Lines of Best Fit — Practice
One line drawn through the middle of the cloud, then read like any other line: slope and intercept, with units.
What these questions 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 passes through (0, 12) and (4, 24). Find its equation.
Worked answer
y = 3x + 12
b = 12
m = 24 - 124 = 3
y = 3x + 12
Problem 2
A line of best fit passes through (0, 50) and (10, 30). Find its equation.
Worked answer
y = -2x + 50
b = 50
m = 30 - 5010 = -2
y = -2x + 50
Problem 3
For y = 4x + 15, predict y when x = 6.
Worked answer
39
4(6) + 15
= 24 + 15 = 39
Problem 4
For y = 2.5x + 8 where x is hours and y is dollars, what does 2.5 mean?
Worked answer
2.50 per hour
Dollars per hour
$2.50 for each extra hour
Problem 5
For y = 6x + 20 where x is days of growth, what does 20 mean?
Worked answer
the starting value at day 0
At x = 0, y = 20
So the plant began at 20
Problem 6
A line predicts y = 34 but the actual value was 40. Find the residual.
Worked answer
6
40 - 34 = 6
Positive, so the point sits above the line
Problem 7
A line predicts y = 88 but the actual value was 81. Find the residual.
Worked answer
-7
81 - 88 = -7
Negative, so the point sits below the line
Problem 8
A line of best fit is y = -1.5x + 60. Predict y when x = 12.
Worked answer
42
-1.5(12) = -18
-18 + 60 = 42
Problem 9
Data covers x from 2 to 15. Predicting at x = 80 is
- extrapolation, and unsafe
- interpolation, and safe
- impossible to compute
- the same as the mean
Worked answer
extrapolation, and unsafe
80 is far outside 2 to 15
That is extrapolation
The pattern may not continue
Problem 10
For the line y = 5x + 10, at what x does y reach 60?
Worked answer
10
5x + 10 = 60
5x = 50
x = 10
Problem 11
Line A has residuals 1, -2, 2, -1. Line B has residuals 6, -5, 4, -5. Which fits better?
- A
- B
- Equally well
- Cannot be determined
Worked answer
A
A misses by at most 2
B misses by up to 6
So A fits better
Problem 12
A best-fit line for temperature (°F) against altitude (thousands of feet) has slope -3.5. Interpret it.
Worked answer
temperature drops 3.5 degrees per thousand feet
Negative means it falls
Each extra thousand feet drops it 3.5°F
Problem 13
A best-fit line passes through (3, 17) and (8, 42). Find its equation.
Worked answer
y = 5x + 2
m = 42 - 178 - 3 = 5
17 = 5(3) + b, so b = 2
y = 5x + 2
Problem 14
A line is y = 3x + 4. A data point is (5, 22). Find the residual.
Worked answer
3
Predicted: 3(5) + 4 = 19
Residual: 22 - 19 = 3
Problem 15
A best-fit line for a child's height against age gives y = 2.4x + 30 inches. Predict the height at age 40 and say whether to trust it.
Worked answer
126 inches, not trustworthy
2.4(40) + 30 = 126 inches
That is over 10 feet
Far outside the data range, so an unsafe extrapolation
Problem 16
Two students fit lines to the same data. One gets y = 2x + 5, the other y = 2.1x + 4.6. Can both be reasonable? Answer yes or no.
Worked answer
yes
Both pass through the middle of the cloud
Slightly different lines can both summarise it well
So yes