Lines of Best Fit — Challenge
One line drawn through the middle of the cloud, then read like any other line: slope and intercept, with units.
What these problems cover
These questions use 9 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Slope — How much y changes for each 1 that x increases.
- Range — The set of outputs that actually occur.
- Line of best fit — A straight line summarising a linear pattern in a scatter plot.
- Interpolation — Predicting inside the range of the data. Usually safe.
- Extrapolation — Predicting outside the range. Risky.
- Residual — The gap between an actual value and the line's prediction.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
A line of best fit for y (plant height, cm) against x (days) is y = 1.4x + 3. Which statement is correct?
- The plant grows about 1.4 cm per day and was 3 cm tall at the start
- The plant grows about 3 cm per day
- After 1.4 days the plant is 3 cm tall
- The plant will be 1.4 cm tall on day 3
- The plant grows 4.4 cm per day
Worked answer
The plant grows about 1.4 cm per day and was 3 cm tall at the start
The slope 1.4 is the change in y for each extra day.
So growth is about 1.4 cm per day.
The intercept 3 is the predicted height at day 0.
Problem 2
Data collected for x from 1 to 12 gives the line y = 6x + 20. Using it to predict y at x = 60 is unwise mainly because
- x = 60 is far outside the range of the data
- the slope is too large
- the intercept is positive
- the line was fitted by eye
- 60 is not a whole multiple of 12
Worked answer
x = 60 is far outside the range of the data
The data only covers 1 to 12.
Predicting at 60 is extrapolation.
The linear pattern may not continue that far.
The arithmetic is fine; the assumption is not.
Problem 3
For a fitted line, one point has actual value 47 and predicted value 52. Its residual is
- -5
- 5
- 99
- 52
- 47
Worked answer
-5
Residual is actual minus predicted.
47 - 52 = -5.
Negative means the point sat below the line.
Problem 4
Adding one extreme outlier far from the rest of the data will usually
- pull the line of best fit towards itself
- leave the line unchanged
- make the line vertical
- make r exactly 1
- have no effect on predictions
Worked answer
pull the line of best fit towards itself
A least-squares line minimises squared distances.
A far-off point contributes a very large squared distance.
So the line tilts towards it to reduce that.
This is why outliers must be reported, not hidden.
Problem 5
Two lines are fitted to the same scatter plot by two students. Line A has residuals 2, -3, 1, -1; line B has residuals 5, -4, 3, -4. Which fits better, and why?
- A, because its residuals are smaller overall
- B, because it has a larger first residual
- A, because its residuals sum to -1
- B, because its residuals are more varied
- Neither, because both sums are close to zero
Worked answer
A, because its residuals are smaller overall
Smaller residuals mean the line passes closer to the data.
A's residuals are 2, -3, 1, -1; B's are much larger.
Compare the sizes, not the signs.
Sum alone tells you little, since positives cancel negatives.
Problem 6
A line of best fit predicts test score from hours slept: y = 5h + 40. A student sleeps 0 hours. The prediction of 40 points is
- probably meaningless, since no one in the data slept 0 hours
- exactly right
- impossible, since scores cannot be positive
- the same as the slope
- proof that sleep causes higher scores
Worked answer
probably meaningless, since no one in the data slept 0 hours
The intercept is a prediction at x = 0.
If nobody in the study slept zero hours, that is extrapolation.
The number exists but has no support from the data.