Bivariate Data & Correlation — Challenge
Fit a line, read its slope in context, and judge how well it fits — then resist the urge to claim a cause.
What these problems cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Slope — How much y changes for each 1 that x increases.
- Range — The set of outputs that actually occur.
- Outlier — A dot far away from the rest of the pattern.
- Extrapolation — Predicting outside the range. Risky.
- Residual — The gap between an actual value and the line's prediction.
- Regression line — The line of best fit computed by a formula rather than by eye.
- Residual plot — A plot of residuals against x, used to judge whether a line was the right model.
- Prove — The statement you must arrive at. Always the last line.
Problem 1
On a scatter plot the points tend to rise as x increases. What does this show?
- A positive association
- A negative association
- No association
- A perfect correlation
- An outlier
Worked answer
A positive association
An association is positive when larger x values go with larger y values.
The rising trend is exactly that, so the association is positive.
It does not have to be perfect: the points may scatter about the trend and still show it.
Problem 2
A line of best fit is y=2x+5. What does it predict for x=10?
- 15
- 20
- 25
- 30
- 52
Worked answer
25
Substitute x=10 into the equation.
y = 2(10)+5 = 25.
This is a prediction rather than an observation: an actual data point at x=10 may differ.
Problem 3
Which correlation coefficient shows the strongest negative linear relationship?
- -1
- -0.5
- 0
- 0.5
- 1
Worked answer
-1
A correlation coefficient runs from -1 to 1, and its sign gives the direction.
Strength is measured by distance from 0, so the strongest negative value is the one closest to -1.
That is -1 itself, which would mean the points lie exactly on a falling straight line.
Problem 4
A line of best fit predicts 10 but the actual value is 12. What is the residual?
- -2
- 0
- 2
- 10
- 22
Worked answer
2
A residual is the actual value minus the predicted value.
12-10 = 2.
A positive residual means the real point lies above the line.
Problem 5
A correlation coefficient of 0 means what?
- A strong relationship
- No linear association
- A perfect relationship
- The data is wrong
- All points are equal
Worked answer
No linear association
A coefficient of 0 means no straight-line trend: knowing x gives no help in predicting y linearly.
It does not mean the variables are unrelated in every way; the points could lie on a strong curve and still give a coefficient near 0.
The coefficient measures linear association only.
Problem 6
A line of best fit is y=-3x+20. What does it predict for x=4?
- 4
- 8
- 12
- 17
- 32
Worked answer
8
Substitute x=4: y = -3(4)+20.
-12+20 = 8.
The negative slope means each extra unit of x lowers the prediction by 3.
Problem 7
A strong correlation between two variables does not by itself establish what?
- Correlation
- Causation
- Association
- A trend
- A measurement
Worked answer
Causation
Correlation shows that two variables move together.
It does not show that one makes the other happen: a third factor may drive both, or the link may be coincidence.
Establishing causation needs more than a scatter plot, usually a controlled experiment.
Problem 8
A line of best fit passes through (0,3) with slope 2. What is its equation?
- y=2x+3
- y=3x+2
- y=2x-3
- y=3x
- y=2x
Worked answer
y=2x+3
The point (0,3) is on the y-axis, so the intercept is 3.
With slope 2 the equation is y = 2x+3.
Check: at x=0 the line gives y=3, as required.