# Developing Confidence Intervals DAY 1

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## Objective

SWBAT use a simulation in order to develop a confidence interval.

#### Big Idea

Reporting a point estimate along with a margin of error provides much more information than the point estimate alone. We can use simulations to develop a confidence interval.

## Warm-up

15 minutes

While I check the simulation assignment with the homework rubric, I place students in pairs (determined in advance) and ask them to get a laptop.  Students will need to access the student data set that we made on the first day of the statistics unit for today's activity.  I ask them to locate the file on our shared drive.  As a warm-up I ask each student to write a paragraph describing the distribution of our "reaction time" data.  This data was generated by students playing the computer game "Sheep Dash" and recording their scores.

## Gathering and Sharing Class Data

40 minutes

I project a dot plot of our class "Sheep Dash" data on the board and ask one or two volunteers to share their descriptions of the data set.  If necessary, I will take a few minutes to review strategies for describing data sets at this point of the lesson.

I then introduce my students to a very large data set called Census at School.  In this data set, values for 13 variables have been collected from hundreds of thousands of students throughout the US. Because one of the variables is "Sheep Dash" time, we can use this data set to see how our average class time compares to the times of students nationally.

Students work in pairs to add our own data to the Census at School data base.  I take time to do this because I think it is important to participate in this national project that supports nation-wide statistical literacy.  I have registered my class in advance and created a password.  I make this information available to students through Edmodo to make it easy for them to enter the site quickly. Each pair needs a measuring tape as some of the questions are related to measurement.  It takes about 15 minutes for students to complete the survey.

When my students have entered their data, we review our class results at Census at School. The Sheep Dash times should match the ones on the board (I tell students to use their original times rather than playing again).

## Simulating the Sampling Distribution

30 minutes

I ask my students to consider how we might summarize the national Sheep Dash times and compare our own times to the population of students who took the survey.  Hundreds of thousands of students have been surveyed through Census at School and summary statistics are not provided. We can, however, take as many samples as we like.  Therefore, we will work together to take lots of samples (with a sample size about the same as the size of our class), calculate the mean of each of these samples, and then consider the distribution OF THE MEANS.  This is the fundamental process behind "sampling distributions," which is a major topic in introductory statistics classes.

The reason sampling distributions are so useful is that they help us make good guesses about characteristics of a population when we only have sample data to work with.  I make a big deal out of this idea of wanting to know the true mean of the population (the parameter) but having to settle for information gained from samples (statistics) because this is such an important theme in AP Statistics.  Algebra 2 is the only prerequisite for this course at our school, so many of my students will take AP Statistics directly after this course.

I distribute Reaction Time to my students and tell them that they will complete this activity with their partner.  The goal of this activity is for students to work through a simulation of the sampling distribution for sample means.  This activity is very hands-on and informal, but it provides my students with a great deal of insight about sampling distributions.

In this activity, students informally develop a 95% confidence interval by calculating the mean and standard deviation of sample means.  In doing so, they revisit describing univariate data sets and using the normal model when appropriate.  In this class period, I intend for students to complete the data gathering part of the activity, which involves collecting 5 samples each from the Census at School database and calculating the mean reaction time of these samples.  The remainder of the activity will be completed in class tomorrow [MP1, MP4].

## Wrap-up and Assignment

5 minutes

I ask for volunteers to share the mean values that they have found so far, so that as a class we can get a feel for typical values.  I reiterate that what we would really like to know is the mean reaction time of ALL the students that have taken the Census at School survey.  If we knew this, we would have no need for a confidence interval!  However the site is set up so that we can only draw samples, so we will have to use statistical techniques to make a good guess at the mean of this large data set.

I tell my students that if they were not able to calculate 5 means from the data they will need to do so at home.  In class tomorrow, we will consider the distribution of these statistics.