Linear Models and Correlation

 Linear Function- A set of ordered pairs (x,y) which can be described by an equation of the form y=mx+b, where m and b are constants.  Consumer Price Index (CPI)- A measure of the cost of living in the US.  Correlation Coefficient (r)- A measure of the strength of the linear relation between two variables.  Positive Relation- Positive value for the correlation coefficient, larger values of one variable are associated with larger values of the other variable.  Negative Relation- Negative value for the correlation coefficient, larger values of one variable are associated with smaller values of the other variable.  Perfect Correlation- A correlation coefficient of 1 or -1.  Strong Relation (correlation)- A relation where most of the data points fall close to a line.  Weak Relation (correlation)- A relation where, although a linear trend can be seen, many points are not very close to the line.

 The sign of r indicates the direction of the relation between the variables (slope), and its magnitude indicates the strength of the relation.  Correlation is between -1≤r≤1  If ∣r∣ is close to 1, then the relation is strong.  If ∣r∣ is close to 0, then the relation is weak.  If r=0, the there is no relation.  Perfect Correlation- when r=1 or r=-1

 Larger numbers of one variable are associated with larger numbers of the other  Positive slope  From left to right, the graph is going up

 Larger values of one variable associate with smaller values of the other  Negative slope  From left to right, the graph is going down

 Not uncommon for points in a scatterplot of a data set to lie near a line.  Linear model may be appropriate, even if the linear function does not contain all data points.  Most often, the best linear model does not contain any of the data points.  Linear models predict values of dependent and independent variables

 While r provides a mathematical measure of linearity, it DOES NOT provide information about cause and effect.  I.E. There is a large positive correlation between shoe size and reading level in children ◦ Does NOT mean that learning to read better causes your feet to grow ◦ Does NOT mean that wearing bigger shoes improves reading skills ◦ Correlation is large because each variable is related to age ◦ Older children generally have bigger feet and higher reading skills  Correlation does not mean causation

 M=slope  B=y-intercept

 Find the slope between the two points Y 2 -Y 1 _________________________ X 2 -X 1  Use equation Y-Y 1 =m(X-X 1 ) ◦ Add Y 1 to both sides ◦ Y=m(X-X 1 )+Y 2  m=slope

 From Page 89  Find equation for line between year 1988 and ◦ Find Slope ◦ Enter slope and (x,y) into point-slope form ◦ Simplify

 Worksheet 2-2