The Simple Linear Regression Model
In the cleaning parts example, we collected data on 50 parts. We fit a regression model to predict Removal as a function of the OD of the parts. But what if we had sampled a different set of 50 parts and fit a regression line using these data? Would this produce the same regression equation? By fitting a regression line to observed data, we are trying to estimate the true, unknown relationship between the variables. This fitted regression equation is just one estimate of the true linear model. In reality, the true linear model is unknown.
In simple linear regression we assume that, for a fixed value of a predictor X, the mean of the response Y is a linear function of X. We denote this unknown linear function by the equation shown here where b0 is the intercept and b1 is the slope. The regression line we fit to data is an estimate of this unknown function.
The equation of the fitted line is denoted by the following equation:
Here, b0 and b1 are estimates of beta0 and beta1, respectively. The notation Y-hat indicates that the response is estimated from the data and that it is not an actual observation. In the cleaning example, the intercept, b0, is 4.099 and the slope, b1, is 0.528.
If we select a different sample of parts, our fitted line will be different. To illustrate, we use the Demonstrate Regression teaching module in the JMP sample scripts directory.