L-REG-1: Solutions — Correlation Analysis

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 — Read Direction and Strength from a Plot (C1 + C3)

This problem draws a new plot each time. Read it in two moves:

1. Direction: follow the points from left to right. Rising means positive; falling means negative.
2. Strength: judge how tightly the points follow a straight path, then name the band:

Magnitude	Course label
	Weak
	Moderate
	Strong

The feedback shows the exact of the plotted points. Expect to read moderate clouds as weaker than they are at first: that is the common bias Figure 2 in the lesson showed.

Problem 2 — Your First Computation of (C2)

The first two steps (the five column sums and the numerator) are worked. For the remaining steps:

For example, with and : the sums are , , , and .

Common slips: leaving out the in a bracket, treating as , and forgetting the square root in the denominator.

Problem 3 — Computing and Interpreting (C5 + C6)

Square the displayed and round only after squaring. If :

In context, the fitted line using accounts for about 45% of the observed variability in . The negative direction belongs to ; is never negative.

Concept Check — When (C4)

The correct conclusion is: there is little or no apparent linear association, but the scatterplot shows a non-linear relationship. A suitably symmetric U-shaped pattern has positive and negative linear contributions that cancel, giving .

Problem 4 — Why Do These Two Measures Move Together? (C7)

Six cards are drawn from a bank of eight. Sort each by its design:

• A third variable could drive both: firefighters and fire damage (fire size), chocolate and Nobel laureates across countries (national wealth; and each point is a country, not a person), shoe size and reading score in children (age).
• The second measure could drive the first: police officers and crime rate (districts with more crime get more officers), painkillers and headache days (more headaches lead to more pills).
• Random assignment supports a cause: seedlings randomly given different amounts of fertiliser, volunteers randomly given different amounts of coffee, classes randomly given different numbers of quizzes.

The size of never decides which bin a card belongs in. The design does.

Problem 5 — From Back to (C5 + C6)

Undo the square with a square root, then take the sign from the plot. For with points that fall from left to right:

Squaring removed the sign, so it must come from the direction of the trend.

Problem 6 — What Does Ignore? (C9)

• Converting height to inches and recomputing : stays . It is built from z-scores, so it has no units.
• Converting height to inches and recomputing the slope: the slope changes. A slope is a rate in units of per unit of ; one inch is 2.54 cm, so the slope per inch is 2.54 times the slope per centimetre.

Same , different slope: is not a slope.

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 — Compute from Raw Data (C1 + C2 + C8)

Each dataset is new. Build the table row by row: , and for every pair, then the five column sums. For six students with sleep hours and quiz scores :

41.5	423	2961.0	293.75	30087

Then look at the plot. About one dataset in five has a curve or one far point; the feedback says when, and then is not a fair summary even though the arithmetic is right.

Problem 2 — Classify and Compute (C3 + C5 + C6)

Direction comes from the sign of , strength from and the bands above, and from squaring the displayed . For : positive, moderate, and .

Problem 3 — Diagnose Correlation Claims (C6 + C7 + C10)

1. read as 70%: the researcher used instead of . , so the line accounts for 49% of the variability.
2. Shoe size and reading: age is a confounder; foot size and reading ability both grow as children get older.
3. read as 63%: square it. , about 40%. The sign is irrelevant to .
4. Two reading studies: across ages 6 to 12 the trend has plenty of room to show, so is large; among 9-year-olds alone the range of age is narrow, so ordinary differences between children dominate and shrinks. The relationship need not differ: that is restricted range.

Problem 4 — Which Points Pull Down? (C2 + C8)

The means are hours and . A point’s product is negative when it sits above one mean and below the other: upper left or lower right of the dashed cross. Five students qualify: , , , and . Points low on both variables, such as , have a positive product and push up. Being below the fitted line is a different idea: it says nothing about the sign of the product.

Problem 5 — Interpret Explained Variability Responsibly (C5 + C6 + C7)

Steps: square the displayed and interpret as the share of observed variability in accounted for by the fitted line, never as a probability or a causal share.

The report: read four things from the study. The strength from ; the direction from the sign; the share from (not , and not ); and the causal verdict from the design:

• if the amounts were only recorded (a survey, records), the data do not show a causal effect;
• if the amounts were randomly assigned (an experiment), the association supports a causal effect within the range studied.

“A correlation can never show a cause” is too strong: random assignment is exactly what licenses the causal claim.

Problem 6 — A Responsible Correlation Analysis (C2 + C8 + C1)

1. Compute from the shown data, as in Problem 1.
2. Judge the conditions from the plot: a roughly straight band with no dominant point (met), a curve (not met), or one point far from the rest (not met).
3. Let the report follow the plot: report and as an association when the conditions hold; describe the curve itself when it bends; report with and without a far point, and investigate it, when one point dominates. Never delete a point just because it lowers .

Mixed Review — Retrieval from Earlier Lessons

Review Problem 1 — A z-Score in Context (L-DS-5)

For , and : , so the score is 2.3 standard deviations above the mean. Typical slips: reporting the raw difference , reversing the subtraction, or dividing by the variance.

Review Problem 2 — The Conclusion of a Test (L-INF-5)

Compare with . If , reject : the data are convincing evidence for the alternative. If not, fail to reject : the data are not convincing evidence for the alternative. A test never “accepts” or proves .

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

A strong answer covers four points:

1. What says: the points fall from upper left to lower right in a fairly tight straight band, so is a strong negative linear association.
2. What says: , so a fitted line using screen time accounts for about 64% of the observed variability in sleep-quality scores.
3. What is left: the other 36% is not accounted for by the line; it is not automatically random noise.
4. Why not cause: nobody assigned the screen time. Stress, workload or a phone in the bedroom could drive both, or poor sleep could lead to more screen time at night.

Question 2 — Novel Application

Compute from the shown pairs with the computational formula, keep the full calculator value, and square it for ; round only at the end. The report that follows is solved as in Independent Practice Problem 5: strength from , direction from the sign, share from , and the causal verdict from the design.

Question 3 — Error Analysis

The analysis contains one of four named errors; find the first step that is wrong:

• Bracket without (step 2): the x-bracket is . Without the it turns negative, which is impossible for real data.
• No square root (step 3): the denominator is . Dividing by the product itself gives a value near 0.
• Squared the rounded (step 4): square the full value of , then round. Squaring the two-decimal can shift by 0.01.
• read as a slope (step 4): has no units, so it cannot be “units of per unit of ”. That rate is the slope (L-REG-2).

Question 4 — What Changes ?

• Units changed, or and swapped: is the same.
• Only part of the range kept (accepted applicants, mild-winter cities): shrinks because the range of is narrower: restricted range.
• “Each extra unit of adds units of ”: wrong, because is not a slope.

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.

Path A — The Chocolate Desk

The number. Compute for all ten fictional countries from the table.

One country. One country sits far from the rest. Recompute without it. Either falls a long way (most of the association rested on that one country) or rises (the country sat far above the trend and hid a tighter one, as Sweden did in Messerli’s data: with it, without). Report both values.

Not a slope. “Every extra kilogram is worth laureates” treats as a slope. has no units.

The second file (answer on the graph). Select the country far from the centre of the cloud and away from the trend the others follow; the country with the most laureates usually sits on the trend and barely moves .

The note to the editor.

1. The strength comes from and the bands.
2. Each point is a whole country, so the data say nothing about whether the people who eat more chocolate are the ones who win prizes (an ecological correlation).
3. National wealth, which pays for both chocolate and research, is a plausible third variable.
4. is a unit-free measure of how closely the points follow a line, not laureates per kilogram.
5. So the column may report the association but must not say that eating chocolate wins prizes.

The hook, settled. Should a country that wants more Nobel prizes eat more chocolate? Not on this evidence: the correlation is real and correctly computed, but countries are not people, wealth is a plausible common cause, one country moved on its own, and is not a rate. Only a design in which chocolate is assigned could answer the causal question.

Path B — The Entrance Test Review

The number. Compute for the accepted students: it is weak.

The problem. Every student in the data scored at least the cut-off, so the scores cover only the top of the range. Restricting the range of shrinks even when the relationship is unchanged.

The fuller evidence. The open-intake year, with nobody screened out, shows the relationship across the full range of scores:

• if that is moderate or larger, the test did track first-year averages, and the small today is mostly a range effect: do not drop the test on this evidence;
• if that is also weak, the test predicted little even across the full range, so there is a real reason to review it.

The memo. The same reasoning: narrow range, shrinks, judge the test by the full-range value. Whether a test is fair to particular groups of applicants is a separate question that a correlation cannot answer.

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.

Challenge 1 — and Unexplained Variance (C5 + C6)

With :

(a) Partially right: the fitted line does not account for 75% of the observed variability, but that share is not necessarily random noise.

(b) It can come from other variables, non-linear structure, measurement variation, or individual differences.

(c) Model answer: “Unexplained” means not accounted for by this one straight line using . That share can come from other variables, curvature the line misses, measurement, or genuine individual variation. It does not mean the 75% has no cause, and it does not mean the line is useless.

Challenge 2 — Transfer Beyond a Straight Line (C1 + C4 + C8)

The plot is drawn afresh each time, in one of four shapes. Match the action to the plot:

• A roughly straight band with no isolated point: report and with the plot, then inspect residuals before predicting.
• A U-shape: do not use alone; model the curve or compare meaningful ranges.
• One influential point: investigate it and report sensitivity with and without it; do not delete it without a documented reason.
• Two separated clusters: stratify by the grouping variable, or fit a model that represents the groups.

The coefficient never overrules the visible structure.

Challenge 3 — Anscombe’s Quartet (C1 + C4 + C8)

(a) Dataset I is the only one for which Pearson is a useful primary summary:

• Dataset I is an approximately linear cloud.
• Dataset II is a smooth curve.
• Dataset III has one extreme point off a nearly perfect line; removing it raises to about .
• Dataset IV is dominated by one high-leverage point at .

(b) Nearly identical correlations (, , and ) can hide completely different patterns. Plot the data before relying on . L-REG-3 opens with these four plots.