11C

Making predictions

Using the least-squares regression line to make predictions, interpolation and extrapolation, the reliability of predictions, residuals of predictions, and the limits of a regression model.

Predicting

To predict, substitute the $x$-value into $\hat{y} = a + bx$.

Example 1●

Using $\hat{y} = 48.2 + 3.66x$, predict the score of a student who studies for $4.5$ hours.

$\hat{y} = 48.2 + 3.66 \times 4.5 = 64.67$, so about $65$ marks.

Interpolation and extrapolation

Interpolation predicts inside the range of the data. It is usually reliable when the correlation is strong.
Extrapolation predicts outside the range of the data. It assumes the trend continues, which may not be true, so treat it with caution.

Example 2●●

The data covered $1$ to $6$ hours. Predict the score for $20$ hours of study, and comment.

$\hat{y} = 48.2 + 3.66 \times 20 = 121.4$.

This is extrapolation far beyond the data, and a score above $100$ is impossible. The linear trend can't continue forever, so the prediction is not reliable.

Flashcards

Interpolation

Predicting inside the range of the data.

Extrapolation

Predicting outside the range of the data. Use caution.

Making a prediction

Substitute $x$ into $\hat{y} = a + bx$.

Practice

Using $\hat{y} = 48.2 + 3.66x$, predict $y$ when $x = 4.5$.
Using the same line, predict $y$ when $x = 20$.
Using the same line, predict $y$ when $x = 3$.
The student who studied $3$ hours scored $61$. Find the residual, using the prediction from the last question.
$\hat{y} = 50 - 2x$. Predict $y$ when $x = 18$.
The data covered $x$ from $1$ to $6$. Predicting at $x = 4.5$ is:
Which prediction is most reliable?
Why was the prediction of $121.4$ marks unreliable?
A regression line shows that taller people weigh more. This means: