Linear Regression and Correlation — HS Statistics practice questions

Cities with more firefighters at a fire have more fire damage. What best explains this?

1. Firefighters cause the damage
2. Damage causes cities to hire firefighters
3. Bigger fires bring more firefighters and cause more damage
4. It is a coincidence

Answer: C — Bigger fires bring more firefighters and cause more damage

C) Correct — Fire size is a lurking variable behind both. A) Confuses correlation with causation. B) Reverses the timing; the firefighters respond to the fire. D) The pattern has a clear common cause.

Which graph shows the relationship between two quantitative variables?

1. A histogram
2. A scatterplot
3. A pie chart
4. A bar chart

Answer: B — A scatterplot

B) Correct — Each point is one individual's pair of values. A) Shows one variable's distribution. C) Shows parts of a whole. D) Shows counts of categories.

An influential point in regression is one that:

1. has a large residual
2. lies closest to (x̄, ȳ)
3. has the smallest x-value
4. noticeably changes the slope or intercept when removed

Answer: D — noticeably changes the slope or intercept when removed

D) Correct — Influential points often have extreme x-values. A) An outlier in y may barely move the line. B) Points near the center have little influence. C) Only extreme x-values far from the pattern tend to be influential.

A student's actual score is 70, and the line predicts 74. What is the residual?

1. −4
2. 4
3. 70
4. 16

Answer: A — −4

A) Correct — Residual = actual − predicted = 70 − 74. B) Subtracts in the wrong order. C) Is the actual value. D) Squares the residual.

A regression of test scores on study hours has r² = 0.64. How should r² be read?

1. 64% of the variation in scores is explained by the linear relationship with hours
2. 80% of the variation is explained
3. Study hours cause 64% of each score
4. 64% of the points lie on the line

Answer: A — 64% of the variation in scores is explained by the linear relationship with hours

A) Correct — r² is the share of variation in y accounted for by the model. B) Uses r = 0.8 instead of r². C) r² is not a causal share of each score. D) r² is not a count of points on the line.

r² (coefficient of determination)?

Proportion of Y variation explained by X. 0 ≤ r² ≤ 1.

Describing a scatterplot?

Direction (positive or negative), form (linear or curved), strength (how closely points follow the form), and unusual features (outliers, clusters), in context.

WORKED EXAMPLE: Conceptually compute the correlation coefficient r for {(1,2), (2,4), (3,5), (4,4), (5,5)} and interpret.

Step 1 — COMPUTE means: x̄ = (1+2+3+4+5)/5 = 3; ȳ = (2+4+5+4+5)/5 = 4. Step 2 — DEVIATIONS (x − x̄): -2, -1, 0, 1, 2. Deviations (y − ȳ): -2, 0, 1, 0, 1. Step 3 — PRODUCTS (x − x̄)(y − ȳ): 4, 0, 0, 0, 2. Sum = 6. Step 4 — SQUARES Σ(x − x̄)² = 4+1+0+1+4 = 10. Σ(y − ȳ)² = 4+0+1+0+1 = 6. Step 5 — FORMULA: r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² · Σ(y − ȳ)²] = 6 / √(10·6) = 6/√60 ≈ 6/7.746 ≈ 0.775. Step 6 — INTERPRET: r ≈ 0.78 → strong positive linear association. As x increases, y tends to increase, though not perfectly linearly. Step 7 — r² ≈ 0.60 → about 60% of variation in y is explained by linear regression on x.

Residual?

Observed y - predicted y. Errors.