Learning+Center+Design

**Learning Center Design** ** Center Information ** **Author:** Samantha Mowery **Title of the Centers:** Where are all the yellow M&Ms? An introduction to the Chi-Square Test for Goodness of Fit **Content Area:** AP Statistics/AP Biology **Types of Centers:** Exploratory, Skill & Enrichment **Organization of Centers:** Sequential - The students will work in groups of 3 or 4 to discover the importance and uses of the Chi-Square test statistic.  Each group will begin with the first center and check with the teacher before moving to the next center. **Number of Centers:** 4 **Time Needed:** approx. 45 min

** Learning Objectives (cognitive and behavioral) ** Students will understand the value of a chi-square test statistic and p-value by learning
 *  The meaning of a chi-square test statistic
 *  About the inverse relationship between a chi-square test statistic and p-value
 *  How a chi-square test statistic and p-value effects one’s ability to prove an alternative hypothesis

Using a real world example, students will collect data to prove or fail to prove an alternative hypothesis.

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Without using a calculator, students will calculate a chi-square test statistic and p-value to form a conclusion about a real world situation.

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Using a calculator, students will calculate a chi-square test statistic and p-value to form a conclusion about a real world situation

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">** Materials ** <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">1 standard size package of M&Ms (provided by the each student) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">A roll of paper towels <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">TI84 Plus calculator <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Activity worksheet <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Writing utensil

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">** Center Activity Description ** <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">**Introduction:** <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">The teacher will introduce the activity by proposing the following question and hypotheses. <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">**Hypothesis Statements** <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Null: The mix of colors of plain M&Ms is 13% browns, 14% yellows, 13% reds, 20% oranges, 16% greens, and 24% blues. <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Alternative: At least one of the proportions is not consistent with the stated proportions. ||  ||
 * || <span style="display: block; font-family: Verdana,Geneva,sans-serif; font-size: 10pt; text-align: center;">Are the proportions of the colors of plain M&Ms the same as advertised by Mars or is at least one of the proportions different?

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">**Center 1: Collecting Data** <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Students will… <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">1) Open and combine their packages of M&M. They will then separate the candies into groups based on color and count how many of each color they have. Once their counts are complete they will fill in the first row (observed) of the table provided. <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">2) Calculate the expected counts by multiplying the total number of M&Ms by each of the advertised proportions then enter the product into the second row (expected). <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">3) Call for the teacher to come and inspect their findings

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Teacher will… <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">1) Check to make sure the group calculations are correct <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">2) Use questioning techniques to ensure that the students have a strong understanding of what has been done so far and motivate them to make a hypothesis about what their findings may show. <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Potential questions: <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">3) Instruct the students to move on to the next center.
 * 1) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">a. Do you see a large difference between the observed and expected values?
 * 2) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">b. Where do you see the largest difference? The smallest?
 * 3) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">c. Do you believe the manufacturers claim about the proportions of M&Ms in each package?

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Table for Center #1: <span style="font-family: Verdana,Geneva,sans-serif;">Let collect some data! Open your bag of M&Ms and count the number of each color and enter into the row for observed (this is your sample data) and calculate the total. Now calculate the expected counts by multiplying the total by each of the advertised proportions and enter in the expected row **<span style="font-family: Verdana,Geneva,sans-serif;">(never round). **

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Time for Center 1: 6 – 8 min
 * Color || Brown || Red || Yellow || Orange || Green || Blue || Total ||
 * Observed(O) ||  ||   ||   ||   ||   ||   ||   ||
 * Expected (E) ||  ||   ||   ||   ||   ||   ||   ||

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">**Center 2: Checking Conditions** <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Students will… <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">1) Check the following conditions <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">All the expected individual counts are at least 1 <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">No more than 20% of the expected individual counts are less than 5. ||  || <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">2) State the name of the test being conducted and state that the conditions either have or have not been met. <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">3) Call for the teacher to come and inspect their findings
 * ||  || <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">SRS
 * ||  || <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">SRS

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Teacher will… <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">1) Check that the group calculation are correct <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">2) Use questioning techniques to ensure that the students have a strong understanding of what the conditions are checking. <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Potential questions: <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">3) Instruct the students to move on to the next center <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Time for Center 2: 5 min
 * 1) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">a. Have all your conditions been met?
 * 2) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">b. What does condition 3 mean?
 * 3) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">c. When checking the conditions, should you look at the observed or expected counts?
 * 4) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">d. What happens if the conditions are not met?

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">**Center 3: Calculating Chi-Square and P-value** <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Students will… <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">1) Calculate each color’s contribution to the chi-square test statistic using the following formula and record the values in the table below then sum up each colors contribution and record the sum in the box for total.

<span style="font-family: Verdana,Geneva,sans-serif;">**See activity handout**

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">2) Calculate the degrees of freedom (df = n – 1, where n is the number of categories) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">3) Use Chi-Squarecdf (on the TI84 Plus) to find the P-value. <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">4) Use STAT/TESTS/D:Chi-SquareGOF-TEST (on the TI84 Plus) to find Chi-Square test statistic and the P-value <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">5) Compare the values found in steps 2 - 3 and 4. (they should be the same) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">6) Call for the teacher to come and inspect their findings

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Teacher will… <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">1) Check that the calculations are correct <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">2) Use questioning techniques to ensure that the students have a strong understanding of the test statistic and p-value represent about the original hypotheses. <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Potential questions: <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">3) Instruct the students to move on to the next center <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Time for Center 3: 10 – 15 min
 * 1) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">a. What is the lowest value you could find for your test statistic? When would this occur?
 * 2) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">b. How do you calculate your degrees of freedom? Why is it n – 1?
 * 3) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">c. Based on your P-value what conclusions do you want to make about the original hypothesis?
 * 4) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">d. How do you plan to conclude?

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">**Center 4: Conclusions** <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Students will… <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">1) Use the P-value to form a conclusion about the initial hypotheses. <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">2) Call for the teacher to come and inspect their findings

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Teacher will… <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">1) Check to make sure the conclusion includes all the correct wording and information <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Time for Center: 10 min

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">** Assessment ** <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">The teacher will be assessing the students throughout the lesson informally by checking to ensure they have the correct data and using questioning techniques ensure the students’ understanding of each part of the Goodness of Fit Test.

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Once the test is complete the students will work individually on a problem similar to the M&Ms project. [|Goodness of Fit Example.pdf] This will act as their closure for the lesson and will be collected at the end of the class.

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">** Handouts ** <span style="font-family: Verdana,Geneva,sans-serif; font-size: 10pt;">Activity Worksheet - [|M&Ms Activity - Goodness of Fit Introduction.pdf]