The mathematical purpose of this activity is for 10 people in the class to perform a simulation to analyze the data from the Warm-up. Students reorganize the original data into two groups using a chance process to determine how likely it is that the original results are due to the way the original groups were organized.

Making statistical technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Students may subtract the means so that the differences are always positive. Tell students that the difference between the means can be negative. If the differences are recorded only as positive, some features of the distribution may be missed.

The purpose of this discussion is for students to understand and interpret a randomization distribution. Ask, “Do you think that the difference between the means of Product A and Product B could have happened by chance even if in the population there was no difference in average ratings? Explain your reasoning.” (I am not sure if the difference could be due to chance. It happened 40% of the time by chance in our simulation, so I am not feeling confident that it did not occur by chance. However, 10 feels like it is not enough trials to know how often we would really expect to see a difference of at least 0.8.)

Here is a dot plot and table showing the same simulation repeated 200 times. Display the dot plot and table, and give students quiet think time.

difference between the means	frequency
-1.6	3
-1.2	9
-0.8	24
-0.4	44
0	49
0.4	36
0.8	22
1.2	12
1.6	1

Here are some questions for discussion:

• “36 trials have a difference of -0.8 or less. 35 trials have a difference of 0.8 or greater. In what percentage of the trials are the means from the two groups 0.8 or more apart?” (35.5% of the time, because )
• “Based on the percentage of trials, is it reasonable to say that the original difference between the groups could be due to chance rather than the actual products?” (Yes, that seems reasonable. Because more than one-third of the trials are at least as different as the original grouping, it is reasonable to suspect that the difference in average rating could be due to chance rather than the products themselves.)
• “If the difference between the satisfaction ratings between Product A and Product B had been 1.2, would you have been more or less confident that this could be due to chance? Explain your reasoning.” (I would have been much less confident because a difference of 1.2 happens only 12.5% of the time in the 200 trials.)
• “How do you think this data would change if the number of trials was 500 instead of 200?” (I think that the data would change a little, but 200 seems like enough trials to understand the likelihood of a difference of at least 0.8 occurring at random. 500 trials might yield a more accurate percentage, but would also take longer to do.)

The mathematical goal of this activity is to understand the importance of randomness in experimental design. In a future lesson, students will collect data and analyze the results of the experiment. In this activity, students assess methods for putting subjects into groups and design an experiment to test whether a treatment will affect the results. Students should learn that a treatment is the variable that is changed between two groups in an experiment.

Direct students’ attention to the reference created using Collect and Display. Ask students to share their design for the experiment. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

The goal of this activity is to get students thinking about the importance of randomness in an experimental design. Here are some questions for discussion:

• “Why is it important to divide students into separate groups using a random process?” (If you don’t use a random process, you won’t be sure if the differences between the groups are due to the treatment or the way you divided the groups.)
• “What other methods could be used to randomly assign subjects to two groups?” (Each student could grab an item from a bag in which half of the items are labeled “count” and the other half “silent.”)

Write students' names on papers, put them in a bag, and shake the bag to mix the papers. Draw half of the names to be included in 1 group. If time permits, do the drawing in front of the students rather than beforehand so they can see the random process used to divide students into groups (although the bag could be set up before class). Record the names for the counting group and the silent group to be used in the next lesson.

Ask, “Do you think that the way the two groups were chosen was done using a random process?” (Yes, because we chose our names out of a bag without looking.)