1. Ch 26 – Comparing Counts Day 1 - The Chi-Square Distribution
Part VII – Exploring and Understanding Data
Ch 26 – Comparing Counts Day 1 - The Chi-Square Distribution

2. The Chi-Square Test The Chi-square test has two uses:
Goodness of fit: to determine if data fits a given distribution
Independence: to determine if there is a relationship between two categorical variables

3. The Chi-square test statistic
The basic idea behind chi-square is to see if what really happened is significantly different from what you expected to happen
The formula for a chi-square test statistic is:

4. Decisions using Chi-square
We use chi-square only for hypothesis tests – no confidence intervals
Once we get a test statistic, we use the chi-square table to make a decision as to whether or not to reject H0
The table works in a similar way to the t-table

6. Goodness of Fit Test
A botanist wants to know if a certain insect has a preference in the color of roses it eats. She records the number of insects who infest each color in her study.
Red
White
Yellow
Pink 53 71 46 30

7. Hypotheses
For this type of test there are no symbols in the hypotheses, only words
The null hypothesis basically states that there is nothing unusual going on
In this case:
H0: the insects have no color preference
Ha: the insects do have a color preference

8. Type of Test Chi-square test for goodness of fit α = .05 df = 3
For goodness of fit tests:
df = # of categories – 1

9. Conditions for Chi-square
Check
Random Sample
Read the problem – assume, if no description of sample is given
All expected counts are ≥ 5
Verify – if no, make a note of this and go on with the test
Make sure you show your expected counts somewhere in your problem

10. Expected Values The expected values are based on your null hypothesis
In this problem, if our null hypothesis was true, all cells would have the same value
Red
White
Yellow
Pink
Observed 53 71 46 30
Expected

11. Conditions for Chi-square (This problem)
Check
Random Sample
Assume
All expected counts are ≥ 5
yes

12. The test statistic

13. Conclusion
Since p < .05, reject H0. There is enough evidence to conclude that the insects do have a color preference.

14. Another example…
An anatomy teacher hypothesizes that the final grades in her classes are distributed as 10% A’s, 23% B’s, 45% C’s, 14% D’s and 8% F’s. At the end of the semester, a random sample of her students has the following grades. Was her hypothesis correct? A B C D F 4 13 24 7 5

15. Expected Values 10% A’s, 23% B’s, 45% C’s, 14% D’s and 8% F’s
Total in sample = 53 A B C D F
Observed 4 13 24 7 5
Expected
5.3
12.19
23.85
7.42
4.24

16. Χ2 test for goodness of fit, α=.05, df =4
H0: The distribution does fit the teacher’s hypothesis Ha: The distribution doesn’t fit the teacher’s hypothesis
Χ2 test for goodness of fit, α=.05, df =4
Condition
Check
Random Sample
All expected counts are ≥ 5
Stated
Since p >.05, fail to reject H0. There is not enough evidence to conclude that the teacher’s grades do not fit her hypothesis.
No – one count is < 5 this concerns us, but we will proceed

17. Homework 26-1
GOF WS