Prediction

jkesler

Linear Regression and Correlation

51 Prediction

[latexpage]

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

$$\hat y=−173.51+4.83(73)=179.08$$

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.

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 12.11

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.

$$\hat y=–173.51+4.83(90)=261.19$$

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 + 2.8x

Assuming you have confirmed that the correlation coefficient is significant, 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?

If you are trying to predict an outcome but the correlation coefficient is not significant, we can simply use the mean of the $y$-data as our predicted value, since the $x$-data and the $y$-data are not well correlated in this case.

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Introductory Statistics with Google Sheets by jkesler is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.