Chi Square Modeling Using M & M’s Candies



M & M’s Candies Chi Square Test

Introduction:

The Chi Square test (X2) is often used in science to test if data you observe from an experiment is the same as the data that you would predict from the experiment. This investigation will help you to use the Chi Square test by allowing you to practice it with a population of familiar objects, M & M candies.

Objectives: Before you start this investigation you should be able to:

• write a null hypothesis that pertains to the investigation;

• determine the degrees of freedom (df) for an investigation;

• calculate the X2 value for a given set of data;

• use the critical values table to determine if the calculated value is equal to or less than the critical value;

• determine if the Chi Square value exceeds the critical value and if the null hypothesis is accepted or rejected.

Materials:

• One quarter cup of M&Ms

• Calculator, writing utensil and an M&M counting partner

• Chi-Square critical value table

Procedure:

1. Dump out your candies onto a piece of paper. WITHOUT COUNTING, record all of the different colors you see (classes) in Table 1 and in Table 2.

2. The following M&M recipe information was obtained by one of Willy Wonka’s spies:

|Color |% In Batch |

|Brown |13 |

|Yellow |14 |

|Red |13 |

|Blue |24 |

|Orange |20 |

|Green |16 |

Again, WITHOUT COUNTING, use the recipe information (percentage out of 100%) of the different colors of M&M as the “Percentage Expected” and record in Table 1

Table 1

|Color of Candy |Number Observed (o) |Percentage Expected |Number Expected (e) |

| | | |(Total number M&Ms X Percentage Expected) |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

| |Total # M&Ms = | | |

1. In the space below, write a null hypothesis, which predicts the percentage of the different colors of candies.

2. WITH YOUR PARTNER, count the number of each color of candy and record the number in Table 1 under “Number Observed.”

3. Add up the numbers of each color counted and record the “Total # M&Ms” where indicated.

4. Calculate the number of each color expected in Table 1 and record under “Number Expected.”

5. Record the numbers expected, and the numbers observed in Table 2.

6. Complete the calculations and determine the Chi Square value.

Table 2

|Classes |Expected |Observed |o-e |(o-e)2 |(o-e)2 |

|(Colors) |(e) |(o) | | |e |

| | | | | | |

| | | | | | |

| | | | | | |

| | | | | | |

| | | | | | |

| | | | | | |

| | | | | | |

Degrees of freedom = _________ ( = __________

Analysis Questions:

1. What is the X2 value for your data? _________

2. What is the critical value (p = 0.05) for your data? _____________

3. Is your null hypothesis accepted or rejected? Explain why or why not.

______________________________________________________________________________________

______________________________________________________________________________________

4. Based on the outcome of your analysis, propose an explanation either for why your estimate was so good,

or for why your observed number of M&Ms did not match your estimate (i.e. is something unexpected

happening at the M&M factory?).

______________________________________________________________________________________

______________________________________________________________________________________

ACT Prep

5. Mars, Inc. decides to change the recipe for their M&Ms and add two more colors to their mix. How will

this affect the degree of freedom for the study?

___________________________________________________________________________________

6. If you repeated the experiment with these additional colors and obtained the same X2 value, would you reach the same conclusion about your null hypothesis? Explain.

__________________________________________________________________________________

__________________________________________________________________________________

7. If you repeated the original experiment with a full pound bag (instead of one quarter cup), what effect would this have on the confidence you have in your conclusion? Explain.

__________________________________________________________________________________

__________________________________________________________________________________

8. One study of grand juries in Alameda County, California, compared the demographic characteristics of jurors with the general population, to see if jury panels were representative. The results for age are shown below.

|Age |Count-wide % |# of jurors observed |# of jurors expected |(O-E) |(O-E)2/E |

|21-40 |42% |5 | | | |

|41-50 |23% |9 | | | |

|51-60 |16% |19 | | | |

|over 60 |19% |33 | | | |

|Total |100% |66 | | | |

Complete the table, calculate the X2 value and determine the degrees of freedom. Were the 66 jurors selected at random from the population of Alameda County? Defend your answer using the results from your Chi Square analysis.

____________________________________________________________________________________

____________________________________________________________________________________

____________________________________________________________________________________

Now, EAT YOUR M&Ms !!!

Teacher Tips:

1. I buy the small, individual bags of M & M’s for each student and then they count their own and add data to a partner’s data. You can also add all the numbers for all the groups to get a larger population and more reliable results.

2. Percentages of colors of M & M’s may vary from season to season. For instance, you may find more brown and orange around Halloween, or more green and red around Christmas.

3. Example of data:

Table 1

|Color of Candy |Number Observed (o) |Percentage Expected |Number Expected (e) |

| | | |(Total number of all pieces of candy X Percentage |

| | | |Expected) |

|Green |40 |20% |40 |

|Blue |20 |10% |20 |

|Orange |30 |20% |40 |

|Brown |40 |20% |40 |

|Red |40 |20% |40 |

|Yellow |30 |10% |20 |

| | | | |

| | | | |

| |Total # candies = 200 | | |

Table 2 - Example

|Classes |Expected |Observed |o-e |(o-e)2 |(o-e)2 |

|(Colors) |(e) |(o) | | |e |

|Green |40 |40 |0 |0 |0 |

|Blue |20 |20 |0 |0 |0 |

|Orange |40 |30 |-10 |100 |2.5 |

|Brown |40 |40 |0 |0 |0 |

|Red |40 |40 |0 |0 |0 |

|Yellow |20 |30 |10 |100 |5 |

| | | | | | |

| | | | | | |

Degrees of freedom = 5 ( = 7.5 (number of classes – 1)

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Name _______________________

Period _______

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