Name ______
AP STATISTICS CHAPTER 4:
EXPONENTIAL GROWTH:
What is the major difference between linear and exponential growth?
Give an example of Linear Growth:
Give an example of Exponential Growth:
REVIEW OF LOGS:
Definition: if and only if
Example: Simplify .
because
Problems: Evaluate the following logs. In these problems, we assume b is 10, when no value of b is given.
1) 2)
3) 4)
5) 6)
7) 8)
9) 10)
11) 12)
13) 14)
15) 16)
LOG LAWS TO KNOW AND LOVE
a)
ex:
b)
ex:
Problems: Simplify
1)
2)
3)
4)
Observe the following chart. Why do the values in this chart exhibit an exponential growth?
Year / y /1980 / 1
1981 / 2
1982 / 4
1983 / 8
1984 / 16
1985 / 32
1986 / 64
1987 / 128
1988 / 256
1989 / 512
Fill in the third column, compute the log (base 2) of each y-value. What effect does taking the log have on the data?
EXPONENTIAL GROWTH MODEL
In an exponential growth model, growth depends on a rather than .
The model for exponential growth is
These problems are characterized by over .
If a variable grows , its grows .
For each of the following, complete the following, using your graphing calculator.
1) Find the LSRE for x vs y. Record the equation and r. Construct a residual plot to verify if a linear model is appropriate.
2) Find a new LSRE for x vs log y. Record this new equation. Compare r’s to see if the correlation has improved.
3) Solve for , our predicting equation.
4) Use this equation to make the desired prediction.
Example: Saving accounts
Predict the account balance at the start of years 6 and 17.
Start of year # / $ in account (1000’s)1 / 5.00
3 / 5.40
5 / 5.85
10 / 7.10
20 / 10.50
40 / 23.10
50 / 34.20
80 / 110.80
Problem 1: Virus cells
(millions)
0 / 60.0
1 / 51.8
4 / 32.7
8 / 18.1
12 / 10.0
15 / 6.1
20 / 2.7
40 / 0.2
Predict the number of cells remaining after 5 and 10 minutes.
Problem 2: Harley-Davidson stock prices
Year1987 = year 1 / Stock price in dollars
1 / .392
2 / .765
3 / 1.18
5 / 2.70
8 / 6.80
10 / 11.60
12 / 23.52
15 / 54.31
Predict the stock’s price in year 4 and year 13.
POWER LAW MODELS
Sometimes, a response variable will grow in to a
of the variable.
Some common examples:
Length (in) / Weight (lbs)58 / 28
61 / 44
63 / 33
68 / 39
69 / 36
72 / 38
72 / 61
74 / 54
74 / 51
76 / 42
78 / 57
82 / 80
85 / 84
86 / 83
86 / 80
86 / 90
88 / 70
89 / 84
90 / 106
90 / 102
94 / 110
94 / 130
114 / 197
128 / 366
147 / 640
Lengths and weights of alligators.
Predict the weights if length is 65 inches, 100 inches.
Intensity of a lightbulb from different distances.
Distance (meters) / Intensity1.0 / 0.2966
1.1 / 0.2522
1.2 / 0.2055
1.3 / 0.1746
1.4 / 0.1534
1.5 / 0.1352
1.6 / 0.1145
1.7 / 0.1024
1.8 / 0.0923
1.9 / 0.0832
2.0 / 0.0734
Predict the intensity from 1.55 meters, 3 meters.
CAUTIONS ABOUT CORRELATION AND REGRESSION
A , a variable not included in a study which may
the interpretation of among the
variables.
Give one example of a lurking variable:
Beware of:
Extrapolation is use of a for beyond the
of the . These predictions are often inaccurate.
Using data for prediction. based on.
are usually too high when applied to .
Example: Fathom infant mortality rates
Give an example of a confounding variable.
EXPLAINING ASSOCIATIONS: CAUSATION
Consider the following statements. In each, does the explanatory variable cause a change in the response variable? Is this the same as a strong association?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name ______
AP STATISTICS CHAPTER 4:
1. Increased humidity (during warm weather), tends to make the air feel less comfortable.
2. People who participate in sports in HS tend to have higher grades.
3. People who have more education tend to have higher incomes as adults.
A comparison of surgery patients at 2 local hospitals.
Hospital A Hospital B
SUMMARY/QUESTIONS TO ASK IN CLASS
Name ______
AP STATISTICS CHAPTER 4:
Died 63 16
Survived 2037 784
Patient in Good Condition Patient in Poor Condition
Hospital A Hospital B Hospital A Hospital B
Died 6 8 Died 57 8
Survived 594 592 Survived 1443 192
Compare mortality rates:
SUMMARY/QUESTIONS TO ASK IN CLASS