1. Chi-Square Goodness of Fit
3/30/2018

2. Using Chi-Square
In this chapter, we are going to do significance tests for categorical variables
Unlike the past few chapters, which have been using quantitative variables
Some things about this chapter are easier
There are no confidence intervals
The null and alternative hypotheses are always the same
Other things are potentially difficult
We are going to learn 3 different types of Chi-Square tests
You have to know which one to use

3. Chi-Square Goodness of Fit Test
For today, we are only going to learn one of the types of Chi-Square test:
Chi-square Goodness of fit
What does the Chi-Square Goodness of Fit Test do?
It tells us whether a certain result is consistent with a hypothetical result (same as other significance tests we’ve done)

4. For Example
Mars, Inc. makes M&M’s. The company claims that the distribution of colors is as follows: 13% brown, 13% red, 14% yellow, 16% green, 20% orange, and 24% blue.
We take a random sample of 60 M&M’s (one bag), and we get the following result:
Is this sample consistent with the company’s claims? If not, maybe we shouldn’t believe the company…

5. Chi-Square Goodness of Fit
With what we have already learned, we could do a one-proportion-Z-Test for each different color.
But this has some potential problems. Such as…

6. Chi-Square Goodness of Fit
With what we have already learned, we could do a one-proportion-Z-Test for each different color.
But this has some potential problems. Such as…
More work—imagine a variable with 35 categories. You would have to do 35 significance tests
What if one category rejects the null and another doesn’t? You probably want to draw a conclusion about the company’s claims overall, not about each color individually
Instead, we can do a single Chi-Square Goodness of Fit test (considers all categories at the same time)

7. Hypotheses
One nice thing about Chi-Square Goodness of Fit tests: we typically state our hypotheses just with words, rather than symbols

8. Hypotheses
We could write our hypotheses with symbols, if we wanted to:
But this is not as clear/intuitive to most people. I would recommend using words

9. Observed vs. Expected

10. Back to our Example
Claimed proportions: 13% brown, 13% red, 14% yellow, 16% green, 20% orange, and 24% blue
Calculate the expected number of each color in a sample of 60 M&M’s

11. Back to our Example
Claimed proportions: 13% brown, 13% red, 14% yellow, 16% green, 20% orange, and 24% blue
Calculate the expected number of each color in a sample of 60 M&M’s

12. Finding the Test Statistic
From these observed and expected values, we can calculate a test statistic (just like Z or T in previous significance tests)
In this case, our test statistic is Chi-Square

13. Finding the Test Statistic
Let’s try it for our example

15. Chi-Square Distribution

16. Using the Test Statistic
We want to find out the probability of getting a result at least as extreme as the one we got

17. Using the Test Statistic
We want to find out the probability of getting a result at least as extreme as the one we got
You could use Table C (feel free to try it)
I don’t particularly like Table C
We can use cdf instead
Just like we did normalcdf for a normal sampling distribution, and tcdf for a T sampling distribution

18. Using the Test Statistic
We can use cdf instead
Just like we did normalcdf for a normal sampling distribution, and tcdf for a T sampling distribution
So, in our example, =10.18
Lower bound:
Upper bound:
Df:

19. Using the Test Statistic
We can use cdf instead
Just like we did normalcdf for a normal sampling distribution, and tcdf for a T sampling distribution
So, in our example, =10.18
Lower bound: 10.18
Upper bound: Big Number
Df: 5
Result: what does this tell us?

20. Using the Test Statistic
Result: what does this tell us?
This is our p-value
There is a probability (7% chance) of getting a result at least as extreme as the one we got, IF THE NULL HYPOTHESIS WERE TRUE
Using our standard cutoff of .05 for statistical significance, we would say that this is NOT a statistically significant difference. We fail to reject the null. We cannot conclude that the company’s claim is incorrect

21. Calculator
Unlike previous tests, we HAVE to enter our data into the STAT-Edit before doing the test (and the older TI-83’s don’t even have it). So it isn’t quite as convenient
If you want to try it, go to STAT—TESTS, and then GOF-Test
You have to enter all of your observed in L1, and all of your expected in L2
But, on the plus side, you don’t have to calculate the test statistic by hand
Particularly as you get more categories, this may or may not be worth it

22. Go ahead and give it a try. Do we get the same p-value?
P.S. make sure you use counts, not proportions

23. Conditions
Note: for large sample size condition, use EXPECTED counts, not observed

24. An Example Problem

25. An Example Problem
p-value of No, we fail to reject the null hypothesis that there is no difference, and we cannot conclude that births have different likelihoods on different days of the week.

26. Another Example
When creating a new form of Tobacco plants, biologists expect the color of the new plant to be distributed as follows: 25% of the offspring will be green, 25% will be albino (white), and 50% will be yellow.
Of 84 plants, 23 were green, 50 were yellow, and 11 were albino
Were the results consistent with the researchers’ expectations?

27. Another Example
When creating a new form of Tobacco plants, biologists expect the color of the new plant to be distributed as follows: 25% of the offspring will be green, 25% will be albino (white), and 50% will be yellow.
Of 84 plants, 23 were green, 50 were yellow, and 11 were albino
Were the results consistent with the researchers’ expectations? NO (p-value of .039)

28. Follow-up Analysis
This is ONLY necessary on an AP Test if they specifically ask for it
As far as I can tell, this has only happened once
So being able to do the test is WAY more important than the follow-up analysis
If we reject the null, we conclude that the expected distribution is incorrect. However, we don’t know which categories contribute to that the most.
See next slide

29. So (even though we didn’t reject the null here), the one contributing the most towards rejecting is yellow, because of the 5.186
In fact, more than half of the total distance is due to the difference in yellow

30. An Example
The # of students with each letter grade in AP Statistics is listed below. If the school wanted 20% A’s, 30% B’s, 30% C’s, 15% D’s, and 5% F’s, would Mr. Wetherbee get in trouble?
If yes, which letter is contributing most to that? A B C D F
# Students 10 9 2 5

31. A B C D F
Observed 10 9 2 5
Expected 7
10.5
5.25
1.75
Ch-Square Contribution
1.286
.214
2.012
6.036
Total χ 2 =9.762
P-value: .045
Yes, Mr. Wetherbee would be in trouble
The # of F’s contributes more 60% of this distance, so this would primarily be due to the high # of F’s
The second largest contribution would be from the lower-than-expected number of D’s.