Bivariate Data & Correlation — Practice
Fit a line, read its slope in context, and judge how well it fits — then resist the urge to claim a cause.
What these questions 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
r = 0.92 describes a relationship that is
- strong and positive
- weak and positive
- strong and negative
- non-existent
Worked answer
strong and positive
Positive sign
Close to 1, so strong
Problem 2
r = -0.88 describes a relationship that is
- strong and negative
- weak and negative
- strong and positive
- non-existent
Worked answer
strong and negative
Negative sign, so it falls
|-0.88| is close to 1, so strong
Problem 3
r = 0.05 means
- almost no linear relationship
- a strong relationship
- a perfect relationship
- a negative relationship
Worked answer
almost no linear relationship
0.05 is very close to 0
So almost no linear relationship
Problem 4
A line predicts 50 and the actual value is 56. Find the residual.
Worked answer
6
56 - 50 = 6
Problem 5
A line predicts 120 and the actual value is 113. Find the residual.
Worked answer
-7
113 - 120 = -7
Problem 6
A regression line is y = 4x + 6. For x = 7 the actual value was 30. Find the residual.
Worked answer
-4
Predicted: 4(7) + 6 = 34
30 - 34 = -4
Problem 7
A regression line is y = -2.5x + 80. Predict y when x = 12.
Worked answer
50
-2.5(12) = -30
-30 + 80 = 50
Problem 8
For y = 1.8x + 12, with x in hours and y in dollars, interpret the slope.
Worked answer
1.80 more per hour
Dollars per hour
Each extra hour is associated with $1.80 more
Problem 9
A residual plot shows a clear curved arch. This means
- a line is the wrong model
- the line fits perfectly
- r must be zero
- the data has no outliers
Worked answer
a line is the wrong model
Structure in the residuals means the model missed something
A curve needs a curved model
Problem 10
A regression line has a negative slope. What can you say about the sign of r?
Worked answer
negative
Negative slope means negative r
Problem 11
A strong r between two observed variables proves
- nothing about cause
- that the first causes the second
- that the second causes the first
- that a third variable is involved
Worked answer
nothing about cause
Correlation measures how they move together
Cause needs a controlled experiment
Problem 12
For a least-squares line, what do all the residuals add up to?
Worked answer
0
Positives and negatives cancel
The sum is 0
Problem 13
A regression line for score against hours studied is y = 6.5x + 41. A student studies 4 hours and scores 74. Find the residual.
Worked answer
7
Predicted: 6.5(4) + 41 = 67
74 - 67 = 7
Problem 14
Study A has r = 0.6; study B has r = -0.8. Which has the stronger linear relationship?
Worked answer
B
|0.6| = 0.6
|-0.8| = 0.8
B is stronger
Problem 15
For y = -0.4x + 32, where x is age in years and y is hours of sleep, interpret the slope in context.
Worked answer
sleep drops 0.4 hours per year of age
Negative, so sleep falls
Each extra year is associated with 0.4 fewer hours
Problem 16
A regression on data for x between 5 and 20 gives y = 3x + 10. Is predicting at x = 200 reliable? Answer yes or no.
Worked answer
no
200 is far outside 5 to 20
That is extrapolation
So not reliable