Lesson 1Connect the DotsDevelop Understanding

Learning Focus

Represent data using a scatterplot.

Understand the meaning of the correlation coefficient.

Describe the difference between correlation and causation.

After collecting data, how can we tell if there is a relationship between two variables?

Open Up the Math: Launch, Explore, Discuss

One reason to understand functions is that they allow us to model data and then use it to understand the world around us. Data and statistics are used to think about science, medicine, business, sports, and about everything in between. There is a whole field of study focused on collecting and analyzing data, and we will touch on only a little here. Statistics is the practice of asking questions and looking for answers in the data. In the next few lessons, we will be using scatterplots and bivariate data to analyze relationships between variables.

Let’s get started by looking at some data. Your teacher will give you some cards, each of which contains a table of data and a scatterplot of the data. Your job is to examine the data and put these cards in order in a way that makes sense to you and that you can justify.

1.

Compare each scatterplot with its correlation coefficient. What patterns do you see?

2.

Use the data in Set A as a starting point. Keeping the same -values, modify the -values to obtain a correlation coefficient as close to as you can.

 $2$ $2.3$ $3.3$ $3.7$ $4.2$ $4.6$ $4.5$ $5$ $5.5$ $5.7$ $6.1$ $6.4$

What did you have to do with the data to get a greater correlation coefficient?

3.

This time, again start with the data in Set A. Keep the same -values, but this time, modify the -values to obtain a correlation coefficient as close to as you can.

 $2$ $2.3$ $3.3$ $3.7$ $4.2$ $4.6$ $4.5$ $5$ $5.5$ $5.7$ $6.1$ $6.4$

What did you have to do with the data to get a correlation coefficient that is closer to ?

4.

One more time: start with the data in Set A. Keep the same -values, and modify the -values to obtain a correlation coefficient as close to as you can.

 $2$ $2.3$ $3.3$ $3.7$ $4.2$ $4.6$ $4.5$ $5$ $5.5$ $5.7$ $6.1$ $6.4$

What did you have to do with the data to get a correlation coefficient that is negative?

5.

What aspects of the data does the correlation coefficient appear to describe?

6.

On the night before the last math test, Shaniqua held a study group at her house. It was a fun night; they ate a lot of pizza, did math, and laughed a lot. Shaniqua scored better on her test than usual and thought it might be related to pizza. She collected the following data from her friends in the study group:

 Number of Pizza Slices Eaten % Increase in Test Score Shaniqua David Susana Ruby Deion Oscar $2$ $6$ $1$ $4$ $3$ $5$ $5$ $9$ $4$ $7$ $6$ $8$

Create a scatterplot of this data, and calculate the correlation coefficient.

Based on this data, would you recommend eating pizza on the night before a test to increase scores? Why or why not?

1.

Describe a situation with two variables that may have a high correlation but are not causally related.

2.

What are some reasons that two variables may be highly correlated but may not have a causal relationship?

Takeaways

It means:

The scatterplot looks like:

The correlation coefficient tells us: the strength of a linear relationship between two variables.

If the correlation coefficient is:

If the correlation coefficient is:

If the correlation coefficient is:

If the correlation coefficient is:

If the correlation coefficient is:

Lesson Summary

In this lesson, we have learned about representing two-variable quantitative data with a scatterplot. We have learned that one of the ways we can judge if a line is a good model for the data is by using the correlation coefficient.

Retrieval

Use the univariate data set for problems 1 through 3.

1.

Find the measures of central tendency (mean and median) for the data set.

2.

Find the range for the data set.

3.

Create a box plot for the data set.

4.

Create an explicit equation for the sequence, and use it to find the value of the term of the sequence.