1. 15.7 Curve Fitting

2. With statistical applications, exact relationships may not exist
 Often an average relationship is used
Regression Analysis: a collection of methods by which estimates are made between 2 variables
Correlation Analysis: tells the degree to which the variables are related
Scatter plot: good tool for recognizing & analyzing relationships
dependent variable
(prediction to be made)
independent variable
(the basis for the prediction)

3. Ex 1) A college admissions committee wishes to predict students’ first-year math averages from their math SAT scores. The data was plotted:
The points are NOT on a clear straight line.
But what can this plot tell us?
- In general, higher SAT scores correspond to higher grades
- Looks like a positive linear (or direct) relationship
* Our graphing calculator can plot scatter plots
Remember to identify which variable is independent & which is dependent.
* Once we have a scatter plot, we identify the line or curve that “best fits” the points

4. Some examples:

5. Statisticians use the method of least squares to obtain a linear regression equation
y = a + bx
*the sum of all the values y is zero
*the sum of the squares of the values y is as small as possible
*On Calculator  2nd (CATALOG)
 Scroll down to Diagnostic On
 Enter, Enter

6. Ex 2) A runner’s stride rate is related to his or her speed.
Speed (ft/s)
15.86
16.88
17.50
18.62
19.97
21.06
22.11
Stride rate (steps/s)
3.05
3.12
3.17
3.25
3.36
3.46
3.55
Plot using a scatter plot
STAT  1: Edit… L1 = enter data for speed (independent)
L2 = enter data for stride rate (dependent)
2nd Y= (STAT PLOT)
 1: Enter
WINDOW
GRAPH  looks like a line!

7. Do a “linear regression”
STAT  CALC  8: Lin Reg (a+bx) Xlist: L1
Ylist: L2
Calculate enter
 y = x
What is r?
 correlation coefficient
It tells us “how good” our regression model fits. The closer it is to 1 (for direct linear) or –1 (for inverse linear), the better our model fits the data.

8. Ex 3) Match the scatter diagrams with the correlation coefficients
r = 0.3, r = 0.9, r = –0.4, r = –0.75 a) b) c)
r = 0.9
rising line
fits points well
r = –0.4
falling line
fits points but not closely
r = 0.3
rising line
fits points but not closely
A straight line is not always the best way to describe a relationship.
The relationship may be described as curvilinear.
Exponential
ExpReg
y = abx
Logarithmic
LnReg
y = a + b lnx
Power
PwrReg
y = axb
All of these can be found in the STAT  CALC menu

9. Scatter plot  STAT  1:EDIT  Enter data in L1 & L2 WINDOW GRAPH
Ex 4) An accountant presents the data in the table about a company’s profits in thousands of dollars for 7 years after a management reorganization.
Year
Profits 1
150 2
210 3
348 4
490 5
660 6
872 7
1400
Scatter plot  STAT  1:EDIT  Enter data in L1 & L2
WINDOW GRAPH
Looks exponential
Let’s try an exponential regression
STAT  CALC  0: ExpReg
 y = (1.439) x
Now use this model to predict the profit for year 8.
y = (1.439)8 =  $1,971,000