y hat (symbol: ŷ) refers to the value for the dependent variable that we obtain from a linear regression model. Note that it is merely the value that we have predicted using our linear model.
The actual value of the dependent variable (denoted y) may be different. The difference between the actual and predicted values (ŷ-y) is called the error.
How to find ŷ (y hat):
- Given the data on the dependent and independent variables, we find the least square regression line.
- The least square regression line obtained is of the form, y = mX + c.
- To find the value of ŷ, substitute the value of X (independent variable) in the linear model above. This will give you the predicted value (y hat) for that value of X.
Example: Consider the following regression model which gives the relationship between the number of hours spent studying per day (independent variable X) and the marks obtained in an exam(dependent variable Y),
Predict the marks obtained by the student (ŷ) if he spends 4 hours studying.
Solution: We substitute the value of the independent variable X=4 in the above equation to get,
Ŷ = 20*4+5 = 80+5 = 85.
So, we predict that the student will get 85 marks in the exam if he spends four hours studying per day.