📝 Chapter 8: Correlation, Regression & Culminating Investigation

Assessment AS Learning — Practice Quiz

🔄 Not Graded — Unlimited Retakes

Purpose: Self-check your work with scatter plots, correlation coefficient, regression, residuals, and two-variable analysis.

Score: 0 / 12

Topic 8.1–8.2 — Correlation

Question 1

A correlation coefficient of \( r = -0.92 \) indicates:

A weak positive linear relationship
A strong negative linear relationship
No relationship
A nonlinear relationship

Solution:

\( r = -0.92 \): negative direction, magnitude near 1 = strong linear relationship.

Question 2

If \( r = 0.5 \) for the relationship between \( x \) and \( y \), what proportion of the variation in \( y \) is explained by \( x \)?

Solution:

The coefficient of determination \( r^2 = (0.5)^2 = 0.25 = 25\% \).

Question 3

A study finds that ice cream sales are positively correlated with drowning incidents. This is most likely an example of:

Cause-and-effect (ice cream causes drownings)
Reverse cause-and-effect
Common cause (e.g., warm weather)
Coincidence with no association

Solution:

Hot weather is a "lurking variable" causing both increased ice-cream sales and more swimming → more drownings. Common cause.

Topic 8.3 — Linear Regression

Question 4

The regression line is \( \hat{y} = 5 + 2x \). Predict \( y \) when \( x = 10 \).

Solution:

\( \hat{y} = 5 + 2(10) = 25 \).

Question 5

For data with \( \bar{x} = 5, \bar{y} = 20, s_x = 2, s_y = 8, r = 0.75 \), the slope of the line of best fit is:

Solution:

\( b_1 = r \dfrac{s_y}{s_x} = 0.75 \cdot \dfrac{8}{2} = 0.75 \cdot 4 = 3 \).

Question 6

For Q5, the y-intercept of the regression line is:

Solution:

The line passes through \( (\bar{x}, \bar{y}) \). \( b_0 = \bar{y} - b_1 \bar{x} = 20 - 3(5) = 5 \).

Topic 8.4 — Residuals

Question 7

A regression predicts \( \hat{y} = 50 \) for \( x = 10 \), but the observed \( y \) was 47. The residual is:

Solution:

Residual = \( y - \hat{y} = 47 - 50 = -3 \). The model overestimated.

Question 8

A residual plot shows a clear curved pattern. This suggests:

The linear model is appropriate.
A non-linear model would fit better.
There is no relationship.
All residuals are zero.

Solution:

A pattern in residuals indicates the model fails to capture the underlying structure. Try a non-linear model.

Topic 8.5 — Causation

Question 9

A study finds that students who spend more time studying have higher grades. This is best classified as:

Cause-and-effect (probably) — but caution: a controlled experiment is needed for certainty
Common cause (no relation between variables)
Reverse causation only
Accidental relationship

Solution:

Probable cause-and-effect (more study → better grades), though confounders exist (motivation, etc.).

Question 10

"Children with bigger feet read better." This is most likely:

Cause-and-effect
Common cause: age (older children have bigger feet AND read better)
Reverse cause-and-effect
Accidental relationship

Solution:

Age is a lurking variable causing both. This is a common-cause relationship.

Mixed

Question 11

For the data \( \{(1,2), (2,4), (3,6), (4,8), (5,10)\} \), what is \( r \)?

Solution:

The points are exactly on the line \( y = 2x \). Perfect positive linear relationship: \( r = 1 \).

Question 12

For Q11, the regression line is \( \hat{y} = ? \). State the slope as a number.

Solution:

The line is \( \hat{y} = 2x \). Slope = 2; intercept = 0.