# Lesson 14

Using Normal Distributions for Experiment Analysis

### Lesson Narrative

In this lesson, students work to decide whether a difference in means for experimental groups is likely due to the way the subjects were grouped or likely due to the treatment. Students begin the lesson looking at distributions created from a small experiment in which all of the possible ways to regroup the data can be analyzed. The original difference in means is compared to all the other possible differences in means from redistributing the data into groups at random to decide whether the way the groups were originally set up is a likely cause of the observed difference in means. Later, students use area from a normal distribution based on simulations of regrouping the data to determine whether the original difference in means is significant.

Students must reason abstractly and quantitatively (MP2) when they consider different levels of abstraction from the original data including condensing the data into a difference of means, regrouping the data to find additional difference of means, examining the distribution of the differences of means, and comparing the original difference to the distribution.  Students model with mathematics (MP4) when they approximate the distribution of simulated differences of means by using a normal distribution. Students are asked to find the area under a normal curve for certain values which requires selection of appropriate tools (MP5).

The question of whether to use a one-tailed or two-tailed analysis is an important one in statistics, but beyond the scope of this course. All of the questions in these materials are designed to be analyzed using two-tailed analysis, although the results should be significant or not under either assumption.

### Learning Goals

Teacher Facing

• Use the results from redistributing data from an experiment into groups to determine whether the original difference in means is significant.

### Student Facing

• Let's determine when the results from an experiment are significant.

### Required Preparation

Students will need tools to find the area under a normal curve that is more extreme than a certain value. Acquire devices that can run GeoGebra (recommended) or other statistical technology. It is ideal if each student has their own device. (A GeoGebra Statistics tool is available under Math Tools.) Consider using GeoGebra's Probability view for an intuitive way to visualize the area while allowing the computer to calculate the area.

### Student Facing

• I can calculate the difference in means between two groups.
• I can justify whether there is evidence for a statistical claim by using proportions in the normal distribution.
• I understand that the difference in means can be modeled by a distribution that is approximately normal in shape.

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