AP Statistics Curriculum 2007 GLM Regress

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General Advance-Placement (AP) Statistics Curriculum - Regression

As we discussed in the Correlation section, many applications involve the analysis of relationships between two, or more, variables involved in the process of interest. Suppose we have bivariate data (X and Y) of a process and we are interested on determining the linear relation between X and Y (e.g., determining a straight line that best fits the pairs of data (X,Y)). A linear relationship between X and Y will give us the power to make predictions - i.e., given a value of X predict a corresponding Y response. Note that in this design, data consists of paired observations (X,Y) - for example, the Longitude and Latitude of the SOCR Eathquake dataset.

Lines in 2D

There are 3 types of lines in 2D planes - Vertical Lines, Horizontal Lines and Oblique Lines. In general, the mathematical representation of lines in 2D is given by equations like \(aX + bY=c\), most frequently expressed as \(Y=aX + b\), provides the line is not vertical.

Recall that there is a one-to-one correspondence between any line in 2D and (linear) equations of the form

If the line is vertical (\(X_1 =X_2\))\[X=X_1\];
If the line is horizontal (\(Y_1 =Y_2\))\[Y=Y_1\];
Otherwise (oblique line)\[{Y-Y_1 \over Y_2-Y_1}= {X-X_1 \over X_2-X_1}\], (for \(X_1\not=X_2\) and \(Y_1\not=Y_2\))

where \((X_1,Y_1)\) and \((X_2, Y_2)\) are two points on the line of interest (2-distinct points in 2D determine a unique line).

Y=2X+1
Y=-3X-5

Linear Modeling - Regression

There are two contexts for regression:

  • Y is an observed variable and X is specified by the researcher - e.g., Y is hair growth after X months, for individuals at certain dose levels of hair growth cream.

Suppose we have n pairs (X,Y), {\(X_1, X_2, X_3, \cdots, X_n\)} and {\(Y_1, Y_2, Y_3, \cdots, Y_n\)}, of observations of the same process. If a scatterplot of the data suggests a general linear trend, it would be reasonable to fit a line to the data. The main question is how to determine the best line?

Airfare Example

We can see from the scatterplot that greater distance is associated with higher airfare. In other words airports that tend to be further from Baltimore tend to be more expensive airfare. To decide on the best fitting line, we use the least-squares method to fit the least squares (regression) line.

SOCR EBook Dinov GLM Regr 021708 Fig1.jpg



References




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