1. M & M Statistics: A Chi Square Analysis
AP BIOLOGY
Mrs. Madalon

2. Objective
Apply understanding of a statistical test known as a ”goodness of fit” test, specifically Chi Square Analysis, to determine differences between observed and expected values.
Use chi-square to determine if the Mars Company insures that each package gets the correct number of each color of M&M.

3. Let’s start by stating our null hypothesis…
Null Hypothesis: (in a statistical test) your expected results and observed results are the same!
…any observed difference being due to sampling or experimental error.
NULL HYPOTHESIS
The number of M&M’s in our bag would be representative of the factory %’s and is due to chance, not selection!

4. Procedure Wash your hands
Working with your partner… open up your bag of M&Ms
DO NOT EAT ANY OF THE M&Ms ... Not yet anyway 
Separate the M&Ms into color categories
Count the # of each color of M&M you have
You may now eat any purple M&Ms you find...
Record your color counts under each category in DATA TABLE 1: Individual Data

5. Observed
Record your number of observed M&Ms for each color in DATA TABLE 1: Individual Data
NOW- also put this information on the board to calculate the class data in DATA TABLE 2: Class Data

6. How to Calculate Expected (e) Value
When calculating Expected (e) data… you must refer to the factory %’s for each color in a plain bag of M&Ms. This information is provided on the front page!
NOTE: You cannot use %... You must use numbers. Therefore, convert the factory expected %’s to a number.
For example, Factory expected % for brown M&Ms is 13%... 13/100 = 0.13
To obtain (e)...(e) x Total M&Ms in bag .... For example: 0.13 x 16 = expected value for brown M&Ms

7. How to Calculate Deviation
Deviation (Difference between expected & observed)… To calculate this, (observed – expected).
Say I only observed 2 brown M&Ms in my bag... Using the expected value from the previous slide... my deviation would then be:
2 – 2.08 = -0.08
Deviation Squared (d2): (-0.08)2 = 0.064
D2/e: 0.064/2.08 = 0.03

8. How to calculate X2
This is the sum of all of the d2/e values!

9. How to calculate degrees of freedom
Degrees of Freedom: (n – 1) … this is the number of categories – 1
You need to determine degrees of freedom before you can accept or reject null hypothesis (interpret X2 on the provided table)
You have 6 scenarios in this activity (brown, blue, orange, green, red, & yellow)
Therefore, your degrees of freedom for this activity are
(6 – 1) = 5
WHY MINUS 1? … This is how many other options you have other than the option you got... Therefore, there are 5 other options.

10. How to interpret X2 number…
Looking at your degrees of freedom… find X2 value on the provided table...
If X2 is > 0.05 then you ACCEPT your null hypothesis
If X2 is < 0.06 then you REJECT your null hypothesis
The lower your chi2 (X2 ) number is... The more accurate your experiment!
The larger the sample size (ie: class data) the better/more accurate your data!

11. Now complete DATA TABLE 2: Class Data using the class observed data on the board!

12. Answer the analysis questions and enjoy your M&Ms! 