent? (Is there a difference between the expected and observed counts?)
North
South
Total
Cut A
100 50
150
Cut B
125
275
Cut C 25 75
300
200
500

22. Have a random sample of people All expected counts are greater than 5.
Assumptions:
Have a random sample of people
All expected counts are greater than 5.
H0: geographic region and beef preference are independent Ha: geographic region and beef preference are dependent
P-value = df = 2 a = .05
Since p-value < a, I reject H0. There is sufficient evidence to suggest that geographic region and beef preference are dependent.
Expected Counts:
N S A B C

23. c2 test for homogeneity
Used with a single categorical variable from two (or more) independent samples
Used to see if the two populations are the same (homogeneous)

24. Assumptions & formula remain the same!
Expected counts & df are found the same way as test for independence.
Only change is the hypotheses!

25. Hypotheses – written in words
H0: the proportions for the two (or more) distributions are the same
Ha: At least one of the proportions for the distributions is different
Be sure to write in context!

26. The following data is on drinking behavior for independently chosen random samples of male and female students. Does there appear to be a gender difference with respect to drinking behavior? (Note: low = 1-7 drinks/wk, moderate = 8-24 drinks/wk, high = 25 or more drinks/wk)
Men
Women
Total
None
140
186
326
Low
478
661
1139
Moderate
300
173
473
High 63 16 79
981
1036
2017

27. Have 2 random sample of students
Expected Counts:
M F L M H
Assumptions:
Have 2 random sample of students
All expected counts are greater than 5.
H0: the proportions of drinking behaviors is the same for female & male students Ha: at least one of the proportions of drinking behavior is different for female & male students
P-value = .000 df = 3 a = .05
Since p-value < a, I reject H0. There is sufficient evidence to suggest that drinking behavior is not the same for female & male students.