The Intersection of Least-Squares Regression and Age vs. Mean Height

That said, the relationship between age and mean height is a classic example in statistics, illustrating the power of least-squares regression. A simple scatter plot reveals an association: as age increases, so does the mean height.

Just graphical, not numerical, this scatter plot allows us to see the trend without getting bogged down in details. It's essential to note that correlation does not tell the exact relationship between two quantitative variables like age and height. The direction and strength of the linear relationship can vary widely.

Topics: Least-Squares Regression - Regression lines - Equation and interpretation of the line - Prediction using the line

Regression lines are a fundamental concept in least-squares regression, and understanding them is crucial for making accurate predictions. A straight line summarizes the relationship between two variables if the form of the relationship is linear. The equation of a straight line can be expressed as y = ax + bx, where x represents the explanatory variable and y represents the response variable.

A regression line describes how a response variable changes as an explanatory variable changes. It's often used as a mathematical model to predict the value of a response variable based on a value of an explanatory variable. In this case, we're interested in predicting mean height at age 32 months.

Equation of a straight Line

The equation of a straight line relating y to x has the form: y = ax + bx, where x represents the explanatory variable and y represents the response variable. The slope (b) and intercept (a) are crucial components that need to be determined for accurate predictions.

For our purpose, we're interested in finding the regression line that minimizes the sum of the squares of the vertical distances between data points and the line itself. Mathematically, this is represented by minimizing over values of the pair (a, b). The least-squares regression line is a best-fit line that optimally predicts the response variable based on the explanatory variable.

How to fit a line

Fitting a line involves determining the slope (b) and intercept (a) of the regression line. Once these parameters are calculated, we can use them to predict the value of y for any given value of x. The formula for the least-squares regression line is: y = ax + bx.

Error

The error in a linear regression model represents the difference between observed and predicted values. It's essential to understand that error is not zero; it's an inherent aspect of modeling, especially when dealing with continuous variables like height.

Least-square regression lines are determined by minimizing over values of the pair (a, b). This process ensures that the line is as close as possible to all data points while still minimizing error. In practice, this may involve some trial and error or iterative calculations.

Least-Square Regression Line

The least-squares regression line of y on x can be represented mathematically as: y = ax + bx, where a and b are the slope and intercept parameters, respectively. The equation is:

y - (a x) - (b x) = 0

Or in a simplified form:

ˆx y s s r b  x b y a  