Least Squares and Exploring Data Essay

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Chapter 6: Exploring Data:
Relationships
Lesson Plan
 Displaying Relationships: Scatterplots

For All Practical
Purposes

Mathematical Literacy in Today’s World, 7th ed.  Regression Lines
 Correlation
 Least-Squares Regression
 Interpreting Correlation and Regression

© 2006, W.H. Freeman and

1

Chapter 6: Exploring Data: Distributions
Displaying Relationships
 Relationship Between Two Variables
 Examine data for two variables to see if there is a relationship between the variables. Does one influence the other?
 Study both variables on the same individual.
 If a relationship exists between variables, typically one variable influences or causes a change in another variable.
 Explanatory variable explains, or causes, the change in another variable.
 Response variable measures the outcome, or response to the change.

Response variable –
A variable that measures an outcome or result of a study (observed outcome). Explanatory variable –
A variable that explains or causes change in the response variable. 2

Chapter 6: Exploring Data: Distributions
Displaying Relationships: Scatterplots
 Data to Be Used for a Scatterplot
 A scatterplot is a graph that shows the relationship between two numerical variables, measured on the same individual.
 Explanatory variable, x, is plotted on the horizontal axis, (x).
 Response variable, y, is plotted on the vertical axis (y).
 Each pair of related variables (x, y) is plotted on the graph.

Example: A study done to see how the number of beers that a young adult drinks predicts his/her blood alcohol content (BAC). Results of 16 people:
Explanatory variable, x = beers drunk Response variable, y = BAC level

Young Adult

1

2

3

4

5

6

7

8

Beers

5

2

9

8

3

7

3

5

BAC

0.10

0.03

0.19

0.12

0.04

0.095

0.07

0.06

Young Adult

9

10

11

12

13

14

15

16

Beers

3

5

4

6

5

7

1

4

BAC

0.02

0.05

0.07

0.10

0.85

0.09

0.01

0.05

3

Chapter 6: Exploring Data: Distributions
Displaying Relationships: Scatterplots
 Scatterplot

BAC vs. number of beers consumed

 Example continued: The scatterplot of the blood alcohol content, BAC, (y, response variable) against the number of beers a young adult drinks
(x, explanatory variable).
 The data from the previous table are plotted as points on the graph (x, y).

Examining This Scatterplot…
1. What is the overall pattern (form, direction, and strength)?  Form – Roughly a straight-line pattern.
 Direction – Positive association (both increase).
 Strength – Moderately strong (mostly on line).

2. Any striking deviations (outliers)? Not here.

Outliers – A deviation in a distribution of a data point falling outside the overall
4
pattern.

Chapter 6: Exploring Data: Distributions
Regression Lines
 Regression Line
 A straight line that describes how a response variable y changes as an explanatory variable x changes.
 Regression lines are often used to predict the value of y for a given value of x.
BAC vs. number of beers consumed

A regression line has been added to be able to predict blood alcohol content from the number of beers a young adult drinks.
Graphically, you can predict that if x = 6 beers, then y = 0.95 BAC.
(Legal limit for driving in most states is BAC = 0.08.)
5

Chapter 6: Exploring Data: Distributions
Regression Lines
 Using the Equation of the Line for Predictions
 It is easier to use the equation of the line for predicting the value of y, given the value of x.
Using the equation of the line for the previous example: predicted BAC
= −0.0127 + (0.01796)(beers) y = −0.0127 + 0.01796 (x)
For a young adult drinking 6 beers (x = 6): predicted BAC = −0.0127 + 0.01796 (6) = 0.095

 Straight Lines
 A straight line for predicting y from x has an equation of the form: predicted y = a + b x
 In this equation, b is the slope, the amount by which y changes when x increases by 1 unit.
 The number a is the intercept, the value of y when x =0.
6

Chapter 6: Exploring Data: Distributions
Correlation
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