Recall the third exam/final exam example.

We examined the scatterplot and showed that the correlation coefficient is significant. We found the equation of the best-fit line for the final exam grade as a function of the grade on the third-exam. We can now use the least-squares regression line for prediction.

Suppose you want to estimate, or predict, the mean final exam score of statistics students who received 73 on the third exam. The exam scores **( x-values)** range from 65 to 75.

**Since 73 is between the**, substitute

*x*-values 65 and 75*x*= 73 into the equation. Then:

We predict that statistics students who earn a grade of 73 on the third exam will earn a grade of 179.08 on the final exam, on average.

### Example 12.11

Recall the third exam/final exam example.

#### Problem

a. What would you predict the final exam score to be for a student who scored a 66 on the third exam?

b. What would you predict the final exam score to be for a student who scored a 90 on the third exam?

#### Solution

a. 145.27

b. The *x* values in the data are between 65 and 75. Ninety is outside of the domain of the observed *x* values in the data (independent variable), so you cannot reliably predict the final exam score for this student. (Even though it is possible to enter 90 into the equation for *x* and calculate a corresponding *y* value, the *y* value that you get will not be reliable.)

To understand really how unreliable the prediction can be outside of the observed *x* values observed in the data, make the substitution *x* = 90 into the equation.

$\hat{y}=\mathrm{\u2013173.51}+4.83\left(90\right)=261.19$

The final-exam score is predicted to be 261.19. The largest the final-exam score can be is 200.

### Note

The process of predicting inside of the observed *x* values observed in the data is called interpolation. The process of predicting outside of the observed *x* values observed in the data is called extrapolation.

### Try It 12.11

Data are collected on the relationship between the number of hours per week practicing a musical instrument and scores on a math test. The line of best fit is as follows:

*ŷ* = 72.5 + 2.8*x*

What would you predict the score on a math test would be for a student who practices a musical instrument for five hours a week?