Lesson 13

The mathematical purpose of this activity is for students to compare two distributions. Students may or may not claim there is an important difference between the two distributions for this activity. In the following activity, students will further analyze the data to determine whether there is a significant difference. Monitor for students who use the means (whether calculated or estimated) to compare the distributions.

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

Arrange students in groups of 2. Identify students who calculate the mean and standard deviation or mention margin of error.

The purpose of this discussion is to introduce the idea of variability by chance when comparing two different groups. Ask previously identified students, “How did you use mean, standard deviation, and margin of error to explain the difference between the groups?” (I found out that the mean for product A was 4.4 and that product B had a mean of 3.6. The standard deviation for product A was slightly lower than the standard deviation for product B. I felt like that was enough information to show that product A had a higher product satisfaction rating, but then I started thinking about standard deviation and the margin of error and I wondered if 4.4 and 3.6 were representative of the population mean.) Ask students to discuss their answers to question 2 with their partner.

Here are some questions for discussion:

• “Do you think the results of the survey would be the same if it was repeated?” (I am not sure. The sample size is small and I have no idea about the composition of the larger population.)
• “What could you do to get a better idea of how the two products compare using the same survey?” (You could increase the sample size or take multiple samples.)
• “Do you think some of the differences between the two data sets is due to chance? Explain your reasoning.” (Yes, I think that some of it is due to chance because there are only 5 values in each data set so there is a chance that the 5 values are not representative of the larger population.)

The mathematical purpose of this activity is for ten 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 might be 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).

Tell students, “We are going to create a randomization distribution and I will assign 10 students to represent the two groups of 5 people surveyed in the previous activity.” Select 10 students and distribute a slip from the blackline master representing a data point to each one. Arrange students so that the students with slips with an A are on one side of the room (the first group) and those with a slip with a B are on the other side of the room (the second group). Ask each group to find the mean for the values in their group. Display the means and the difference between the means for all to see.

Remind students to hold on to their slips. Tell students, “I have a bag containing 5 slips that say ‘first group’ and 5 slips that say ‘second group.’” Rearrange students by having each one select one of the slips from the opaque bag at random go to the appropriate side of the room. Find the mean of the first group and the second group. Tell students to record these values in the table. Collect the slips that assign the groups and return them to the bag. Repeat this process 9 more times for a total of 10 trials. Tell students to complete the activity.

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 only recorded 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 happened by chance? Explain your reasoning.” (I am not sure if the difference is due to chance. If it occurred 30% of the time by chance in our simulation I am not feeling so confident that it did not occur by chance. However, 10 feels like it was not enough trials to know how often we would really expect to see 0.8 occur.)

Here is a dot plot and table showing the same simulation repeated 200 times. Display the dot plot and table for all to see 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:

• “In what percentage of the trials are the means from the two groups 0.8 or more apart?” (35.5% of the time since \(\frac{3+9+24+22+12+1}{200} = 0.355\)).
• “Based on the percentage of trials, is it reasonable to say that the original difference between the groups is due to chance rather than the actual products?” (Yes, that seems reasonable. Since 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 is 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 was due to chance? Explain your reasoning.” (I would have been much less confident because a difference of 1.2 happens 12.5% of the time.)
• “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.

Tell students that they will do an experiment involving heart rates in a later lesson. When students finish the question asking about methods for dividing the class, ask students to pause. Select students to share their responses and reasonings for how to divide the class. Discuss the drawbacks of the methods.

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 student 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 one 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.)