##
* *Reflection: Developing a Conceptual Understanding
Assessing Statistical Significance DAY 1 - Section 3: Simulating Possible Results by Hand

How can we distinguish between an occurrence that is statistically significant and one that is merely a coincidence? Our coin toss experiment provides a good introduction to using a simulation to explore the difference. Students gather their simulation results (gathered from individual coin toss trials) and notice that while they are not all the same, they are mound shaped and symmetrical. How does my claim of a ‘special coin’ compare to the probabilities illustrated in our simulated trials? How much difference is significant? The basketball simulation takes this idea a step father. First, we run an experiment and record the result. Then we are tempted to jump to conclusions: “Yes, being distracted does have a negative impact on a student’s ability to score a free throw!” (connect this to: “I got 75/100 heads once, so my coin is special.”) Is it possible that this is a coincidence? Carrying the class through this scaffolded simulation process where distractions are randomized helps them to internalize both the process and the rationale for determining statistical significance.

*Using Simulations to Help Students Understand Significance*

*Developing a Conceptual Understanding: Using Simulations to Help Students Understand Significance*

# Assessing Statistical Significance DAY 1

Lesson 8 of 13

## Objective: SWBAT randomize the results of an experiment in order to assess the statistical significance of these results.

#### Warm-Up and Data Exploration

*20 min*

The purpose of this Warmup is to:

- Practice describing univariate distributions
- Foreshadow the use of simulations in determining whether differences between parameters are statistically significant.

Last night's homework was to flip a fair coin 100 times twice, recording the number of heads that came up in both sets of 100 flips. When my students enter the classroom, I ask them to put their initials on the dot plot on the board to indicate how many heads turned up in 100 flips of a fair coin. Each student's initials should appear on the dot plot twice - once for each of their trials.

When the dot plot is complete, we describe the distribution of student responses using the SOCS pneumonic we learned in an earlier lesson. I remind my students of the special coin I mentioned the previous day and bring it out for them to see. I tell them that once I flipped this coin 100 times and it landed on heads 75 times. I ask if ask anyone doubts that the coin is "special."

This line of questioning always initiates a good discussion about probability. I encourage my students to refer to our class results for evidence of what is likely to happen with a fair coin, and eventually lead them to estimate the probability of 75/100 heads happening with a fair coin based on our results.

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Today's warm-up was the first simulation my students have encountered in my course, so I use this as a bridge to our next topic. Common Core standards **S-IC.4** and **S-IC.5** ask students to use simulations to assess significance and develop margins of error. This can be a very difficult topic for students so I take some time to develop the concept of a simulation.

In the homework/warm-up, we were using a simulation to decide if the difference between the "special" coin's proportion of heads was significantly different from the proportion of a fair coin. Today's basketball activity is also focused on standard S-IC.5.

Before I distribute any paper to students or place them in groups, I ask for 3 volunteers who think they can make a free throw in the class hoop. When these volunteers are selected, I ask for 3 more volunteers who are willing to try to distract the student shooting the basket. I explain that each volunteer will take 10 shots in all, 5 distracted and 5 not distracted, so that we can see if being distracted hindered the ability to make the shot.

Before they begin, I ask the class if they see any problem with having the first 5 shots "distracted" and the second 5 shots "non-distracted." My students always seem to have a keen sense of "fairness" and quickly point out the need for **randomization**. Students may come up with a good idea for randomizing the treatment for this experiment, but if not I suggest that we put 5 pieces of paper with "D" and 5 with "ND" into a hat and draw one out before each shot. This way we are sure to get 5 of each without affecting the probability of any one shot being "distracted."

All three students complete the experiment, we record the results on the board, and then we informally discuss whether the distraction seemed to make a difference. Usually the differences in the percent of baskets that were made is subtle, so there is some question about whether the distraction made a difference.

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Now that my students have had some hands-on experience with the basketball scenario, we use this setting to explore the idea of a simulation. In Basketball Experiment Part 1, students are presented with data from an experiment similar to the one we completed in class. In this experiment, the student made 80% non-distracted shots and 30% distracted shots. By conducting a hands-on simulation, we will explore the likelihood of this happening by chance alone. By comparing the students result (a 50% difference in percentages) to the differences we observe through randomization we will assess the statistical significance of the result.

Students will work individually to read through the first page of Basketball Experiment Part 1 and answer the questions. This page that explains how the simulation helps us assess the significance of the results, so students should be given plenty of quiet time to read this material.

Students will then work in pairs to randomize the results of the experiment and record the resulting statistics. Each pair of students will need one of each of the tables in Basketball Experiment Tables and some scissors.

By the end of the class period, each pair of students should submit their results table so the data can be shared in the following class.

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- LESSON 1: Introduction to Statistics
- LESSON 2: Looking at One-Variable Data Sets
- LESSON 3: Describing Single-Variable Data Sets
- LESSON 4: One-Variable Distribution Activity
- LESSON 5: Bell-Shaped Distributions and the Normal Model
- LESSON 6: Quiz on Distributions and the Empirical Rule
- LESSON 7: Using Technology with Normal Model
- LESSON 8: Assessing Statistical Significance DAY 1
- LESSON 9: Assessing Statistical Significance DAY 2
- LESSON 10: Developing Confidence Intervals DAY 1
- LESSON 11: Developing Confidence Intervals DAY 2
- LESSON 12: Review of One-Variable Statistics
- LESSON 13: Unit Assessment: One-Variable Statistics