Section 3.2: Least Squares Regressions Correlation measures the direction and strength of a LINEAR relationship. A line that summarizes the overall pattern of the relationship is called a regression line. Only applicable to bivariate data with explanatory and response variables. The line MODELS the form of the data and can be used to PREDICT response values (y) given an explanatory value (x).

General Equation of any Regression Line y = a + bx y-intercept the value of y when x = 0 …if appropriate for the data slope the approximate amount by which y changes when x increases by one unit

Non exercise activity (NEA) vs. Fat gain This data measure calories burned in everyday activity (non-exercise) compared to the fat gained (in kilograms) Fat gain = a + b(NEA) Predicting values beyond the scale is called extrapolation…these predictions are not always accurate

A good regression line best models the data by MINIMIZING the vertical distance of each point from the line. This line is called the LEAST SQUARES REGRESSION LINE (or LSRL) This is symbolized as where and the line passes through

Using Calculator to find the LSRL: STAT  CALC  #8 LinReg (a +bx) Listx, Listy, Y1 This will store the LSRL for graphing and further analysis

Residual = observed y – predicted y How well does the LSRL model the data? We use RESIDUALS. Residual = observed y – predicted y or Calculator: Must have LSRL stored in Y1 L3 = LRESID Residual for this data value

To investigate all the residuals of the data graphically, we can use a RESIDUAL PLOT. This is a scatterplot of the explanatory variable (L1) vs. residual values (L3). Be sure that Y1 (the LSRL) and any other plot is turned off. You are looking that the points are closely clustered about the x-axis.

To numerically analyze the residuals (and in turn, how well the LSRL models the data) there are two calculations we can use: Standard deviation of the residuals: this will give you a “margin of error” for any predicted value based upon the LSRL. 2) Coefficient of determination (r2): this will give the proportion of data values that are explained by (fit the model of) the LSRL. On Calculator: Do 1 Var Stat on residuals list and find Sx On Calculator: This is listed with the Lin Reg output

Further interpreting the r2 value… If r2 = 1, all data values fall on the LSRL…the LSRL accurately represents the data If r2 = 0, The LSRL does not predict values any better than the mean of the response variable