Final answer:
The statement that predicted y values in a linear regression do not necessarily match observed y values is true. Predicted values, represented as y hat, are estimations based on the regression line and can differ from actual data.
Step-by-step explanation:
The values of the predicted y on a linear model, often denoted as ŷ or 'y hat', do not necessarily coincide with the values of the observed y for any given x. This statement is true. In linear regression, ŷ is the estimated value of y obtained by using the regression line, which represents the best fit for the data points in the sample.
Although the regression line is based on minimizing the distance between the observed data points and the line itself, there are typically discrepancies between predicted and actual values due to natural variations and measurement errors.
As an example, if the linear model states that for x=1985, ŷ=25,52, it means that the predicted value of y is 25.52, while the observed value might be slightly different.
Additionally, making predictions beyond the observed range of x values in the data can be unreliable, a concept known as extrapolation.
For instance, if the observed x values range from 65 to 75 and a prediction is made for x=90, the predicted y value would be highly uncertain, as illustrated by the substitution ŷ=173.51 +4.83(90) = 261.19, which lies outside the original data domain.