LINER Conditions for Inference on Slope

Unit 9 · Inference for Quantitative Data: Slopes

What AP Stats asks here

Before running a slope CI or test, verify LINER: Linearity, Independence, Normality of residuals, Equal SD, Random. Two of those (I and R) come from the design; the other three (L, N, E) come from plots. AP rubrics award explicit points for naming each condition AND citing the specific evidence (scatterplot, residual plot, residual histogram).

LINER checklist

L — Linearity

\text{scatterplot + residual plot}

I — Independence

\text{random / 10% / no time-series}

N — Normality of residuals

\text{residual histogram or NPP}

E — Equal SD of residuals

\text{residual plot, constant spread}

R — Random

\text{stated in design}

AP Tip: Memory aid: L and E both live in the residual plot. N lives in the residual histogram or NPP. I and R come from the design.

Caution: Consecutive time-series observations (daily prices, repeated measures on the same subject) violate independence even if everything else looks clean.

Type 1

Identify each LINER condition

Each letter maps to a specific verification source. Knowing the mapping is half the rubric.

Example 1

The residual plot shows constant vertical spread across the range of

x

. Which LINER condition is being verified? (A) Linearity (B) Independence (C) Equal SD of residuals (D) Normality of residuals

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Type 2

Diagnose the failing condition from a residual plot

Curvature (U-shape or arch) ⇒ Linearity fails. A fan or funnel ⇒ Equal SD fails.

Example 2

A residual plot opens like a fan to the right — narrow on the low end, wide on the high end. Which LINER condition fails? (A) Linearity — the data must be curved. (B) Independence — observations are correlated. (C) Normality of residuals. (D) Equal SD of residuals — the spread of residuals grows with x.

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Type 3

Design conditions vs plot conditions

L, E, N are diagnosed from plots. I and R come from the sampling/assignment design.

Example 3

Closing stock prices of one company are regressed on day number across 100 consecutive trading days. Which LINER condition is most likely violated? (A) Linearity — stock prices grow linearly. (B) Independence — consecutive daily prices are autocorrelated, so observations are not independent. (C) Equal SD — variance is constant by definition. (D) Normality — residuals are always normal in finance.

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