Bivariate Data & Correlation — Drills
Bivariate Data & Correlation has 11 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 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.
Readr
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
r = 0.05 describes a relationship that is
- strong and positive
- almost no linear relationship
- strong and negative
- weak and negative
Worked answer
almost no linear relationship
0.05 is very close to 0.
So almost no linear relationship.
Problem 2
r = -0.79 describes a relationship that is
- strong and negative
- weak and negative
- almost no linear relationship
- strong and positive
Worked answer
strong and negative
Sign: negative.
|-0.79| is close to 1, so strong.
Residual
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A regression line is y = 5x + 18. For x = 12 the actual value was 76. Find the residual.
Worked answer
-2
Predicted: 5(12) + 18 = 78
76 - 78 = -2
Problem 2
A regression line is y = 7x + 13. For x = 6 the actual value was 48. Find the residual.
Worked answer
-7
Predicted: 7(6) + 13 = 55
48 - 55 = -7
Predicting
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A regression line is y = 6.1x + 58. Predict y when x = 5.
Worked answer
88.5
6.1(5) = 30.5
+ 58 = 88.5
Problem 2
A regression line is y = -4.7x + 46. Predict y when x = 13.
Worked answer
-15.1
-4.7(13) = -61.1
+ 46 = -15.1
Slope
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For y = 0.6x + 56, with x in kilometres and y in minutes, interpret the slope.
Worked answer
0.6 minutes per kilometre
Units: minutes per kilometre
Each extra kilometre is associated with 0.6 more minutes.
Problem 2
For y = 2.6x + 31, with x in kilometres and y in minutes, interpret the slope.
Worked answer
2.6 minutes per kilometre
Units: minutes per kilometre
Each extra kilometre is associated with 2.6 more minutes.
Resplot
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A residual plot shows a clear curved arch. This means
- r must be zero
- a line is the wrong model
- the line fits perfectly
- the data has no outliers
Worked answer
a line is the wrong model
Structure in the residuals means the model missed something systematic.
The data curves, but the line cannot.
Sign
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A regression line has a positive slope. What is the sign of r?
Worked answer
positive
A rising line means r is positive.
Problem 2
A regression line has a negative slope. What is the sign of r?
Worked answer
negative
A falling line means r is negative.
Cause
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A strong r between two observed variables proves
- nothing about cause
- that the first causes the second
- that a third variable is involved
- that the second causes the first
Worked answer
nothing about cause
Correlation measures how the variables moved together.
Cause needs a controlled, randomised experiment.
Residsum
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For a least-squares regression line, what do all the residuals add up to?
Worked answer
0
Points above the line give positive residuals, points below give negative.
They cancel exactly, so the sum is 0.
Comparing
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Study A has r = 0.7; study B has r = -0.5. Which has the stronger linear relationship?
- A
- Cannot be determined
- They are equal
- B
Worked answer
A
|0.7| = 0.7
|-0.5| = 0.5
0.7 > 0.5, so A.
Problem 2
Study A has r = 0.4; study B has r = -0.78. Which has the stronger linear relationship?
- B
- A
- They are equal
- Cannot be determined
Worked answer
B
|0.4| = 0.4
|-0.78| = 0.78
0.78 > 0.4, so B.
Interpret
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For y = -0.3x + 30, where x is age in years and y is hours of sleep, interpret the slope in context.
Worked answer
sleep drops 0.3 hours per year of age
The slope is negative, so sleep falls as age rises.
Each extra year is associated with 0.3 fewer hours.
Problem 2
For y = -0.5x + 27, where x is age in years and y is hours of sleep, interpret the slope in context.
Worked answer
sleep drops 0.5 hours per year of age
The slope is negative, so sleep falls as age rises.
Each extra year is associated with 0.5 fewer hours.
Reasoning it through
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A regression on data for x between 3 and 19 gives y = 3x + 10. Is predicting at x = 156 reliable? Answer yes or no.
Worked answer
no
156 is far outside 3 to 19.
That is extrapolation, so not reliable.
Problem 2
A regression on data for x between 5 and 19 gives y = 3x + 10. Is predicting at x = 84 reliable? Answer yes or no.
Worked answer
no
84 is far outside 5 to 19.
That is extrapolation, so not reliable.