CONFIDENTIAL — UNIT 9

The Case of the Vanishing Slope

Inference for Quantitative Data · Slopes

Every good detective knows: two variables can look like partners in crime, standing close together on a scatterplot, pointing in the same direction. But a scatterplot is only a rumor. To prove a real linear relationship exists in the population — not just in this one sample — you need evidence. That's where this agency comes in.

In this file, Lead Detective Kim Min-ho walks a new trainee through four cases:

1. Checking whether the evidence is even admissible (LINER conditions)
2. Estimating the true slope with a confidence interval
3. Putting a suspicious slope on trial (hypothesis test)
4. Reading how strong the evidence really is (interpreting the t-statistic)

No calculator tricks, no shortcuts skipped — every number in this file has been checked three times. Tap Case 1 above (or the button below) whenever you're ready to start investigating.

CASE FILE 01

Is the Evidence Even Admissible?

LINER Conditions

"Before I let any regression line testify in court," Detective Kim says, tapping a scatterplot pinned to the board, "it has to pass five background checks. I call them the LINER conditions. Skip even one, and the judge — reality — will throw the whole case out."

The Five Checks

LLinear

The true relationship between x and y is actually a straight line. Check: the residual plot should look like random scatter with no curve or pattern.

IIndependent

Individual observations don't influence each other. Check: data comes from random sampling/assignment, and if sampling without replacement, the sample is less than 10% of the population.

NNormal

At each x-value, the residuals are approximately normally distributed. Check: a histogram or normal probability plot of the residuals looks roughly symmetric/straight.

EEqual SD

The spread of residuals stays about the same for every x-value (no "fan" shape). Check: the residual plot has consistent vertical spread across all x.

RRandom

The data itself was produced by a random sample or a randomized experiment — otherwise you can't generalize to a wider population.

EXHIBIT A

Detective Kim shows the trainee a residual plot. It's shaped like a smile — a clear U-curve, dipping down and coming back up.

Since the residuals show a systematic curved pattern instead of random scatter, the straight-line model is missing something.

Violation: the Linear condition fails. A line is the wrong shape for this data.

EXHIBIT B

The residual plot fans out — points cluster tightly near small x-values, then spread widely for large x-values.

The variability of the residuals is changing as x increases, instead of staying constant.

Violation: the Equal SD condition fails (this is sometimes called "heteroscedasticity").

TRAINEE TEST 1.1

A study only surveys students who happened to walk by the school library on a Tuesday afternoon. Which LINER condition is most directly threatened?

Random fails (and Independent is shaky too) — this is a convenience sample, not a random one, so we can't safely generalize results to all students.

TRAINEE TEST 1.2

The residual plot shows points scattered with no pattern and roughly the same spread across all x. A histogram of residuals is bell-shaped and symmetric. Sample was randomly selected and is under 10% of the population.

All five LINER conditions check out! CASE ADMISSIBLE — it's safe to proceed with inference for the slope.

CASE FILE 02

Drawing the Boundary of Truth

Confidence Interval for the Slope

"We never know the *true* slope, $\beta$ — that number belongs to the entire population, hiding just out of reach," says Detective Kim. "All we get is one sample's estimate, . So instead of pretending we know the exact truth, we draw a boundary — a range we're confident contains it."

The Formula

Confidence Interval for the Slope

Degrees of Freedom

is the sample slope from computer/calculator output, is its standard error, and is the critical value from the t-distribution table using .

CASE FILE — EXAMPLE

Computer output from a study of 15 study-partners (n = 15) gives slope b = 2.5 and SE_b = 0.4. Build a 95% confidence interval for the true slope.

df = n − 2 = 15 − 2 = 13.
For 95% confidence with df = 13, the critical value is t* ≈ 2.160.
Margin of error = t* × SE_b = 2.160 × 0.4 = 0.864.
Interval: 2.5 ± 0.864 → (1.636, 3.364).

We are 95% confident the true slope is between 1.64 and 3.36. In context: for every extra hour studied, mean test score is estimated to rise between about 1.64 and 3.36 points.

SECOND WITNESS

A different case: n = 10 (df = 8), b = 1.8, SE_b = 0.3. Build a 90% confidence interval.

For 90% confidence, df = 8 → t* ≈ 1.860.
Margin of error = 1.860 × 0.3 = 0.558.
Interval: 1.8 ± 0.558 → (1.242, 2.358).

We're 90% confident the true slope lies between 1.24 and 2.36.

TRAINEE TEST 2.1

A 95% CI for the slope comes out to (1.64, 3.36) — notice it doesn't contain 0. What does that tell the detective about the relationship between the variables?

Since 0 is not a plausible value for the slope, there's convincing evidence of a real (nonzero) linear relationship in the population — the case isn't a coincidence.

CASE FILE 03

Putting the Slope on Trial

Hypothesis Test for the Slope

"Sometimes a confidence interval isn't enough — we need a verdict," Detective Kim explains. "So we put the boring explanation on trial: 'There is no relationship at all.' We call that the null hypothesis, and we only overturn it if the evidence against it is overwhelming."

The Hypotheses

Null — "innocent until proven guilty"

Alternative — the detective's suspicion

The Test Statistic

t-statistic for the slope

Just like any t-test, we compare the sample statistic (b) to its hypothesized value (0), measured in standard errors. If |t| is large, the sample slope is far from 0 relative to its own uncertainty — suspicious!

CASE FILE — EXAMPLE

n = 20 people, computer output gives b = 3.2, SE_b = 1.1. Test H₀: β = 0 vs Hₐ: β ≠ 0 at α = 0.05.

df = 20 − 2 = 18.
t = (3.2 − 0) / 1.1 = 2.91.
Critical value for α = 0.05 (two-sided), df = 18, is t* ≈ 2.101.
Since |2.91| > 2.101, the result falls in the rejection zone. The two-sided p-value ≈ 0.009.

Since p ≈ 0.009 < 0.05, we reject H₀. GUILTY — RELATIONSHIP CONFIRMED There is convincing evidence of a real linear relationship.

SECOND WITNESS

n = 25, b = −0.85, SE_b = 0.40. Test at α = 0.05.

df = 25 − 2 = 23.
t = (−0.85 − 0) / 0.40 = −2.125.
Critical value, df = 23, α = 0.05 two-sided ≈ ±2.069.
|−2.125| > 2.069 → p-value ≈ 0.044.

p ≈ 0.044 < 0.05, so we reject H₀ — there's evidence of a real (negative) linear relationship, though the case was closer than the first.

TRAINEE TEST 3.1

n = 12, b = 0.6, SE_b = 0.5. Compute t and decide: does this evidence convict H₀ at α = 0.05? (df = 10, critical t* ≈ 2.228)

t = 0.6 / 0.5 = 1.2. Since |1.2| < 2.228, we fail to reject H₀. CASE DISMISSED — not enough evidence of a real relationship.

CASE FILE 04

Reading the Strength of a Clue

Interpreting the t-Test Statistic

"A rookie mistake," Detective Kim warns, "is thinking a bigger slope always means stronger evidence. It doesn't. What matters is how many standard errors away from zero that slope sits. That's exactly what t measures."

What t Actually Means

The t-statistic tells you: how many standard errors is the sample slope away from the hypothesized value (usually 0)? A bigger |t| means the sample slope would be a rarer, more surprising result if there really were no relationship — so it's stronger evidence against H₀.

SIDE-BY-SIDE COMPARISON

Suspect A: b = 4, SE_b = 2 → t = 4/2 = 2
Suspect B: b = 4, SE_b = 0.5 → t = 4/0.5 = 8

Both suspects have the exact same estimated slope (b = 4) — but Suspect B's estimate is far more precise (a much smaller standard error), so it sits many more standard errors away from 0.

Even though b is identical, Suspect B provides much stronger evidence against H₀ (larger |t|, smaller p-value) than Suspect A. The size of the slope alone never tells the whole story — precision matters just as much.

TRAINEE TEST 4.1

Two studies both find b = 10. Study X has t = 1.5. Study Y has t = 5.0. Which study gives more convincing evidence of a real linear relationship, and why?

Study Y. A larger |t| means the sample slope is farther (in standard errors) from 0, which corresponds to a smaller p-value and stronger evidence — regardless of the two studies sharing the same b.

TRAINEE TEST 4.2

True or false: if t is close to 0, that means the true population slope must be exactly 0.

False. A small |t| just means our sample didn't provide convincing evidence against β = 0 — we "fail to reject," which is not the same as proving H₀ true. The real slope could still be nonzero; we simply lack sufficient evidence.

CLOSED CASE — FINAL REPORT

Case File Summary

Unit 9 · Inference for Slopes — Quick Reference

"Every good detective keeps a one-page summary pinned above the desk. Here's yours." — Kim Min-ho

LINER Conditions

Linear — residual plot has no pattern.
Independent — random sampling/assignment, <10% condition.
Normal — residuals roughly normal at each x.
Equal SD — constant residual spread, no fan shape.
Random — data from random sample/experiment.

Confidence Interval for Slope

df = n − 2. Find t* from the t-table at the chosen confidence level.

Hypothesis Test for Slope

Reject H₀ when p-value < α (evidence of a real linear relationship).

Interpreting t

t = how many standard errors the sample slope is from 0. Larger |t| → smaller p-value → stronger evidence against H₀. A small |t| means "not enough evidence," never "proof of no relationship."

Case closed. Go back through any file above to review the story, the worked examples, or retry the trainee tests.