Unit 8, Lesson 12
Experimenting
Algebra 2
12.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
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. The important aspect is to get students thinking about how they would approach the comparison and about their justification. In a later 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).
Launch
Arrange students in groups of 2. Identify students who calculate the mean and standard deviation or mention margin of error.
Activity
NoneTen people are selected at random to be given either Product A or Product B to test. After using the product, they rate the product on a scale from 1 to 5, with 1 being the worst and 5 the best.
The dot plots represent their ratings.
- Which product has a higher overall satisfaction rating? Explain your reasoning.
- Do you think that 2 different random samples of 5 people would lead you to the same conclusion?
Student Response
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Building on Student Thinking
Activity Synthesis
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 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 has 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 are representative of the population mean.) Ask students to discuss their answers to the second question.
Here are some questions for discussion:
- “Do you think the results of the survey would be the same if the survey was repeated?” (I am not sure. The sample size is small, and I know little 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?” (I could increase the sample size.)
- “Other than the actual product, what other factors could be at play in the ratings?” (A person might not be willing to give a 5 unless the product is awesome. Personal preference always plays a part in these kinds of ratings. Someone may be having a bad day during the survey and rate the product lower for unrelated reasons.)
- “Do you think the difference between the two data sets is due to chance? Explain your reasoning.” (Yes, I think that some of it is due to chance. 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. If the person who rated Product B with a 2 was replaced with someone who rated the product with a 5, then there would not seem to be a difference in the products. One data value seems like it has a lot of importance.)
Student Lesson Summary
Experiments provide a way to test how a variable affects a situation. It is important to design the experiment carefully so that other variables that might have an effect on the response are accounted for.
One important aspect in experimental design is the use of randomness to assign groups for the experiment. Random assignments are used so that there is less of a chance that other variables will affect the data.
For example, a researcher wants to study the question, “Does the size of the aquarium in which frogs are kept affect the sizes of the frogs?”
The researcher selects 20 young frogs to be used in the experiment. The frogs are then numbered, and a random number generator is used to separate the frogs into two groups of 10. One group will be put in a 10-gallon aquarium to grow, while the other group is placed in a 100-gallon aquarium to grow. The researcher will measure the sizes of the frogs, using their weight, at the end of a year.
The size of the aquarium is one of the variables, and a treatment is the value of the variable that is changed between the two groups in an experiment. In this experiment, there are two treatments—a smaller aquarium and a larger aquarium.
The researcher finds that the frogs in the smaller aquarium have a mean weight of 111.2 grams and the frogs in the larger aquarium have a mean weight of 169.3 grams. The difference of 58.1 grams seems large, but is it enough to say the aquarium size is the cause of the difference? Even if all the frogs were in the same aquarium, we expect there to be some variability in the weight of the frogs. If some of the frogs were switched at the beginning of the experiment, would there still be a large difference? A simulation can be used to see if differences we see in our groupings are also consistent with groupings that did not depend on our treatment.
In this kind of simulation, all the data is grouped together, then data is randomly redistributed among two groups, and the difference between the means of these new groups is computed. This process is repeated several times to create a randomization distribution: a distribution of the differences between the means for the treatment groups containing randomly redistributed data. The mean difference from the experiment is then compared to this distribution to determine whether such differences are also consistent with groupings that did not depend on the treatment. To investigate whether the aquarium size is the important factor in the difference or whether the difference could be due to the chance ways the frogs were separated into treatment groups, we can use a simulation to examine what results we might expect from chance. For the moment, we assume the aquarium does not play a part in the size of the frogs, and we put all 20 frog weights into one group. We can separate the weights into two groups in a random way and determine the difference in mean weights between the groups.
Doing this many times will produce a distribution of weight differences that could be the result of how the frogs happened to be divided into groups. We can then compare the actual difference in mean weights from the treatment groups to the randomization distribution to determine if the 58.1-gram difference is unusual (which would suggest the aquarium size played an important role) or whether this difference is also typical of what we might see from random assignment to groups.
Randomly breaking the data into two groups 30 times produces the histogram here.
Notice that, based on this distribution, a difference of either 58.1 grams or greater or -58.1 or less is unusual. This suggests that the original difference of 58.1 grams is highly unlikely to occur if the aquarium size played no role. The researcher has evidence to support the hypothesis that aquarium size has an impact on the weight of frogs growing in it.
If the original difference had been about 15 grams, there would have been reason to believe that the weight difference observed might be due to random chance because 11 out of the 30 differences in the randomization distribution are at least 15 grams away from 0.