standard_description: Know and apply the Binomial Theorem for the expansion of (x + y)^(n )in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)
problem_activity: Activity
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 based on redistributing the data. Students must also select appropriate tools (MP5) to find the area under the normal curve for the analysis.
Launch
Provide access to devices that can run GeoGebra or other statistical technology.
Reading, Listening, Conversing: MLR6 Three Reads.
Use this routine to support reading comprehension of this word problem. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (researchers regroup data to study the effects of captivity on the weight of birds). After the second read, students list any quantities that can be counted or measured, without focusing on specific values (total number of birds, number of groups, group size, the number of ways to regroup). After the third read, ask students to brainstorm possible strategies to answer the questions. This helps students connect the language in the word problem and the reasoning needed to solve the problem while keeping the cognitive demand of the task.
Design Principle(s): Support sense-making
Representation: Internalize Comprehension.
Represent the same information through different modalities by using images which show the area under the curve for this situation and the one in the previous activity for comparison. Some students may benefit from a checklist or list of steps to be able to use the appropriate technology to find the areas.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
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 will be 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
\(10^{29}\)
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.
[IMG]
The original groups have a difference of means of 0.27 grams. Researchers run 1,000 simulations regrouping 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.
[IMG]
They determine that the mean of the differences of means from the simulations is 0.0021 grams and the standard deviation for the differences of means from the simulations is 0.112 grams.
What features of the distribution in the histogram let you know that modeling with a normal distribution is reasonable?
Model the simulations using a normal distribution with a mean of 0.0021 and a standard deviation of 0.112. What is the area under this normal curve that is more extreme than 0.27?
How can this area be used to compare the difference of means from the simulations to the difference of means from the original groups?
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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Student Facing
Are you ready for more?
Suppose we decide that if the probability of observing our difference or a more extreme difference happens less than 5% of the time in simulations we will conclude that it was captivity that caused the difference, and if the probability is greater than 5% we will not draw any conclusions. In what ways could the conclusion we make (or decide not to make) be wrong?
Student Response
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Activity Synthesis
The purpose of this discussion is to show that areas under normal curves can be used to help decide whether the difference in means from an experiment are significant.
Select students to share their solutions and reasoning. To promote discussion, consider asking: "In this problem, a 1.5% chance of the difference in means from the original groups happening by random chance seems low enough to conclude that the difference in means is likely due to the treatment rather than the way the subjects were grouped. What percentage might make you change your opinion on whether the groupings were the cause of the original difference in means?"
Tell students that standard cutoffs used in statistics are 1%, 5%, or 10% before the results are called into question.
Ask students, "Why do you think it would be important to set your expectations for a cutoff
before
doing the experiment?" (Setting the expectations can help decide how many subjects to include in the study and does not unfairly bias the results. Some experimenters may continue to run new simulations until their results become significant.)


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