Lesson 15

In this warm-up, students use their understanding of normal distributions to determine whether the difference is likely due to the random groupings or the treatment.

The purpose of the discussion is to recall how the mean and standard deviation determine a normal distribution which can then be used to determine whether the observed difference in means is likely due to the random groupings or whether there is evidence that it is due to the treatment.

Ask students,

• "Why is it useful to know the standard deviation for the differences in means from randomized groupings?" (This number gives a sense of how spread out we expect the differences to get based on the groupings. It can be used to determine a normal distribution to determine how likely it is that the observed difference in means is due to chance groupings.)
• "How could the 'mean of differences in means from randomized groupings' be calculated using the original data?" (Redistribute the original data into two groups using a chance process to create new randomized groupings. Find the mean of each group and record the difference in means. Repeat the process many times to collect many differences in means for the groups. Find the mean of those recorded differences.)

This info gap activity gives students an opportunity to determine and request the information needed to assess whether the results from an experiment are either significant or due to chance groupings.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Tell students they will continue to examine data from experiments to determine if there is a significant difference in means or whether the difference is due to random groupings. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it. Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “What questions did you ask that were helpful? Explain your reasoning.” (It was helpful to start by asking about the difference in means from the original groupings. That gave me a benchmark against which I could measure the information about the trials in which the data was randomly regrouped.)
• “Select a piece of information from one of the data cards. How could people collect or compute that information?” (After collecting the data from the experiments, find the mean of each group and subtract them.)

This activity is optional since it involves skills only needed in the absence of technology for finding areas under the normal curve.

In this activity, students explore the use of tables to find area under a normal curve bounded by certain values. Students are first introduced to z-scores and then use a table to find the area under the normal curve that is less than the value associated with that z-score.

Distribute the tables that show the areas under the standard normal curve based on z-scores. Demonstrate how to use the table to find the area for z-scores

• -1.96 (area of 0.0250)
• -0.50 (area of 0.3085)
• 1.65 (area of 0.9505)

Some students may struggle to recognize the area given by the table using \(z\)-scores and think the area given is always the more extreme portion. Ask students to think about the value in the table and remind them that the entire area is 1, so if the value is greater than 0.5 it must be a region that takes up more than half of the area under the curve.

The purpose of the discussion is for students to understand how to use a table to find areas under a normal curve.

Select students to share their solutions for all of their questions including their reasoning for the area under the normal curve between 142 and 149 grams.

Some additional questions for discussion:

• "Why are all the areas for positive z-scores greater than 0.5?" (Since positive z-scores represent values greater than the mean and the mean is the center of the distribution, at least half of the area is less than the value.)
• "Remember that the area under the entire normal curve is 1. How could you find the area of the region greater than a value that has a z-score of 0.95?" (The area under the normal curve and less than the value with a z-score of 0.95 is 0.8289. Since the entire area is 1, the area greater than that value is 0.1711 since \(1 - 0.8289 = 0.1711\).)