Consider the graph shown:

Is there a relationship between the years since purchased and the value in thousands of dollars? Explain.

Which of the lines on the graph is the line of best fit?

A **line of best fit** (or trend line) is a straight line that best represents the data on a scatterplot. We can use lines of best fit to help us make predictions or conclusions about the data.

We previously approximated a line of best fit by trying to balance the number of points above the line with the number of points below the line. This can result in multiple different models.

3 points above, 3 points below

5 points above, 4 points below

In the context of a line of best fit, the slope-intercept form represents

\displaystyle y=mx+b

\bm{m}

the rate of change for y with respect to x

\bm{b}

the starting value of y when x is 0

For example, this graph models a plant's growth over several weeks.

These terms describe the range in which we make predictions:

**Interpolation**: Prediction within the range of x-values in the data**Extrapolation**: Prediction outside the range of x-values in the data

Using the previous example of the plant height over time:

The reliability of predictions depends on the strength of the relationship, whether the data is interpolated or extrapolated, and the number of points in the data set.

A larger sample size increases reliability.

Interpolation with a strong correlation implies a reliable prediction.

Interpolation with a moderate or weak correlation leads to a less reliable prediction.

Extrapolation generally leads to an unreliable prediction. The further outside the range of known values, the less reliable it is.

Natalia collected data to answer the question, "What is the relationship between the years since purchasing a car and its value?" Her data is shown in the table.

Time since purchase (years) | 0.5 | 0.8 | 1.2 | 1.3 | 1.5 | 1.7 | 1.8 | 2.1 | 2 | 2.5 |
---|---|---|---|---|---|---|---|---|---|---|

Value (thousands of dollars) | 29 | 28.5 | 28.5 | 27.4 | 28.5 | 27 | 25.9 | 25.9 | 24.7 | 26.4 |

Time since purchase (years) | 2.6 | 2.8 | 3.1 | 3.4 | 3.6 | 3.9 | 4.05 | 4.6 | 4.8 | |

Value (thousands of dollars) | 24.6 | 23.5 | 24.6 | 23.3 | 21 | 21 | 22 | 21 | 20.1 |

a

Find the equation of the line of best fit.

Worked Solution

b

Interpret the slope and y-intercept of the line.

Worked Solution

c

Make a prediction about the value of a car after 3 years.

Worked Solution

d

Make a prediction about the value of a car after 10 years.

Worked Solution

e

Is the prediction for the car's value after 3 years or after 10 years more reliable?

Worked Solution

A teacher recorded the number of days since a student last studied for an exam and their score out of a possible 80 points on the exam.

Days since studying | 3 | 2 | 6 | 4 | 4 | 1 | 6 | 3 | 4 | 2 |
---|---|---|---|---|---|---|---|---|---|---|

Exam score | 64 | 59 | 42 | 57 | 58 | 72 | 33 | 63 | 55 | 62 |

a

Formulate an investigative question that can be answered by the data.

Worked Solution

b

Was the data most likely collected through measurement, observation, a survey or an experiment?

Worked Solution

c

Describe the relationship between the number of days since studying and the exam score.

Worked Solution

d

Calculate the line of best fit using technology.

Worked Solution

e

Answer the question formulated in part (a).

Worked Solution

f

If a student studied the same day as the exam, what would we expect their score to be?

Worked Solution

Idea summary

A **line of best fit** for a set of data can be used to interpret a given situation and make predictions about values not represented by the data.

A line of best fit has an equation of the form y=mx+b. We can use technology to perform the linear regression analysis.

In the context of a line of best fit, the slope-intercept form represents

\displaystyle y=mx+b

\bm{m}

the rate of change for y with respect to x

\bm{b}

The starting value of y when x is 0

These terms describe the range in which we make predictions:

**Interpolation**: Prediction within the range of x-values in the data**Extrapolation**: Prediction outside the range of x-values in the data

The reliability of predictions depends on the strength of the relationship, whether the data is interpolated or extrapolated, and the number of points in the data set. In general, interpolation is more reliable than extrapolation.