Simple Linear Regression

Deterministic Relationship If the value of y (dependent) is completely determined by the value of x (Independent variable) (Like an equation in the form y = 2x + 10, or f(x) = 5x-1) However, in most situations, the variables of interest are not deterministically related! For example, the value of y = 1 st year college GPA is certainly not determined solely by x = high school GPA.

Probabilistic Model

Let x * denote the value of x…. Without the random deviation e, all observed (x, y) points would fall exactly on the population regression line. The inclusion of e in the model equation recognizes that points will deviate from the line.

Simple Linear Regression Model:

Slope Population Regression Line

Summary

X * denotes a specified value of the predictor variable x …. So has 2 different interpretations  It is a point estimate of the true mean y value when x = x *.  It is a point predictor of an individual y value that would be observed when x = x *.

Find the point estimate of the mean y-value for the following: Age (x) Weight (y) So what’s the point estimate for an 18 year old mom?

Point estimate and point prediction are identical – only the interpretation is different. Prediction – weight of single baby who mom is 18 Estimate – average weight of all babies born to 18 year- olds

Answer the following: Explain the slope in context of the problem Explain the y-intercept in context of the problem.

Find SS Resid. On calculator – every time you calculate a linear regression – it calculates the residuals. Put them in list 3 and square them & add the list.

It represents the typical deviation in the y-variable from the least squares line.

Find the residual for a mother who is 19.

Find the probability that a 19 year old mother has a baby that is more than 3000 g.

Coefficient of determination (r 2 ) It’s the amount of variation in the y-variables that can be explained by the least squares line.

Homework Worksheet