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Unit 7, Lesson 13

Using Normal Distributions for Experiment Analysis

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Integrated Math 3

13.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Notice and Wonder

Materials

None

Activity Narrative

The purpose of this Warm-up is to elicit the idea that these distributions are symmetric and approximately normal, which will be useful when students study the origin of the distributions in a later activity. While students may notice and wonder many things about these distributions, the symmetry, shape, and center are the important discussion points.

Launch

Arrange students in groups of 2. Display the images for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.

Activity

None

What do you notice? What do you wonder?

<p>Dot plot from negative 3 to 3 by point 3’s, difference in means. Beginning at negative 3, number of dots above each increment is 1, 0, 1, 0, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1.</p>

<p>Dot plot from negative 4 to 4 by point 5’s, difference in means. Beginning at negative 4, number of dots above each increment is 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1.</p>

Student Response

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Building on Student Thinking

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the images. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

13.2

Activity

Instructional Routines

None

Materials

None

Activity Narrative

In this activity students begin to examine how their understanding of normal distributions can help to determine whether the results from an experiment are significant. For this activity, students recognize that the results from randomizing the groupings for data tend to be approximately normally distributed and centered around 0. In the next activity, students will use the properties of normal distributions to determine how consistent the observed difference in means is with the hypothesis that the treatment does not cause any impact on the measured response.

Students must reason abstractly and quantitatively (MP2) when the analysis leads them to look at the differences in means from possible groups created from the original experiment data.

Launch

None

Activity

None

13.3

Activity

Instructional Routines

Aspects of Mathematical Modeling MLR6: Three Reads

Materials

To Gather

Statistical technology

Activity Narrative

In this activity, students work through a case in which looking at all the possible combinations of regrouping experimental data is too much to consider. In many cases, even the simulations are too many to use to find exact proportions of mean differences to compare with the original difference in means. In these cases, we can use the approximately normal distribution of the simulations to estimate a proportion of redistributed data that would be at least as extreme as the original difference in means.

Students model with mathematics (MP4) when they use a normal distribution to model the simulated difference in means after redistributing the data. Students must also select appropriate tools (MP5) to find the area under the normal curve for the analysis.

This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.

Lesson Synthesis

The purpose of the discussion is to help students understand what they can do to analyze the significance of results from an experiment. To promote discussion, consider asking:

"When the number of subjects in an experiment is large, finding all the different recombinations of data is difficult. What can you do to determine if the original difference in means is significant in this situation?" (Use simulations to determine differences in mean for many possible groupings, then find the mean and standard deviation of those values to create a normal distribution. The normal distribution can then be used to estimate the likelihood of the original difference in means happening even if we assumed there was no impact of the treatment.)
"Why is a normal distribution a good model for differences of means from the simulated regrouping of data?" (The differences in means should produce a distribution that is approximately symmetric and bell-shaped, which is a type of distribution that is usually represented fairly well by a normal distribution.)
"What does the total area on either end of the normal distribution that is more extreme than the original difference in means represent?" (The area represents the approximate proportion of the regroupings that have a difference of means at least as large as the one from the original groups.)
"If you want the analysis to show that there is evidence that the difference in means from the experiment is due to the treatment, would you want the areas on either end of the normal distribution to be large or small? Explain your reasoning." (Small, because the area represents the likelihood of seeing a difference that large or even more extreme when we assume the treatment had no impact. If that likelihood is small, then the difference is likely due to the treatment.)

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Student Lesson Summary

To analyze the significance of the data collected from an experiment, a randomization distribution can be used. In some cases, in which the number of subjects is small, all of the possible ways to regroup the data can be used to compare the original difference in means. When the difference in means is more extreme than most of the differences seen from the randomized regroupings (usually more than 90%, 95%, or 99%, depending on the situation), we can say that we have evidence that the difference in means is due to the treatment.

The more subjects included in the experiment, the greater number of possible regroupings. For example, 14 subjects divided into 2 groups of 7 can have their data redistributed into groups 3,432 different ways. When there are 60 subjects divided into 2 groups of 30, there are more than 118 quadrillion () different ways to redistribute the data into groups of 30. This large number of ways to regroup the data makes looking at the distribution of every possible regrouping difficult.

In these cases, we often do a simulation and redistribute the data many times to get a sense of the true distribution of all possibilities. For example, this histogram shows the difference of means for 1,000 simulations of redistributing 60 data values into 2 groups of 30 each.

<p>Histogram. Horizontal axis from negative 30 to 30 by 10’s, difference of means. Vertical axis from 0 to 50 by 5’s. The histogram shows an approximately normal distribution with a center near 0.</p>

The simulations should produce approximately normal distributions with a center near 0. This allows us to use our understanding of normal distributions to estimate the proportion of regroupings that are at least as extreme as the original difference in means from the experiment. When the proportion is small enough, we should conclude that there is enough evidence to say that the difference in means from the original groups is most likely due to the treatment.

For example, using the values from the histogram, the mean is 0.04 and the standard deviation is 9.07. That provides enough information to create a normal distribution that models the data. In the model image, we see the normal distribution and the regions for which the difference of means might be significant since there is only a 5% chance of the original difference in means being in the shaded region (less than -17.78 or greater than 17.78).

<p>Normal distribution graph, origin O. difference of means.</p>

If the original difference in means is 20, for example, then we can conclude that there is evidence to show that the difference in means is due to the treatment. On the other hand, if the original difference in means is 10, for example, then we should say that there is not enough evidence to conclude that the difference in means is due to the treatment, because the results are also consistent with a model that assumes the treatment had no impact.

To see what might be happening when we regroup data, consider an experiment that takes 12 subjects and divides them into 2 groups at random. The control group contains 6 subjects, and the treatment group contains 6 subjects. To explore what's possible, assume the control group results in these data: 1, 3, 4, 6, 8, and 10. The treatment group results in these data: 2, 5, 7, 9, 11, and 12.

Find the difference in means for the original group by subtracting the control group mean from the treatment group's mean.
With a smaller data set like this, we can actually consider all of the different arrangements of the data. There are 924 distinct ways to separate the 12 values into 2 groups of 6. The frequency table shows all non-negative differences in means and how often they occur. Due to symmetry, the negative differences should match. Notice that a difference in means of 4.33 occurs 7 times, so a difference of -4.33 also occurs 7 times. The dot plot shows the same information.

difference in means 6 5.67 5.33 5 4.67 4.33 4

frequency 1 1 2 3 5 7 11

difference in means 3.67 3.33 3 2.67 2.33 2

frequency 13 18 22 28 32 39

difference in means 1.67 1.33 1 0.67 0.33 0

frequency 42 48 51 55 55 58

Dot plot from negative 6 to 6 by point 3’s, difference in means. Beginning at negative 6, number of dots above each increment is 1, 1, 2, 3, 5, 7, 9, 11, 13, 19, 22, 28, 32, 39, 42, 48, 51, 55, 55, 58, 55, 55, 51, 48, 42, 39, 32, 28, 22, 19, 13, 11, 7, 5, 3, 2, 1, 1.

What proportion of possible groupings have a difference at least as great as the difference in means for the original groups? Explain or show your reasoning.
The proportion you calculate represents the probability that, if we assumed the treatment had no impact on the response, we would still see a difference at least as large as what we saw with our original grouping. Based on the proportion you calculated for this situation, which description is most accurate? Explain your reasoning.
1. Because the proportion is so low, it is unlikely that the difference in means is due to the randomized groupings. This means that the difference in means is most likely caused by the treatment.
2. Because the proportion is not that low, it is possible that the original difference in means is due to the random groupings. This means that there is not enough evidence to determine that the difference in means is likely caused by the treatment.

difference in means	6	5.67	5.33	5	4.67	4.33	4
frequency	1	1	2	3	5	7	11

difference in means	3.67	3.33	3	2.67	2.33	2
frequency	13	18	22	28	32	39

difference in means	1.67	1.33	1	0.67	0.33	0
frequency	42	48	51	55	55	58

Launch

Statistical technology is needed for every student.

Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and distribution, without revealing the questions.

For the first read, read the problem aloud, and then ask, “What is this situation about?” (comparing the weight of wild birds to those raised in captivity) Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured. (the weights of different birds, how many birds are used, the size of each group, the number of ways to regroup the birds)
After the third read, reveal the first two questions and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referencing quantities from the second read. (Use the shape of the distribution to justify the normal distribution model. Then use technology to find the area under the normal model.)

Representation: Internalize Comprehension. Represent the same information through different modalities by using images that compare the area under the curve in this situation and the area under the curve in the previous activity.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

Activity

None

Researchers want to know the effect of captively raising birds on the weight of the birds. The researchers begin with 100 birds divided into 2 groups of 50 each. One group of 50 is raised in captivity and the other 50 are tagged and released into the wild. After 5 years, all 100 birds are collected and weighed.

There are more than different ways to regroup the 100 birds into groups of 50 again, so looking at all the combinations would be too time consuming to reproduce. In this case, we can run simulations to determine how the original difference in means compares to those from regrouping the data.

The original groups have a difference of means of 0.27 gram. Researchers run 1,000 simulations that regroup the data into 2 groups at random and record the differences in means for the groups in each simulation. The histogram shows the differences in means from the simulations.

<p>Histogram, difference of means, from negative 0 point 4 to 0 point 4 by 0 point 1.</p>

The researchers determine that the mean of the differences of means from the simulations is 0.0021 gram and the standard deviation for the differences of means from the simulations is 0.112 gram.

What features of the distribution in the histogram let us know that modeling with a normal distribution is reasonable?
Model the simulations using a normal distribution with a mean of 0 and a standard deviation of 0.112. What is the area under this normal curve that is greater than 0.27?
What does this area mean in this situation?
Based on the area under the normal curve, is there evidence that the original difference in means is likely due to where the birds spent the 5 years? Explain your reasoning.

Student Response

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Lesson 13

Using Normal Distributions for Experiment Analysis

Warm-up

Standards Alignment

Building On

Addressing

HSS-ID.A

Building Toward

HSS-IC.B.5

HSS-ID.A.4

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSS-ID.A.4

Building Toward

HSS-IC.B.5

Standards Alignment

Building On

HSS-ID.A.4

Addressing

HSS-IC.B.5

Building Toward