A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
In 1973 the statistician Frank Anscombe published four small data sets. Each has 11 points. Each has the same mean of , the same mean of , the same correlation , and the same least-squares line, . If all you saw were those numbers, the four sets would be identical.
Here they are. Before you read on, make a prediction. Nothing is scored.
Every set: n = 11, x̄ = 9, ȳ = 7.50, r = 0.816, r² = 0.67, and the same least-squares line ŷ =
3.00 + 0.500x (drawn in ink). The dashed line marks x = 10, where every line predicts ŷ = 8.0.
Predict first · not scored
Which set would you trust to predict y at x = 10?
Anscombe's quartet. Data: F. J. Anscombe (1973), "Graphs in statistical analysis", The American Statistician 27(1), 17–21; the plots are drawn here from the published values.
Every line says at . Whether you should believe it is a different question, and the summary numbers cannot answer it: and measure how tightly points hug a line, not whether a line is the right model. This lesson gives you the tools that can answer it. You will check the sets one at a time in C2, C4 and C5, settle your prediction in Example 3, and audit a fresh quartet in the Boss Fight.
After this lesson, you will be able to:
Say what a fitted line claims: the slope as a predicted average change, the intercept only when zero lies inside the data, and predictions only inside the observed range.
Read a residual plot and name what it shows: no pattern, a curve, a fan, or one point far from the rest.
Predict how one point moves a line, and tell an outlier from a high-leverage point and an influential one.
Test whether a population correlation is zero, only when the residual plot allows it, and tell a significant relationship from a useful one.
Section 2: Prerequisites
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
What you need coming in, and why it matters today:
The least-squares line (L-REG-2): you can find and , and you have read a slope and an intercept in context. Today starts by retrieving those readings, then asks when to believe them.
Residuals (L-REG-2): you computed single residuals. Today you plot all of them at once and read the picture.
Conditions for a regression line (L-REG-2): two quantitative variables, a roughly linear pattern, no extreme influential outliers, and independent observations. Today adds one new condition, an even spread around the line, and shows how a residual plot checks all of them.
and (L-REG-1): measures the strength and direction of a linear relationship, and is the share of the variation in that the line accounts for.
The five steps of a hypothesis test (L-INF-5): hypotheses, conditions, test statistic, decision, conclusion in context. Reject when the evidence is strong enough; “fail to reject” never means “accept”.
Quick check: can you still say what a slope and an intercept claim?
Build the Conclusion
Say what the line claims
Choose a phrase for each blank so that the sentence states the correct conclusion.
Bridge to today: In L-REG-2 you built the line. Here you decide when to believe it. The tool is the residual plot: the same residuals you computed one at a time, drawn all together. It tells you whether the line is the right shape, whether its misses are the same size everywhere, and whether one point is steering it. The five steps from L-INF-5 come back at the end, with one rule added: the residual plot is the licence for the test.
Section 3: Core Concepts
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
How this section is organized: one short retrieval concept, then the new tools, then two enrichment concepts.
C1: What a fitted line claims (retrieved from L-REG-2), with extrapolation seen for real in Figure 1
C2–C3: Residual plots: a curve, a fan, and noise that only looks like a pattern (Figures 2 and 3; settles set II)
C4–C5: Outliers, leverage and influence: when one point steers the line (Figure 4; settles sets III and IV)
C6–C7 (enrichment): Testing whether the correlation is zero, and why significant is not the same as useful
C1 — What a Fitted Line Claims
A fitted line makes three claims, and each has a limit: the slope is a predicted average change, the intercept is a supported baseline only when zero lies inside the data, and a prediction is backed by data only inside the observed range of .
Recap from L-REG-2:
Slope: “For each additional [unit of ], the predicted increases (or decreases) by [units of ],on average.” Association, not cause.
Intercept: is always the line’s value at . It is a baseline the data support only if lies inside the observed range. Outside it, the intercept is an extrapolation.
Interpolation and extrapolation: a prediction at an inside is interpolation; outside it, even just past the edge, it is extrapolation. The equation gives a number anywhere; only inside the range do data stand behind it.
Figure 1 shows why the range matters. Ten students practised for between 1 and 8 hours before a test. Predict the score of a student who practised 12 hours before you reveal anything.
students the line was fitted to (1–8 h) students who practised longer (revealed)
Figure 1: Predict first, then reveal. The line ŷ = 49.87 + 4.85x is fitted only to students who practised
1 to 8 hours (simulated data). Past 8 hours the dashed line is the same equation with no data behind
it; the revealed students who practised longer show what actually happened.
The line kept climbing because nothing in the 1–8 hour data told it to stop. Scores cannot pass 100, and the students who practised longer levelled off near 90. Extrapolation is not wrong arithmetic; it is an assumption that the pattern continues where nobody looked.
“The intercept is the baseline, the value when is zero, so it is always meaningful.” It is always the line’s value at zero, but it is a baseline you can report only when lies inside the observed range. If the data start at 4 km, at 1 year or at 9 °C, the intercept is an extrapolation, however sensible the number looks.
C2 — Residual Plots: Is a Straight Line the Right Shape?
A residual plot puts each point’s residual on the vertical axis against (or against ), so whatever the line failed to capture is left on its own to be seen.
Residual plot
A plot of the residuals against (or against the fitted values ), with a horizontal line at .
If a straight line is the right shape, the residuals scatter above and below zero with no curve: about as many above as below at every part of the range.
If the residuals sit on one side of zero at both ends and on the other side in the middle, the data bend and the straight line cuts across the bend. Every straight line through curved data leaves that footprint, however high is. That is what is wrong with Anscombe’s set II: its points lie on a smooth arch, and its residual plot is an arch too.
” is close to 1, so a straight line is the right model.” measures how closely points hug the best straight line; it cannot tell you whether a straight line is the right shape. Anscombe’s set II has , exactly like set I, and it is a curve. Only the plot shows that.
C3 — Residual Plots: Even Spread, and Noise That Looks Like a Pattern
A good line leaves residuals of about the same size everywhere; when the vertical spread of the residuals grows (or shrinks) steadily along , the plot shows a fan, and the line’s misses are bigger in some places than others.
This condition is new: L-REG-2 checked the shape of the pattern, but not whether the scatter around the line is even. A fan does not mean the line is the wrong shape. It means one typical error does not describe every prediction, and the test in C6, which assumes an even spread, cannot be trusted.
Reading a residual plot
Look for four things, in this order:
No pattern: an even band around zero. This supports a straight line with even spread; it does not prove it.
A curve: the residuals bend. The straight line is the wrong shape (C2).
A fan: the band widens or narrows steadily along . The misses depend on .
One point: a single residual far from all the others. That is a regression outlier (C4).
Before you open Figure 2, answer this for yourself: which tab shows a model you should not use, and why? Each tab is named only by a letter; choose a diagnosis and the data set’s name appears.
Data set:
Scatter plot and fitted line
Residual plot (same scale on every tab)
Scatter plot and fitted line
Residual plot (same scale on every tab)
Scatter plot and fitted line
Residual plot (same scale on every tab)
Scatter plot and fitted line
Residual plot (same scale on every tab)
What does this residual plot show?
Figure 2: Pick a tab, read its residual plot, then choose a diagnosis; the name of the
data set appears only after you choose. Three of the tabs are Anscombe's sets I, II and III
(1973); one is simulated. All four residual plots use one scale, from −4 to +4.
Reading residual plots has a trap of its own. Pure noise is lumpy: points bunch up, leave gaps, and run above zero for a stretch, purely by chance. A real pattern has to stand out from that. Statisticians test their own eyes with a line-up: hide the real plot among plots of pure noise and see whether it can be picked out.
Five of these residual plots are pure noise from a correct model. One is not. Which?
Figure 3: A line-up. Every panel plots 24 residuals against x on the same scale, from −6
to +6. Five panels are simulated from a correct straight-line model with constant spread, so any
clump or bump in them is chance; one panel comes from data whose spread grows with x. The method
follows Buja et al. (2009) and Wickham et al. (2010).
If you found the fan, notice how much the other five plots wobble. A bump in a residual plot counts as a pattern only when it runs through the whole plot, as a bend or a steady widening does.
“Three residuals in a row above zero, or a little clump, means the line is wrong.” Pure noise does that all the time. Call it a pattern only when it runs through the whole plot: a bend from one end to the other, or a band that widens steadily. If you are unsure, imagine the plot in a line-up of noise: would it stand out?
C4 — Regression Outliers
A regression outlier is a point with an unusually large residual: it sits far above or below the line compared with how far the other points typically miss.
Regression outlier
An observation whose residual is large compared with the typical residual size. You need that comparison scale: a residual of 22 is huge when the others are within and ordinary when they are within .
Anscombe’s set III is an exact straight line of ten points with slope 0.345, plus one point at that sits far above it. That single point tilts the fitted line to slope 0.500 and drags down from 1 to 0.816. In tab D of Figure 2 its residual stands alone, and the other ten residuals form a straight sloping row: the footprint of a line pulled off course. Without the outlier, the line predicts at , not 8.0.
“An outlier spoils the fit, so delete it.” First check it: a typing error should be corrected, and a point from a different population (a broken sensor, a different machine) may be set aside with a stated reason. A real observation stays. Deleting it to raise hides how variable the data are. The honest report shows the fit with and without the point.
C5 — Leverage and Influence
A point’s leverage comes from its value alone: the farther it lies from , the more it can tilt the line; a point is influential when removing it would noticeably change the slope or the intercept, which takes leverage and a large miss from the trend of the other points.
Before reading on, predict with Figure 4: three candidate points, three positions. Which one, added on its own, moves the slope most?
Add just one of the three points and refit. Which one moves the slope most?
Show the line refitted with:
Figure 4: Three ways to add one point. After you predict, show each point's refitted line
(ink) against the line without it (grey, dashed); the added point is ringed in orange. Leverage
comes from x, the residual from y, and a point needs both to move the slope.
Leverage and influence
Leverage is potential: a point far from has high leverage whatever its .
Influence is effect: a point is influential if removing it substantially changes or . Check it by fitting the line with and without the point.
A point can be an outlier with little influence (C in Figure 4: at , it lifts the line without tilting it), have high leverage with no influence (A: far out but on the line), or be both far out and off the trend (B: influential).
Now Anscombe’s set IV: ten points share , and one point sits alone at . With ten points at a single , those ten alone cannot give a slope at all. The whole slope of 0.500 comes from the one point at : move it and the line follows. And at there are no data nearby, so is interpolation in name only. Of the four sets, only set I can back the prediction.
“Least squares weighs every point equally, so no single point can matter much.” Each squared residual counts once, but a point far out in has more say over the slope than a point in the middle. Influential points often have only moderate residuals, because they pull the line toward themselves. Compare the fits with and without the point instead of judging by the residual alone.
C6 — Testing Whether the Correlation Is Zero
A sample correlation can be non-zero by chance even when the population correlation is zero; a t-test of asks whether the sample gives enough evidence of a linear relationship in the population, and the residual plot decides whether that test can be trusted at all.
Enrichment. This concept and C7 go beyond the course standards: no standard assesses them. They connect the regression line to the hypothesis tests of L-INF-5.
The test follows the five steps from L-INF-5:
Hypotheses. (no linear relationship in the population) against , or a one-sided chosen before seeing the data.
Conditions, in plain words. The pairs are independent, and the residual plot shows no curve, no fan and no point pulling the line.
Test statistic.
Decision. Read from the t-table at (two tails together 0.05 is the 95% column). Reject when , which is the same as .
Conclusion in context. Say what the evidence shows about the population relationship, without claiming a cause.
The residual plot is the licence. When the plot shows a problem, the computed still exists, but its p-value does not mean what it claims. Say it this way: The test assumes the points scatter evenly around a straight line. This residual plot shows a curve (or a fan), so the test’s p-value can’t be trusted here. Fix the model first.
Why a t-distribution with degrees of freedom? The line uses up two numbers from the data, a slope and an intercept, so pieces of information are left to judge the scatter around it.
Student's t-Distribution Table
Critical values (t*) for given degrees of freedom (df) and tail area.
Student's t-Distribution Table
df
Confidence
80%
90%
95%
98%
99%
99.9%
0.10 (1)0.20 (2)
0.05 (1)0.10 (2)
0.025 (1)0.05 (2)
0.01 (1)0.02 (2)
0.005 (1)0.01 (2)
0.0005 (1)0.001 (2)
1
3.078
6.314
12.706
31.821
63.657
636.619
2
1.886
2.920
4.303
6.965
9.925
31.599
3
1.638
2.353
3.182
4.541
5.841
12.924
4
1.533
2.132
2.776
3.747
4.604
8.610
5
1.476
2.015
2.571
3.365
4.032
6.869
6
1.440
1.943
2.447
3.143
3.707
5.959
7
1.415
1.895
2.365
2.998
3.499
5.408
8
1.397
1.860
2.306
2.896
3.355
5.041
9
1.383
1.833
2.262
2.821
3.250
4.781
10
1.372
1.812
2.228
2.764
3.169
4.587
11
1.363
1.796
2.201
2.718
3.106
4.437
12
1.356
1.782
2.179
2.681
3.055
4.318
13
1.350
1.771
2.160
2.650
3.012
4.221
14
1.345
1.761
2.145
2.624
2.977
4.140
15
1.341
1.753
2.131
2.602
2.947
4.073
16
1.337
1.746
2.120
2.583
2.921
4.015
17
1.333
1.740
2.110
2.567
2.898
3.965
18
1.330
1.734
2.101
2.552
2.878
3.922
19
1.328
1.729
2.093
2.539
2.861
3.883
20
1.325
1.725
2.086
2.528
2.845
3.850
21
1.323
1.721
2.080
2.518
2.831
3.819
22
1.321
1.717
2.074
2.508
2.819
3.792
23
1.319
1.714
2.069
2.500
2.807
3.768
24
1.318
1.711
2.064
2.492
2.797
3.745
25
1.316
1.708
2.060
2.485
2.787
3.725
26
1.315
1.706
2.056
2.479
2.779
3.707
27
1.314
1.703
2.052
2.473
2.771
3.690
28
1.313
1.701
2.048
2.467
2.763
3.674
29
1.311
1.699
2.045
2.462
2.756
3.659
30
1.310
1.697
2.042
2.457
2.750
3.646
40
1.303
1.684
2.021
2.423
2.704
3.551
50
1.299
1.676
2.009
2.403
2.678
3.496
60
1.296
1.671
2.000
2.390
2.660
3.460
80
1.292
1.664
1.990
2.374
2.639
3.416
100
1.290
1.660
1.984
2.364
2.626
3.390
∞
1.282
1.645
1.960
2.326
2.576
3.291
C7 — Significant Is Not the Same as Useful
A significant test says the relationship is probably not exactly zero; how much it matters is a separate question, answered by the size of the effect (, the share of variation the line explains) and by what the prediction is for.
A social media company models daily engagement from posting frequency for 5,000 users. The result is highly significant (,,), and : the line explains 4% of the variation in engagement and leaves 96% unexplained. With 5,000 users, even a very weak relationship is easy to detect. Whether 4% is worth acting on depends on the decision, not on the p-value.
is an effect size: a measure of how big a relationship is that does not grow just because the sample does. L-REG-4 has an effect size for tables too.
Two traps with significance:
(1) “Significant means a good model.” With , even gives and , yet : the line explains almost nothing.
(2) ” means predictions are 85% accurate.” is the share of variation explained, not a hit rate for individual predictions.
Section 4: Worked Examples
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Example 1 — Fully Worked: What This Line Claims
Context: A city bike-share scheme records, for 60 days, the daily high temperature (, °C) and the number of rentals (). The temperatures ranged from 4 °C to 29 °C. The least-squares line is , and its residual plot shows no curve, no fan and no stray point.
Full solution with reasoning:
Slope. rentals per °C. “For each additional degree of daily high temperature, the predicted number of rentals increases by 18.6, on average.” Association only: warmer days go with more rentals; the data do not show that heat causes them.
Intercept. is the line’s value at 0 °C. The data run from 4 °C to 29 °C, so 0 °C lies outside them: 412 rentals is an extrapolation, not a baseline the data support.
A prediction at 20 °C. is inside : interpolation. rentals. The residual plot shows no pattern, so the line has evidence here.
A prediction at 38 °C.: extrapolation. The arithmetic gives , but on a 38 °C day riders may stay indoors; no day in the data was that hot. Report the number only with that warning, or not at all.
Example 2 — Partial Scaffold: Testing
Context: A random sample of students gives between weekly hours spent on practice problems and the score on a statistics quiz. The pairs are independent, and the residual plot shows no curve, no fan and no point pulling the line. Test against at .
Your turn: Check the licence, compute and , read from the t-table in C6, decide, and write the conclusion. Then compute and say what it adds.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Show Solution
Step 1: against ,.
Step 2: The pairs are independent and the residual plot shows no curve, no fan and no point pulling the line, so the test can be trusted.
Step 3:.
Step 4: The t-table gives at . Since , reject (two-sided ).
Step 5: There is evidence of a positive linear relationship between practice hours and quiz score among students like these.
What adds:, so the line explains about 38% of the variation in quiz scores. Significant, with a moderate effect: useful for describing the group, still loose for predicting one student.
Example 3 — Checkpoint: Settling the Quartet at
Context: Go back to your prediction in the Introduction. Every quartet line says at .
Your turn: For each set, decide whether you would defend at , and name the reason in one phrase. Use C2, C4 and C5.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Show Solution
Set I: defend it. The residuals show no pattern, and lies inside the data with points on both sides. (The observed point at is .)
Set II: do not. The residual plot is an arch: the data bend, and the straight line cuts across the bend. The point observed at is , well above 8.0. A curved model is needed.
Set III: not as it stands. One outlier at tilts an otherwise exact line. Without it, the line is and gives at , which is exactly the observed value there. Check the outlier before choosing.
Set IV: do not. Ten points sit at and one at ; nothing lies near , and the slope exists only because of the single far point.
Only set I backs the prediction. The four sets share , so could never have told them apart.
Example 4 — Find the Errors
A road-safety researcher fits to independently sampled drivers, where is years of driving experience (observed from 1 to 25 years) and is a hazard-perception score out of 100. The sample gives ; the residual plot shows no curve and no fan, and the test of gives ,.
The researcher reports:
Researcher’s analysis:
“The regression proves that driving experience causes better hazard perception."
"Since , the model is significant, so we can trust every prediction it makes."
"A driver with 70 years of experience will score . This is a reliable prediction."
"One driver with 2 years of experience scored only 31, which lowered , so we deleted that driver before reporting.”
Identify the four errors in the quoted claims.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Show Solution
Error 1 — Causation. An observational regression shows association. More experienced drivers differ in many ways (age, training, the kind of driving they do). Say “experience is associated with higher scores”.
Error 2 — Significance is not trust in every prediction. is evidence that . It says nothing about predictions outside the data or about how close any one prediction will be.
Error 3 — Extrapolation. is far beyond the observed 1–25 years, and the result, 103.0, is above the highest possible score of 100. The line has no data there.
Error 4 — Deleting an outlier to raise . The driver is a real observation. Check that the score was recorded correctly; if it was, keep it and report the fit with and without it.
Note: , so inside the 1–25 year range the line explains about two-thirds of the variation. None of that carries to 70 years.
Section 5: Guided Practice
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Problem 1 — What the Line Claims (C1)
Build the Conclusion
Complete what the line claims
Choose a phrase for each blank so that the sentence states the correct conclusion.
Problem 2 — A Prediction, Step by Step (C1)
Problem 3 — Read the Residual Plot (C2 + C3 + C4)
Problem 4 — Leverage or Influence? (C4 + C5)
Problem 5 — Test (C6)
Work through the steps, or answer the test statistic on your own if you feel ready.
Section 6: Independent Practice
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Each problem starts with your own attempt. Numbers are untidy from here on: correlations have two decimals and either sign, and sample sizes vary.
Problem 1 — A Forecast Request (C1)
Problem 2 — Test, Then Size (C6 + C7)
Then compare two studies. Both results are significant.
Structural Difference
Both significant. Which relationship helps you predict?
Identify what key word or condition separates Case A from Case B, then select the method for each.
Scenario A
Study A · 25 students. Weekly sleep hours and quiz score: r = 0.45, t ≈ 2.42 with df = 23 (t* = 2.069). Significant. r² ≈ 0.20.
Scenario B
Study B · 6,200 app users. Notifications per day and minutes of use: r = 0.05, t ≈ 3.94. Significant, with the smaller p-value. r² ≈ 0.0025.
Problem 3 — Find the Error (C1 + C2 + C3 + C5 + C7)
Problem 4 — What Kind of Unusual Point? (C4 + C5)
Sort into groups
What kind of unusual point is it?
A line with slope 1.88 was fitted to 20 points with x from 10 to 40 (x̄ = 25); every residual is within ±3. Each card adds one new point (its residual is measured from that original line) and refits. Sort each card.
Drag an item into its group, or select it and then choose the group. On a keyboard: select an item and press the group’s number (1–3); 0 sends it back to Unsorted. Each item also has its own “move” buttons.
Unsorted Drop here
Every item is sorted.
Problem 5 — Which Point Is Holding the Line? (C5)
Answer on the Graph
Which point is holding the line?
Twelve stores: weekly advertising spend (x, thousand dollars) and weekly sales (y, thousand dollars). The least-squares line through all twelve has slope about 1.09. Select the one store whose removal would change the slope the most.
Select the point or points that answer the question. Select a point again to clear it. Keyboard: arrow keys move through the figure; Enter or Space selects.
No point selected yet.
Problem 6 — From Summaries to a Decision (C1 + C6)
Mixed Review — Retrieval from Earlier Lessons
Section 7: Mastery Check
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Question 1 — Feynman Test
Explain to a classmate why and cannot tell you whether a straight line is the right model, and what you would look at instead. Use Anscombe’s quartet as your example, and add one sentence on why a significant correlation can still be useless for prediction.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Compare with a model answer
measures how tightly the points hug the best straight line, and is the share of the variation that line explains. Neither says whether a straight line is the right shape, whether its misses are even, or whether one point is steering it. Anscombe’s four sets all have and the same line, yet one is a curve, one is an exact line tilted by an outlier, and one gets its whole slope from a single far point. The residual plot shows each problem: a bend, a fan, or one residual far from the rest. And a significant test only says the correlation is probably not zero: with thousands of cases, a line that explains 1% of the variation is significant and still nearly useless for predicting one case.
Question 2 — What the Line Claims
Build the Conclusion
Complete the claims
Choose a phrase for each blank so that the sentence states the correct conclusion.
Question 3 — What Should the Team Do?
Question 4 — Which Runner Is Holding the Line?
Answer on the Graph
Which runner is holding the line?
Twelve club runners: weekly hours of training (x) and 5 km race time (y, minutes). The least-squares line through all twelve has slope about −0.38. Select the one runner whose removal would change the slope the most.
Select the point or points that answer the question. Select a point again to clear it. Keyboard: arrow keys move through the figure; Enter or Space selects.
No point selected yet.
Question 5 — The Test Statistic
Question 6 — Significant, but How Big?
Build the Conclusion
Complete the report
Choose a phrase for each blank so that the sentence states the correct conclusion.
Self-Assessment
Need more practiceReady to transfer
If you’re below 60%: run Figure 3 again with “Another line-up” until you can find the fan quickly, then replay Figure 4 and show each point’s line. Those two skills, seeing a real pattern and seeing what moves a line, carry the rest of the lesson.
Section 8: Boss Fight
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Context: Path A puts you on a forecast desk where requests arrive with a residual plot attached and you decide what to send back. Path B hands you four branch reports with identical statistics and asks which one can be trusted. Each starts with your own complete attempt; open the steps only if you need to find where it went wrong.
📈 Path A: The Forecast Desk
Compute the forecast, read the plot, and decide whether it goes out.
🔍 Path B: The Quartet Audit
Four branches, one summary. Which data can carry a forecast?
📈 Path A: The Forecast Desk
Request 1.
Request 2: the memo. A second request arrives; complete the memo that goes back.
Build the Conclusion
Write the memo
Choose a phrase for each blank so that the sentence states the correct conclusion.
🔍 Path B: The Quartet Audit
Then, on paper: go back to the prediction you made in the Introduction. In two or three sentences, say which quartet set you would now trust at and why the shared could never have told you.
Compare with a model answer
Only set I: its residuals show no pattern and it has data on both sides of . Set II is a curve, set III is an exact line tilted by one outlier, and set IV gets its whole slope from one far point, with nothing near . They share because measures how tightly points hug the best straight line, not whether that line is the right model; only the plots tell them apart.
Section 9: Challenge Problems
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Optional stretch: these go beyond the lesson outcomes. Commit to each answer before reading its feedback.
Problem 1 — How Much Is One Point Holding Up the Slope?
Problem 2 — Same Correlation, Four Sample Sizes
Complete the table
Same r, four sample sizes
Fill in every empty cell, then check the whole table once. The cells depend on one another.
Build the Conclusion
Evidence or size?
Choose a phrase for each blank so that the sentence states the correct conclusion.
Problem 3 — Delete the Outlier?
Then write the note for a second analysis of the same kind.
Build the Conclusion
Write the analyst's note
Choose a phrase for each blank so that the sentence states the correct conclusion.
Section 10: Solutions Reference
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
If you’re stuck: reread the matching concept, replay its figure, then open the worked solutions for Section 4 and the reference notes for the generated practice.