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AP Statistics Chapter 4 - More about Relationships between Two Variables
4.1: Transforming Relationships
Properties of Logarithms
Exponential Growth
Linear growth increases by a fixed amount in each equal time period.
Exponential growth increases by a fixed percentage of the previous total.
Exponential Functions
An exponential function has the form  . If B>1, then the data shows exponential growth. If 0<B<1, then the data shows exponential decay. The graph of an exponential function always contains the point (0, A).
Finding an Exponential Model
If the two-variable data (x, y) follows an exponential model, then (x, log y) follows a linear model. To obtain the coefficients A and B for the exponential function, find a linear regression model for (x, log y). The equation is of the form y = a + bx. Use the coefficients of the linear equation as follows:
 and 
Power Functions
A power function has the form  .The graph of a power function will contain the point (0, 0) as long as B is positive.
Finding a Power Model
If the two-variable data (x, y) follows a power model, then (log x, log y) follows a linear model. To obtain the coefficients A and B for the power function, find a linear regression model for (log x, log y). The equation is of the form y = a + bx. Use the coefficients of the linear equation as follows:
and 
4.2: Relations between Categorical Variables
Two-Way Tables
A two-way table organizes counts from two categorical variables into rows and columns. They are often used to summarize large amounts of data by grouping outcomes into categories.
Marginal Distributions
The row totals and column totals give the marginal distributions of the two individual variables. These numbers tell us nothing about the relationship between the two variables. They are simply used to summarize the data.
Conditional Distributions
To find the conditional distribution of the row variable, begin by focusing on a single column. Then find each entry in the column as a percent of the column total. These percentages will tell us about the association between the two variables in the table.
4.3: Establishing Causation
The Question of Causation
An association between two variables doesn't necessarily mean that x causes y. Consider 4 possibilities in this situation:
1. Causation - x does indeed cause y
2. Common Response - a 3rd variable, z, causes changes in both x and y
3. Confounding - both x and a 3rd variable, z, cause changes in y
4. Reverse Causation - y actually causes x
In the above, z would be a lurking variable
Confounding
Two variables are said to be confounded when their effects on a response variable cannot be distinguished from each other.
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