M&M Statistics/A Chi Square Analysis
Notes From the teacher
Day 1:
Before class:
· Review the lab.
· Complete the pre-lab – typically you will complete this BEFORE class.
· Title and date of the lab (remember to add this lab to your table of contents).
· Purpose à 1-2 sentences describing the overall goal of the lab; use complete sentences
· Hypothesis à write a clear and concise NULL HYPOTHESIS
· Lab Procedure à Write a procedure for the lab. First list the materials that you will be using, then use your own words to describe the steps in experiment in paragraph form or in a numbered list.
· Pre Lab Questions à Copy and answer in your lab notebook.
In class:
· Complete the procedure for the lab.
o Record your data using a table similar to the one provided. (Another option for this lab is to cut out and tape the data tables and the key into your lab notebook.)
· Complete the analysis section.
o Copy and then answer the analysis questions à you need to number the question, rewrite the question, and then answer it for full credit)
o Include the class data table.
M&M STATISTICS/A CHI SQUARE ANALYSIS
Learning Objective à We will be calculating a statistical value and using a table to determine the probability that any difference between observed data and expected data is due to chance alone.
Background Information à Have you ever wondered why the package of M&Ms you just bought never seems to have enough of your favorite color? How do they determine what colors go in each bag?! Well, according to the M&M website:
Brown / 13% / 12% / 17% / 13% / 10% / 10%
Yellow / 14% / 15% / 17% / 13% / 20% / 20%
Red / 13% / 12% / 17% / 12% / 10% / 10%
Green / 16% / 15% / 16% / 12% / 20% / 20%
Blue / 24% / 23% / 17% / 25% / 20% / 20%
Orange / 20% / 23% / 16% / 25% / 20% / 20%
So, do those percentages look correct to you? One way that we could determine if the Mars Co. is true to its word is to sample a package of M&Ms and do a type of statistical test known as a Chi Square (X2) Analysis. This type of statistical test allows us to determine if any differences between our observed measurements (counts of colors from our M&M sample) and our expected (what the Mars Co. claims) are simply due to chance or some other reason (i.e. the Mars company’s sorters aren’t doing a very good job of putting the correct number of M&M’s in each package).
We begin our analysis by stating the null hypothesis. A null hypothesis is the prediction that something is not present, that a treatment will have no effect, or that there is no difference between treatment and control. Another way of saying this is the hypothesis that an observed pattern of data and an expected pattern are effectively the same, differing only by chance, not because they are significantly different.
To test this hypothesis we will need to calculate the X2 statistic, which is calculated in the following way:
X2 = Σ(sum of) (O-E)2
E
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Pre Lab à Done BEFORE you come to class
1. Title and date of the lab (remember to add this lab to your table of contents).
2. Purpose à 1-2 sentences describing the overall goal of the lab; use complete sentences
3. Hypothesis à write a clear and concise NULL HYPOTHESIS
4. Lab Procedure à Write a procedure for the lab. First list the materials that you will be using, then use your own words to describe the steps in experiment in paragraph form or in a numbered list.
5. Pre Lab Questions à Copy and answer in your lab notebook.
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PreLab Questions à
1. What is a null hypothesis?
2. What does a chi square test tell us?
3. Write the equation to determine chi square.
4. In what case would you accept or reject the null hypothesis?
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Procedure à
Materials: Calculator Bag of M&Ms
1. Lay out a large sheet of paper—you’ll be sorting M&Ms on this.
2. Open up a bag of M&Ms.
3. DO NOT EAT ANY OF THE M&M’S (for now!)
4. Separate the M&M’s into color categories and count the number of each color of M&M you have.
5. Record your data.
Data Table 1: M & M Individual Raw Data
Color Categories
Brown / Blue / Orange / Green / Red / Yellow /Total
Observed(o)
Expected (e)
6. Record your data on the Individual Raw Data Table (Data Table 1) and also on the Class Data Table (Data Table 2) on the teacher’s computer.
7. Once your data has been recorded you may eat the M & M’s
8. Complete the data table to determine your expected results.
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Analysis Sectionà Copy the questions into your lab notebook and answer. Make sure to show all your work when doing the math.
1. Calculate the Chi square value. Show ALL of your work!
Now you must determine the probability that the difference between the observed and expected values occurred simply by chance. The procedure is to compare the calculated value of the chi-square to the appropriate value in the table below. First examine the table. Note the term “degrees of freedom”. For this statistical test the degrees of freedom equal the number of classes (i.e. color categories) minus one:
degrees of freedom = number of categories –1
2. In your M&M experiment, what is the number of degrees of freedom?
The reason why it is important to consider degrees of freedom is that the value of the chi-square statistic is calculated as the sum of the squared deviations for all classes. The natural increase in the value of chi-square with an increase in classes must be taken into account.
Scan across the column corresponding to your degrees of freedom. Values of the chi-square are given for two different probabilities, ranging from 0.05 to 0.01. Note that the chi-square increases as the probability decreases. If your exact chi-square value is not listed in the table, then estimate the probability.
*This is a copy of the Chi-Square Table that you will have on the AP Biology Exam
Scientists, in general, are willing to say that if their probability of getting the observed deviation from the expected results by chance is greater than 0.05 (so our chi square is LESS than the P value at 5%), then we can accept the null hypothesis. In other words, there is really no difference in actual ratios….…any differences we see between what Mars claims and what is actually in a bag of M&Ms just happened by chance sampling error. Five percent! That is not much, but it’s good enough for a scientist.
If, however, the probability of getting the observed deviation from the expected results by chance is less than 0.05 (so our chi square is GREATER than the P value at 5%) then we should reject the null hypothesis. In other words, for our study, there is a significant difference in M&M color ratios between actual store-bought bags of M&Ms and what the Mars Co. claims are the actual ratios. Stated another way…any differences we see between what Mars claims and what is actually in a bag of M&Ms did NOT just occur by chance sampling error.
BASICALLY, IF YOUR CHI SQUARE VALUE IS LOWER THAN THE P VALUE (FROM THE TABLE) YOU ACCEPT THE NULL. IF YOUR CHI SQUARE VALUE IS HIGHER THAN THE P VALUE (FROM THE TABLE) YOU REJECT THE NULL.
3. Based on your individual sample, should you accept or reject the null hypothesis? Why?
Now that you completed this chi-square test for your data, let’s do it for the entire class, as if we had one huge bag of M&Ms. Using the information reported on the board, copy Data Table 2 and complete it.
Data Table 2: M & M Class Raw Data
Color Categories
Brown / Blue / Orange / Green / Red / Yellow /Total
ObservedExpected
4. Calculate the Chi square value. Show ALL of your work!
5. Based on the class data, should you accept or reject the null hypothesis? Why?
6. Were your calculated chi square values consistent between your individual and the class data? Why do you believe this to be true?