Math II Unit 6Statistics: Finding the Best Model

Wednesday March 30, 2010

Standards

MM2D2. Students will determine an algebraic model to quantify the association between two quantitative variables.

a. Gather and plot data that can be modeled with linear and quadratic functions.

b. Examine the issues of curve fitting by finding good linear fits to data using simple methods such as the median-median line and “eyeballing.”

c. Understand and apply the processes of linear and quadratic regression for curve fitting using appropriate technology.

d. Investigate issues that arise when using data to explore the relationship between two variables, including confusion between correlation and causation.

Examine RelationshipsEQ: How are scatter plots created?What are the basic properties of correlation?What is the difference between correlation and cause-and-effect relationship?

Bivariate Data

A dataset with two variables contains what is called bivariate data.

Quantitative Variable

Variables that differ in amounts or scale and can be ordered ◦(e.g. weight, temperature, time).

Ex: x can represent...Weight of a person

Non-Ex: x can represent colors

Scatter Plot

A scatter plot is a good visual picture of a set of data.

Each relationship contributes one point to the scatter plot, on which points are plotted but not joined!

Correlation

Correlation is a statistical technique that can show whether a pair of variables are related and how strong that relationship is.

Positive Correlation

A positive correlation – As the x-value increases the y-value increases.

On a scatter plot, there will be an upward trend (positive slope)

Negative Correlation

A negative correlation – As the x-value increases, the y-value decreases.

On a scatter plot, there will be a downward trend (negative slope).

No Correlation

A relation where there is NO correlation would produce a scatter plot that does not indicate any trends whatsoever.

Causation

A dependency between two variables, where one is the cause and the other the effect.

Causation vs Correlation

An action or occurrence can causes another (such as smoking causes lung cancer), or it can correlate with another (such as smoking is correlated with alcoholism).

If one action causes another, then they are most certainly correlated. But just because two things occur together does not mean that one caused the other, even if it seems to make sense.

Examples of Causation vs Correlation

1. "People who own red cars are twice as likely to have an accident as people who own blue cars.“

Is this an example of Causation or Correlation?

2. Independent Variable: Temperature of a day in Manhattan

Dependent Variable: Number of ice cream vendors out on that day.

Is this an example of Causation or Correlation?

Linear ModelsEQ: How is a best-fitting line determined?

Linear Regression

Regression - The process of finding a function whose graph approximates a set of data.

Linear regression - When we find a linear function whose graph approximates a set of data.

Visual linear regression - The method of

approximating lines of best fit…”eyeballing”

GO #1: How do we “eyeball” a line best fit a scatter plot?

There are several lines passing through the scatter plot below. Which one do you think best fits the data?

Guidelines for “Eyeballing” a Best-fit line…

1. ___________________________________________________________________

2. ___________________________________________________________________ 3. ___________________________________________________________________

Steps for Writing Equation to Best-fit Line… 1. __________________________________ 2. _________________________________ 3. __________________________________ 4. __________________________________

Work for Example Above…

Guidelines for “Eyeballing” a Line of Best-fit

1. Show direction of points: Sketch the smallest rectangle that will contain all points to determine the general direction of the points.

2. The line should divide the points equally: Draw a line so that there are about as many points above the line as below the line.

3. Draw line where it will go through or at least touch two points: You will need two points to calculate the equation to your best-fit line.

Steps for Writing Equation to Best-fit Line

1. Select two points on your line. (x1, y1) and (x2, y2)2. Use slope formula and your two points to find the slope of the line. 3. Slope-intercept form: y = mx + bUsing your slope and one of the identified points,

substitute the slope in for m and the point in for x and y, and solve for b (y-intercept)

4. To write equation: y = mx + bSubstitute slope in for m and y-intercept (found in

step 3) in for b.Leave x and y as variables.

12

12xxyym

Median-Median LineEQ: How do we find the Median-Median Line?

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

GO # 2: How do we find the median-median line from a set of data?

Example:

Steps for finding Median-Median Line

Step 1: Divide the points into 3 equal groups.

If there were 1 extra point, it would go in the center section. If there were 2 extra points, 1 would be in each of the two outside sections.

Step 2: Find the median x-coordinate and the median y-coordinate in each group of points. This point may or may not be on your graph.

Median-Median Line

Step 3: Draw a line through the two points you found in the outside sections. (This line may or may not pass through the original points on the graph.)

Step 4: Draw a line passing through the point you found in the center section. This line should be drawn parallel to the line you just drew through the outside points.

Median-Median Line

Step 5: Draw a line between and parallel to the two line you have just drawn.

The new line should be 1/3 of the distance from the first line to the second line. (In other words, it should be closer to the line through the outside points.)

This is the median-median line.

CALCULATOR TIME!!!EQ: How do we use the graphing calculator to find the equation to the Linear Regression Line or Median-Median Line?

Pierce (1949) measured the frequency (thenumber of wing vibrations per second) of chirps made by a ground cricket, at various ground temperatures.  Since crickets are ectotherms (cold-blooded), the rate of their physiological processes and their overall metabolism are influenced by temperature.  Consequently, there is reason to believe that temperature would have a profound effect on aspects of their behavior, such as chirp frequency.

Biological Data(or the realities of working with real-life data)

Data:  The following data shows the relationship between chirps per second of a ground cricket and the corresponding ground temperature. 

Chirps/Second Temperature (º F)

20.0 88.616.0 71.619.8 93.318.4 84.317.1 80.615.5 75.214.7 69.717.1 82.015.4 69.416.2 83.315.0 78.617.2 82.616.0 80.617.0 83.514.1 76.3

1. Determine a linear regression model equation to represent this data. Y = ______________

2. Graph the new equation using the calculator steps.

3. Decide whether the new equation is a “good fit” to represent this data.

3.244x + 26.012

Interpolate To estimate values of (data or a function)

between two known values.

Extrapolate To estimate values of (data or a function)

outside known values.

Correlation Coefficient

How well does your regression equation truly represent your set of data?

  One of the ways to determine the answer to this question is

to exam the  correlation coefficient.

The correlation coefficient measures the direction and the strength of the linear association between two numerical paired variables.  

(be sure the Diagnostics are turned on ---2nd Catalog (above 0), arrow down to DiagnosticOn, press ENTER twice.)

Correlation Coefficient

The linear correlation is represented by the variable r.

The value of r will be a value where -1 < r < +1. 

The + and – signs are used for positive linear correlations and negative linear correlations, respectively. 

A perfect correlation of ± 1 occurs only when the data points all lie exactly on a straight line. 

Correlation Coefficient

r = +1, the slope of this line is positive r = -1, the slope of this line is negative

If there is no linear correlation or a weak linear correlation, r is close to 0

A correlation greater than 0.8 is generally described as strong.

A correlation less than 0.5 is generally described as weak.

Correlation Coefficient

Quadratic ModelsEQ: What does a data set look like if the best-fit curve is quadratic?

On Tuesday, May 10, 2005, 17 year-old Adi Alifuddin Hussin won the boys’ shot-putt gold medal for the fourth consecutive year. His winning throw was 16.43 meters. A shot-putter throws a ball at an inclination of 45° to the horizontal.

The following data represent approximate heights for a ball thrown by a shot-putter as it travels a distance of x meters horizontally.

Distance(m)

Height(m)

7 8

20 15

33 24

47 26

60 24

67 21

What would be the height of the ball if it travels 80

meters?

1. Enter the data into your calculator.

2. Determine an appropriate window setting:

3. Graph the data:

4. The graph looks like a parabola.

5. The Quadratic regression equations is…

6. The graphed equation with the data points…

The ball has traveled 80 m (which means we are given the x-value).

We are trying to predict the height of the ball (or the y-value).

We can use the regression equation OR the graph…

The screen below shows the results on the graphing calculator.

What would be the height of the ball if it travels 80

meters?

Answer: Approximately

12.8 meters

Linear or Quadratic?EQ: How is a best-fitting curve determined?How are data gathered and plotted for quadratic models?When are quadratic models more appropriate for a given set of data?

Step 1.  Enter the data into the lists.

Step 2.  Create a scatter plot of the data.

Step 3.  Visually look for a pattern from the graph.

Steps for Determining a Model

Step 4. Compare to see which regression model appears to best represent the scatter plot graph.

or

Lineary = a + bx

Quadraticy = ax2 + bx + c

Step 5. Choose the appropriate Regression Model and calculate the equation.

Step 6.  Graph the Regression Equation from Y1.

Step 7.  Is this model a "good fit“?

Steps for Determining a Model

"The best choice (of a model) depends on the set of data being analyzed and requires an exercise in judgment, not just computation."                               

"Modeling the US Population" by Shelly Gordon

Think about your answer.Is your choice realistic?  Don't use a model that will

lead to predicted values that are totally unrealistic.