standard_description: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
problem_activity: Warm-up
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.
Student Facing
A scientist divides 30 strawberry plants into two groups at random. One group of 15 plants will represent the control group and is grown in standard greenhouse conditions. The second group of 15 plans will represent the treatment group and will grow under the same conditions except they are grown in a different type of soil. After 6 weeks, the total weight (in grams) of the strawberries are measured for each plant. The scientist then performs a randomized experiment to compare the groups.
The data are summarized by these statistics and histogram.
Mean for the control group: 238.67 grams
Mean for the group with different soil: 347.47 grams
Mean of differences in means from randomized groupings: -0.540 grams
Standard deviation of differences in means from randomized groupings: 29.83 grams
[IMG]
Is there evidence that the difference in means from the original groupings is due to the different soil or is it likely that the difference is due to the way the plants were grouped? Explain your reasoning.
Student Response
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Activity Synthesis
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.)


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