Learning Targets

• Determine if a statistic is an unbiased estimator of a population parameter.

• Describe the relationship between sample size and the variability of a statistic.

Experience First

To prep for this lesson, each group will need a paper bag with 78 popsicle sticks in it, labeled with integers from 1 to 78. We chose 78 as the magic number, but you could choose any value (but you might have to adjust the scales for the dotplots on question #4 of the activity). You will also need to create some posters for these dotplots and students will need sticker dots. 

 

Begin the lesson by telling students that there is an unknown number of popsicle sticks in the paper bag (the value is the same for every group). The nice thing is that every popsicle stick has a number from 1 up to N, where N is the total number of sticks in the bag. Unfortunately, students are not allowed to look at all of the sticks in the bag; instead they will pick a random sample of 7 sticks. 

 

Using a bag that you have created for demonstration purposes, have a student randomly select 7 sticks and record the values on the board. Students are now ready to work in their groups on questions #1-2 on the activity. 

 

Students have a STOP sign after question #2. This allows you to have students share out their proposed methods and estimates for the total number of popsicle sticks. After the brainstorm, you will reveal to students the two methods that we will try out:

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1. Find the maximum of the sample.

2. Double the median of the sample.

 

Assign half of the groups in the room to test Method 1 and half the groups in the room to test Method 2. Students can then work through the remainder of the activity. As each group gets to question #5, you will need to reveal to them the true magic number (78). 

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Formalize Later

The goal of the lesson is for students to understand there are different types of statistics and some are good estimators and others are not. The notion that there are statistics other than mean, median, and mode is itself a difficult concept for students to grasp. You’ll want to make sure to really hammer home the definition of a statistic as a number that is calculated from a sample. 

 

When debriefing #4 in the activity, be sure to ask students “What does this dot represent?”

 

Hopefully you still have your dotplot poster from Lesson 4.2 (remember Beyonce?), where students learned that a poor sampling method can lead to a statistic that is a biased estimator. In this lesson, they learn that a poor formula for a statistic can lead to a biased estimator. 

 

We like to present the definition of unbiased estimator in two ways:

• A statistic is unbiased if it does not consistently underestimate or consistently overestimate the true parameter.

• A statistic is unbiased if the mean of the sampling distribution is equal to the true parameter. 

 

Hopefully you also still have your dotplots from Lesson 4.4 (Beyonce again!) where students learned that increasing the sample size decreases the variability of the estimates. 

 

The popsicle stick problem is based on the German Tank Problem.