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					                                  CHAPTER 4 NOTES                                            pg 1
Modeling Nonlinear Data:

Linear Data is data modeled by an equation of the form y = a + bx.

Linearization is the process of transforming nonlinear data into linear data. We use
properties of logarithms to linearize certain types of data.



              PROPERTIES OF LOGARITHMS:


                   1.         log ab  log a  log b
                                  a
                   2.
                              log  log a  log b
                                  b
                   3.         log x p  p log x



Examples:


log 2 x  log 2  log x
    2
log  log 2  log x
    x
log 2 x  x log 2
Case 1: Consider the following set of Linear Data representing an account balance as a
function of time:

x: time (months)        0         48           96          144          192            240
    y: account
                        100       580         1060        1540         2020          2500
   balance ($)

Describe the pattern of change…
The relationship between x and y is linear if, for equal increments of x, we add a fixed
increment to y.
                                  CHAPTER 4 NOTES                                              pg 2
Case 2: Consider the following set of Nonlinear Data representing an account balance as a
function of time:

x: time (months)        0          48           96          144          192             240
    y: account
                       100       161.22       259.93       419.06       675.62       1089.30
   balance ($)

The relationship between x and y is exponential if, for equal increments of x, we multiply a
fixed increment by y. This increment is called the common ratio.


We want to find the best fitting model to represent our data.

      For the linear data, we use least-squares regression to find the best fitting line.

      For the nonlinear data, the best fitting model would be an exponential curve.

PROBLEM: We cannot use least-squares regression for the nonlinear data because least-
squares regression depends upon correlation, which only measures the strength of linear
relationships.

SOLUTION: We transform the nonlinear data into linear data, then use least-squares
regression to determine the best fitting line for the transformed data.

Finally, do a reverse transformation to turn the linear equation back into a nonlinear
equation which will model our original nonlinear data.




                             Linearizing Exponential Functions:


(We want to write an exponential function of the form    y  a  b x as a function of the
form   y  a  bx ).

 y  a  bx          (x,y are variables and a,b are constants)


log y  log(a  b x )

log y  log a  log b x
                                    CHAPTER 4 NOTES                                       pg 3

log y  log a  x log b
var2 = con1 + (var1)(con2)

This is in the general form    y  a  bx , which is linear.

So, the graph of (var1, var2) is linear. This means the graph of    x, log y  is linear.
CONCLUSIONS:


1.      If the graph of    x, log y  is linear, then the graph of  x, y  is exponential.
2.      If the graph of    x, y  is exponential, then the graph of  x, log y  is linear.
Once we have linearized our data, we can use least-squares regression on the transformed

data    x, log y  to find the best fitting linear model.
PRACTICE:
Linearize the data for Case 2 and find the least-squares regression line for the transformed
data.

Then, do a reverse transformation to turn the linear equation back into an exponential
equation.



log y  2  0.0043x
    ˆ                                                y  100  10
                                                     ˆ                     0.0043 x
                                                                                    
10log y  1020.0043 x
      ˆ

                                                     y  100 1.01
                                                                                x
                                                     ˆ
 y  1020.0043 x
 ˆ

 y  102 100.0043 x 
 ˆ
                                  CHAPTER 4 NOTES                                        pg 4


So our exponential model for Case 2 is:


 y  100  100.0043 
                                 x
 ˆ
***Compare this to the equation the calculator gives when performing exponential
regression on the Case 2 data.



                                Linearizing Power Functions:


(We want to write a power function of the form   y  ax b as a function of the form
 y  a  bx ).

 y  ax b     (x,y are variables and a,b are constants)


log y  log  axb 

log y  log a  log xb

log y  log a  b log x
var2 = con1 + (con2)(var1)


This is in the general form   y  a  bx , which is linear.

So, the graph of (var1, var2) is linear. This means the graph of    log x, log y  is
linear.

Case 3: Consider the following set of Nonlinear Data representing the average length and
weight at different ages for Atlantic Ocean rockfish:

  x: age (years)        0             4        8           12           16          20
y: weight (grams)       0            48       192         432          768         1200
                                                CHAPTER 4 NOTES                                       pg 5

      PRACTICE:
      Linearize the data for Case 3 and find the least-squares regression line for the transformed
      data.

      Then, do a reverse transformation to turn the linear equation back into a power equation.
          log y  0.4771  2log x
              ˆ                                               10       ˆ
                                                                   log y
                                                                            10   0.4771
                                                                                       10   log x 2


          log y  0.4771  log x 2
              ˆ                                               y  100.4771 x 2
                                                              ˆ
                              0.4771 log x 2
          10       ˆ
               log y
                        10                                   y  3x 2
                                                              ˆ

So our power model for Case 3 is:

 y  3x 2
 ˆ

***Compare this to the equation the calculator gives when performing power regression on the Case
3 data.



Interpreting Correlation and Regression:

Extrapolation is the use of a regression equation to make predictions outside the domain of the
explanatory variable.

Caution! These predictions are not reliable.

Example: The following data represent Sharon’s typing speed (words per minute) as a function of the
time she has practiced (hours).

  Practice
                         0                10             20                30              40
   (hrs.)
Speed (wpm)             30                40             48                55              61

Use the linear regression model to predict Sharon’s typing speed after 50 hours of practice. After 60
hours of practice? Are these predictions reliable?
                                             CHAPTER 4 NOTES                                          pg 6
A lurking variable effects the relationship between the variables being studied, although it is not part
of the study.

Example: A study compared the SAT scores of high school seniors who took an SAT-Prep course to
high school seniors who did not take the Prep course. The study found no significant difference in the
SAT scores.

Is it fair to conclude that taking an SAT-Prep course has no effect on SAT scores? Identify any
lurking variables.


Association vs. Causation

An association between two variables, even if it is very strong, does not imply that changes in one
variable cause changes in the other variable.

Association DOES NOT imply causation!



Example: A study found a strong negative correlation between student failure rate and years of
teaching experience at East Larson High School. The study found that failure rates for first-years
teachers were significantly higher than failure rates for tenured (more experienced) teachers.

Does longer teaching experience cause lower failure rates??? Identify any lurking variables.



A strong association between two variables x and y could reflect the following relationships:
             Causation
             Common Response
             Confounding
Causation: changes in x cause changes in y.


               x                                               y


Common Response: both x and y are caused by a lurking variable z.


               x                                                y




                                          z
                                       CHAPTER 4 NOTES                                               pg 7
Confounding: changes in x cause changes in y, but y is also caused by a lurking variable z.




                            x                                  y




                                          z




Relations in Categorical Data:

Because we cannot perform direct calculations on categorical data, we use the counts or percents of
individuals by category.

The count or percents of individuals in each category of one variable is called a marginal distribution.


Example: Here are data from eight high schools on smoking among students and among their
parents:

                            Student smokes        Student does not smoke             Total
 Both parents smoke              400                       1380
 One parent smokes               416                       1823
   Neither parent
                                  188                       1168
       smokes

         Total


Find the marginal distributions for parent smoking behavior and student smoking behavior by count.
                                         CHAPTER 4 NOTES                                            pg 8
Note: Counts cannot be directly compared when the sizes of the groups are unequal. Instead,
compare the percents.

Example: Find the marginal distributions for parent smoking behavior and student smoking behavior
by percent.

                                                    Student does not
                            Student smokes                                       Total
                                                        smoke
 Both parents smoke
 One parent smokes
   Neither parent
       smokes

         Total                                                                    100




The count or percents of individuals in each category of one variable that are also in a given category
of the other variable is called a conditional distribution.

Example: Find the conditional distribution for student smoking behavior given that neither parent
smokes.

                        Student smokes          Student does not smoke           Total
Neither parent
   smokes


Simpson’s Paradox

When data from several groups are combined to form a single group, the association between
variables can drastically change.


Example:
Upper Wabash Tech has a Business school and a Law school. The following table shows the
number of applicants admitted to and denied by each school.

                                  Upper Wabash Tech Applicants
                 Business                                               Law
                  Admit       Deny                                     Admit        Deny
  Male             480        120                        Male            10          90
 Female            180         20                       Female          100         200
                                         CHAPTER 4 NOTES                                      pg 9
By combining the data from the Business school and the Law school, we have the following two-way
table:

                               Applicants for both schools combined
                                   Admit                  Deny              Total
          Male                      490                   210                700
         Female                     280                   220                500
          Total                     770                   430               1200

From this table, we have the following result:

         490
              70%       of males are granted admission
         700
         280
              56%       of females are granted admission.
         500
So Wabash admits a higher percentage of male applicants.



Now consider each of the schools separately:

                        Business
                     Admit       Deny            Total
Male                  480        120              600
Female                180         20              200
Total                 660            140          800


480
     80% of males are granted admission
600
180
     90% of females are granted admission.
200

                             Law
                     Admit          Deny         Total
Male                   10            90           100
Female                100           200           300
Total                 110           290           400
                                          CHAPTER 4 NOTES                                         pg 10
 10
     10% of males are granted admission
100
100
     33.3% of females are granted admission.
300
So each school admits a higher percent of female applicants!


If each school admits a higher percent of females, how can both schools combined admit a higher
percent of males?

This is an example of Simpson’s Paradox.


600                                                                  200
     86%          of males apply to the Business school whereas
                                                                          40%       of females apply
700                                                                  500
to the Business
school.

Likewise, 14% of males apply to the law school whereas 60% of females apply to the law school.


                              660
The Business school admits
                                   83%         of its applicants, while the Law school only admits
                              800
110
     28%          of its applicants.
400
Because a higher percent of males apply to the school that’s easier to get into, a higher percent of
males are admitted overall, even though each school admits a higher percent of female applicants.

Consider the following data on 326 cases in which the defendant was convicted of murder:

                        White Defendant                          Black Defendant
                         Death Penalty                                        Death Penalty
                  Yes          No                                      Yes          No
  White                                                 White
                  19            132                                     11            52
  Victim                                                Victim
  Black                                                 Black
                   0             9                                       6            97
  Victim                                                Victim


The following two-way table shows the combined data (combining the victim’s race) for defendant’s
race versus death penalty.
                                           CHAPTER 4 NOTES                                        pg 11

                       Death Penalty
                       Yes         No           Total
White Defendant         19         141           160
Black Defendant         17         149           166
      Total             36          290             326


11.9% of white defendants receive the death penalty

10.2% of black defendants receive the death penalty

             White Defendant
                    Death Penalty
                     Yes        No              Total
  White Victim        19        132              151
  Black Victim        0           9               9
      Total             19          141             160


12.6% receive death penalty for killing white victim

0% receive death penalty for killing black victim

             Black Defendant
                     Death Penalty
                     Yes         No             Total
  White Victim        11          52              63
  Black Victim         6          97             103
      Total             17          149             166


17.5% receive death penalty for killing white victim

5.8% receive death penalty for killing black victim

Simpson’s Paradox:
Although a higher percent of white defendants receive the death penalty overall, a higher percent of
black defendants receive the death penalty for killing a white victim and for killing a black victim.

Out of the 214 white victims killed, 14% of the defendants received the death penalty.

Out of the 112 black victims killed, only 5.4% of the defendants received the death penalty.



Since whites killed whites 151 out of 160 times (94%) this group has greater influence on the
      combined results.

				
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