Lines of Best Fit — Drills
Lines of Best Fit has 10 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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.
Equation
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A line of best fit passes through (0, 34) and (10, 44). Find its equation.
Worked answer
y = 1x + 34
b = 34
m = 44 - 3410 = 1
y = 1x + 34
Problem 2
A line of best fit passes through (0, 47) and (6, 35). Find its equation.
Worked answer
y = -2x + 47
b = 47
m = 35 - 476 = -2
y = -2x + 47
Predicting
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For y = 6x + 44, predict y when x = 5.
Worked answer
74
6(5) = 30
30 + 44 = 74
Problem 2
For y = -5x + 34, predict y when x = 12.
Worked answer
-26
-5(12) = -60
-60 + 34 = -26
Slope
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For y = 3.7x + 37, where x is in years and y in points, what does 3.7 mean?
Worked answer
3.7 points per year
Units: points per year
Each extra year is associated with 3.7 more points.
Problem 2
For y = 4.9x + 19, where x is in days and y in centimetres, what does 4.9 mean?
Worked answer
4.9 centimetres per day
Units: centimetres per day
Each extra day is associated with 4.9 more centimetres.
Intercepts
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For y = 6x + 18, where x is hours studied, what does 18 mean?
Worked answer
the value when x = 0
At x = 0, y = 18.
So 18 is the predicted starting value.
Problem 2
For y = 3x + 36, where x is hours studied, what does 36 mean?
Worked answer
the value when x = 0
At x = 0, y = 36.
So 36 is the predicted starting value.
Residual
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A line predicts y = 72 but the actual value was 74. Find the residual.
Worked answer
2
74 - 72 = 2
Positive, so the point sits above the line.
Problem 2
A line predicts y = 96 but the actual value was 93. Find the residual.
Worked answer
-3
93 - 96 = -3
Negative, so the point sits below the line.
Extrapolate
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Data covers x from 3 to 14. Predicting at x = 102 is
- impossible to compute
- interpolation, and safe
- extrapolation, and unsafe
- the same as the mean
Worked answer
extrapolation, and unsafe
102 is far outside 3 to 14.
That is extrapolation.
The pattern may not continue that far.
Problem 2
Data covers x from 4 to 13. Predicting at x = 69 is
- extrapolation, and unsafe
- the same as the mean
- interpolation, and safe
- impossible to compute
Worked answer
extrapolation, and unsafe
69 is far outside 4 to 13.
That is extrapolation.
The pattern may not continue that far.
Solving
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For the line y = 6x + 22, at what x does y reach 106?
Worked answer
14
6x + 22 = 106
6x = 84
x = 14
Problem 2
For the line y = 8x + 18, at what x does y reach 82?
Worked answer
8
8x + 18 = 82
8x = 64
x = 8
Fit
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Line A has residuals 1, -3, 2, -1. Line B has residuals 5, -5, 8, -5. Which fits better?
- B
- A
- Equally well
- Cannot be determined
Worked answer
A
A misses by at most 3.
B misses by up to 8.
So A fits better.
Problem 2
Line A has residuals 2, -2, 1, -3. Line B has residuals 7, -4, 6, -9. Which fits better?
- B
- Equally well
- Cannot be determined
- A
Worked answer
A
A misses by at most 3.
B misses by up to 9.
So A fits better.
Reasoning it through
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A best-fit line for a child's height against age gives y = 2x + 35 inches. Predict the height at age 45 and say whether to trust it.
Worked answer
125 inches, not trustworthy
2(45) + 35 = 125 inches
That is over 10 feet.
Far outside the data range, so an unsafe extrapolation.
Problem 2
A best-fit line for a child's height against age gives y = 3x + 32 inches. Predict the height at age 32 and say whether to trust it.
Worked answer
128 inches, not trustworthy
3(32) + 32 = 128 inches
That is over 10 feet.
Far outside the data range, so an unsafe extrapolation.
Comparing
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two students fit lines to the same data: y = 2.9x + 5 and y = 3x + 4.6. Can both be reasonable? Answer yes or no.
Worked answer
yes
Both can pass through the middle of the cloud.
Slightly different lines can both summarise it well.
So yes.
Problem 2
Two students fit lines to the same data: y = 1.9x + 5 and y = 1.9x + 4.6. Can both be reasonable? Answer yes or no.
Worked answer
yes
Both can pass through the middle of the cloud.
Slightly different lines can both summarise it well.
So yes.