The chapter opens by fixing a convention. Linear regression for two variables is based on a linear equation with one independent variable, written y = a + bx — with the intercept first and the slope second, the reverse of the algebra form y = mx + b. Every formula in the rest of the chapter follows that convention, so reading a as the intercept and b as the slope from their positions rather than from their letters is what this section exists to establish. The graph of such an equation is a straight line, and any line that is not vertical can be described this way: b greater than zero slopes upward, b equal to zero is horizontal, and b less than zero slopes downward. Beyond the algebra, the section asks for something new — an interpretation of both numbers in the units of the problem, in complete sentences, which is exactly what sections 12.3 and 12.5 will demand of a fitted regression line.
Subject: Statistics · 65 slides · symbolic lesson
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Title
Statistics · Chapter 12 — Linear Regression and Correlation
Linear Equations
Objectives
Five outcomes, and the last one is the one the rest of the chapter builds on.
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, pp. 617-620 — the section these objectives are drawn from
Warm-up
Every algebra course writes a line as y equals m x plus b.
Discussion prompt
A tutor charges a fee of 25 dollars per session plus 15 dollars for each hour. Write the total earnings as an equation, then say what each number means.
Hint: One amount is charged once; the other is charged per hour.
Answer:
In algebra you would probably write y = 15x + 25 — slope first, intercept last. That is correct, and this chapter will write the same line as y = 25 + 15x instead.
\[ y = a + bx: \qquad a = 25, \quad b = 15 \]
The reordering is not cosmetic. Every formula in chapter 12 names the intercept a and the slope b, so a reader who reaches for the algebra habit will read 15 as an intercept and 25 as a slope. Reading the POSITIONS rather than the letters is the whole point of this section.
The second half of the question is the genuinely new part. The 25 is what the tutor earns before working any hour, and the 15 is what each additional hour adds — and saying so in sentences is exactly what sections 12.3 and 12.5 will ask about a fitted line.
Concept
Linear regression for two variables is based on a linear equation with one independent variable, of the form y equals a plus b x, where a and b are constant numbers. The variable x is the independent variable and y is the dependent variable, and typically you choose a value to substitute for x and then solve for y.
y = a + bx — The statistics convention. Here a is the y-intercept and b is the slope — the reverse of the positions the letters occupy in the algebra form.
\[ y = a + bx, \qquad a = \text{intercept}, \quad b = \text{slope} \]
The direction of the substitution matters too. You choose x and solve for y, never the other way around — which is what makes x the independent variable and y the dependent one. That asymmetry will become important in section 12.3, where the regression line is built to predict y from x and is NOT the same line that would predict x from y.
Figure (svg): A card contrasting the algebra form of a line with the statistics form
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, p. 617
Section
Section 1
Concept
For the linear equation y equals a plus b x, b is the slope and a is the y-intercept. From algebra, recall that the slope is a number that describes the steepness of a line, and the y-intercept is the y coordinate of the point where the line crosses the y-axis.
reading the positions — The constant term is a, the intercept; the coefficient of x is b, the slope. Their letters are the reverse of the algebra form's, so the positions are what to trust.
\[ y = \underbrace{a}_{\text{intercept}} + \underbrace{b}_{\text{slope}}\,x \]
It is worth being explicit about why statistics chose this order. Writing the constant first matches how the formulas are built in section 12.3, where the intercept is computed FROM the slope — the slope comes out of the data first, and the intercept is then whatever makes the line pass through the point of means. Ordering the equation that way keeps the derivation and the notation aligned.
Figure (svg): A card contrasting the algebra form of a line with the statistics form
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, pp. 617-619 — the form, and the slope and intercept
Picture it
The same line, written by algebra and by statistics.
Figure (svg): A card contrasting the algebra form of a line with the statistics form
The letter b carries opposite meanings in the two forms, which is why this is worth twenty seconds of deliberate attention now rather than a confused half hour in section 12.3.
Worked example
A small business charges 32 dollars per hour plus a one-time charge of 31.50 dollars. Find the equation expressing total cost in terms of hours.
\[ \$32 \text{ per hour}, \; \$31.50 \text{ once} \]
Name x
Why: What is chosen.
Name y
Why: What follows.
The fixed part
Why: Charged once.
\[ 31.50 \]
The per-hour part
Why: Times the hours.
\[ 32 x \]
Figure (svg): The solution to Worked example Example 12.3, the tax returns shown as a ladder of expressions, one row per legal move
\[ y = 31.50 + 32x \]
Verify: confirm the equation by testing a value
Why: Two hours should cost the fixed 31.50 plus two hours at 32, which is 31.50 plus 64, or 95.50 — and substituting x = 2 gives exactly that. Checking one convenient value catches a swapped intercept and slope immediately, since the wrong equation y = 32 + 31.50x would give 95 at x = 2 and diverge quickly after.
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, p. 619
Faded example
The equation y = 31.50 + 32x.
Fill in the blanks
a = 31.50, \qquad b = 32
Why: The one-time charge is the intercept and the hourly rate is the slope, which is the pattern every cost situation in this section follows.
Worked example
For Example 12.4's y = 25 + 15x.
\[ y = 25 + 15x \]
Find the constant term
Why: No x attached.
\[ 25 \]
That is a
Why: The intercept.
\[ a = 25 \]
Find the coefficient of x
Why: Multiplying x.
\[ 15 \]
That is b
Why: The slope.
\[ b = 15 \]
Figure (svg): The solution to Worked example reading a and b off an equation shown as a ladder of expressions, one row per legal move
\[ a = 25, \qquad b = 15 \]
Verify: confirm against the algebra reading, and note the clash
Why: Written as y = 15x + 25 the same line has m = 15 and b = 25 — so the letter b names 15 in one convention and 25 in the other. Only the positions agree: the constant term is always the intercept and the coefficient of x is always the slope, whatever they are called.
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, pp. 619-620
Trap
\[ y = 25 + 15x \;\Rightarrow\; b = 25 \]
Reach for the algebra position of b
Why: There it was the constant.
\[ \text{but the constant here is } a \]
Every formula in the chapter treats b as the slope, so this reversal corrupts everything downstream.
\[ a = 25 \text{ (the constant)}, \qquad b = 15 \text{ (the coefficient of } x\text{)} \]
Read the positions, not the letters
Why: Constant term is a; coefficient of x is b.
The consequences arrive quickly. Section 12.3's interpretation of the slope, section 12.4's test of significance and section 12.5's predictions all take b as the slope. Getting it backwards once here produces answers that look plausible and are wrong throughout.
Two truths and a lie
All three concern the form.
Eliminate the wrong options
Two are true. Knock those out and keep the false one.
Survives elimination: B
Why: The survivor is false and is the section's central caution. In y = a + bx the letter b is the SLOPE. The algebra form y = mx + b uses b for the intercept, which is exactly the clash to watch for.
Prediction
Commit before reasoning.
Predict first
Which lines cannot be written as y = a + bx?
Correct: Vertical lines.
Why: A vertical line has the same x for every y, so no choice of a and b produces it — the book says any line that is NOT vertical can be described by this equation. Horizontal lines are fine, with b equal to zero, and negative slopes are fine too.
Sorting
Each is a quantity from a described situation.
Sort into buckets
Sort by which role it plays.
The word per is the reliable signal: a quantity stated per something is a slope, and a quantity stated as a fixed or one-time amount is an intercept.
Section
Section 2
Concept
The graph of a linear equation of the form y equals a plus b x is a straight line, and any line that is not vertical can be described by this equation. If b is greater than zero the line slopes upward to the right; if b equals zero the line is horizontal; if b is less than zero it slopes downward to the right.
what each constant does — The slope b sets the direction and steepness; the intercept a slides the whole line up or down without changing its tilt.
\[ b > 0 \;\nearrow\; \qquad b = 0 \;\rightarrow\; \qquad b < 0 \;\searrow\; \]
The horizontal case is worth flagging early, because section 12.2 returns to it with a warning. A horizontal line fits its points perfectly in one sense, yet it says y does not depend on x at all — so a perfect-looking fit there actually indicates NO relationship. That is the one exception to reading closeness-to-a-line as strength.
Figure (svg): Three small graphs showing a line sloping upward, a horizontal line, and a line sloping downward
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, pp. 618-619 — the graph is a straight line; the three cases
Picture it
The book's Figure 12.4.
Figure (svg): Three small graphs showing a line sloping upward, a horizontal line, and a line sloping downward
Only the sign of b distinguishes them. Changing a would shift each of these three pictures vertically while leaving its shape and its case entirely unchanged.
Worked example
Graph y = -1 + 2x.
\[ y = -1 + 2x \]
Read a
Why: The constant.
\[ -1 \]
Plot the intercept
Why: Where x is zero.
\[ (0, -1) \]
Read b
Why: The coefficient.
\[ 2 \]
Step from the intercept
Why: Right one, up two.
\[ (1, 1) \]
Figure (svg): The solution to Worked example Example 12.2, graphing a line shown as a ladder of expressions, one row per legal move
\[ \text{through } (0,-1) \text{ with slope } 2 \]
Verify: confirm with a second point computed from the equation
Why: Substituting x = 3 gives -1 plus 6, which is 5, so (3, 5) should lie on the line — and stepping right three from the intercept and up six lands exactly there. Two points determine a line, so checking a third computed point catches an intercept-slope swap: the wrong reading would put the line through (0, 2) with slope -1 and miss (3, 5) entirely.
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, p. 618
Matching
Match each equation to its direction.
Match the pairs
Why: The fourth is Try It 12.1's equation, and it is linear despite both constants being negative — a is -0.125 and b is -3.5, which is a perfectly ordinary line sloping downward.
Worked example
Four equations, sorted by their pictures.
\[ \text{four lines} \]
y = 25 + 15x
Why: b is 15.
y = 100 - 4x
Why: b is -4.
y = 7
Why: b is 0.
y = -1 + 2x
Why: b is 2.
Figure (svg): The solution to Worked example predicting the shape without graphing shown as a ladder of expressions, one row per legal move
\[ \text{sign}(b) \text{ decides the case} \]
Verify: confirm the intercepts play no part in this
Why: The intercepts here are 25, 100, 7 and -1, spanning positive and negative values, yet they had no bearing on any answer. Changing y = 100 - 4x to y = 5 - 4x would move the line far down the page while leaving it sloping downward at exactly the same steepness — which is what it means to say a controls position and b controls direction.
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, p. 619
Error analysis
Which are correct?
Annotate
On: \( \begin{aligned} &(1)\; \text{the graph is always a straight line} \\ &(2)\; \text{a larger } a \text{ makes the line steeper} \\ &(3)\; \text{a negative } b \text{ slopes the line downward} \\ &(4)\; \text{vertical lines have a very large } b \end{aligned} \)
Errors (2) and (4) both come from conflating the two constants' jobs. Position and direction are controlled separately, which is precisely why two numbers are needed to specify a line.
Faded example
The line y = 12 - 3x.
Fill in the blanks
a = 12, \; b = -3 \;\Rightarrow\; \textdownward 12 \text___ ___
Why: The intercept places the line and the negative slope tilts it down. Each step right lowers y by 3.
Two truths and a lie
All three concern the two constants.
Eliminate the wrong options
Two are true. Knock those out and keep the false one.
Survives elimination: B
Why: The survivor is false. A horizontal line is y = a, which is y = a + 0x — perfectly expressible. Only VERTICAL lines fall outside the form.
Prediction
Commit before reasoning.
Predict first
Why will section 12.2 warn about points falling exactly on a horizontal line?
Correct: It means y does not depend on x at all.
Why: Everywhere else, points hugging a line indicates a strong relationship. A horizontal line is the exception: it says y takes the same value whatever x does, which is the definition of no relationship — so the usual reading of a tight fit gets exactly the wrong answer there.
Section
Section 3
Concept
The variable x is the independent variable and y is the dependent variable. Typically you choose a value to substitute for the independent variable and then solve for the dependent variable.
independent and dependent — The independent variable is the one whose value is set or observed first; the dependent variable is computed from it. Which is which comes from the situation, not from the algebra.
\[ \text{choose } x \;\longrightarrow\; \text{solve for } y \]
The assignment is a modelling decision rather than a mathematical one. Nothing stops you from solving y = 25 + 15x for x, but doing so would answer a different question — how many hours produce a given payment, rather than what a given number of hours pays. Section 12.3 makes this consequential: the line that best predicts y from x is not the line that best predicts x from y.
Figure (svg): A line through the point zero comma twenty-five rising by fifteen dollars for each hour tutored
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, pp. 617-620 — the variables, and Example 12.4's identification
Picture it
Svetlana's earnings against hours tutored.
Figure (svg): A line through the point zero comma twenty-five rising by fifteen dollars for each hour tutored
Hours are chosen and dollars follow, so hours are independent. The orange point is the intercept — what she earns having tutored zero hours — and each step right adds fifteen dollars.
Worked example
Svetlana charges a one-time fee of 25 dollars plus 15 dollars per hour, so y = 25 + 15x. Identify the variables, the intercept and the slope, and interpret them in complete sentences.
\[ y = 25 + 15x \]
The independent variable
Why: What she chooses to work.
The dependent variable
Why: What results.
The intercept
Why: a = 25.
The slope
Why: b = 15.
Figure (svg): The solution to Worked example Example 12.4, naming and interpreting shown as a ladder of expressions, one row per legal move
\[ a = 25 \text{ at } x = 0; \qquad b = 15 \text{ per hour} \]
Verify: confirm the intercept describes a real situation here
Why: The intercept is the value at x = 0, which means a session in which no hours are tutored — and Svetlana would still collect her fee of 25 dollars, so the number describes something that genuinely happens. That is not always true: an intercept can fall outside the range where the model makes sense, which is a caution section 12.5 develops under the name extrapolation.
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, p. 620
Sorting
Each pairs two quantities.
Sort into buckets
Sort by whether the first-named quantity is the independent variable.
Items (b) and (d) are reversed on purpose: the third exam comes before the final, and age drives vocabulary rather than the other way round. The order in which quantities are named carries no information.
Worked example
Emma's Extreme Sports pays instructors 50 dollars per class plus 20 dollars per student in the class.
\[ \$50 \text{ per class}, \; \$20 \text{ per student} \]
Name x
Why: What varies per class.
Name y
Why: What Emma pays.
The fixed part
Why: Per class, not per student.
\[ a = 50 \]
The per-unit part
Why: Multiplies students.
\[ b = 20 \]
Figure (svg): The solution to Worked example Try It 12.3, a per-student rate shown as a ladder of expressions, one row per legal move
\[ y = 50 + 20x \]
Verify: confirm the units of the slope
Why: The slope's units are always y-units per x-unit, so here they are dollars per student rather than dollars per hour — even though the situation looks like the tutoring one. Reading the units off the two variables is what prevents describing this slope as a rate per hour, and it becomes essential in section 12.3 where x and y carry unfamiliar units.
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, p. 619
Trap
\[ \text{cost is known, so let } x = \text{cost} \]
Choose x to be whatever is given in the question
Why: It looks like the known quantity.
\[ \text{but the situation sets the roles} \]
Cost DEPENDS on hours, so cost is y no matter which one a particular question happens to supply.
\[ x = \text{hours}, \quad y = \text{cost}, \quad \text{then solve } 95.50 = 31.50 + 32x \]
Assign the roles from the situation, then solve for whichever is unknown
Why: The equation can be used in either direction once written.
Being asked how many hours produce a given cost is a perfectly good question; it is answered by solving the same equation for x, not by rewriting the model with the roles swapped. Keeping the roles fixed matters in section 12.3, where the fitted line depends on which variable was treated as dependent.
Faded example
For Ethan's appliance repair, y = 25 + 20x with x in hours.
Fill in the blanks
\texthour 20 \text___ ___ \text___
Why: The slope's units are dollars per hour, taken directly from the units of y over the units of x. Naming both units is what makes the sentence say something.
Two truths and a lie
All three concern the variables.
Eliminate the wrong options
Two are true. Knock those out and keep the false one.
Survives elimination: B
Why: The survivor is false. A question may give either variable and ask for the other; the equation can be solved in both directions. What fixes the roles is which quantity depends on which in the situation being modelled.
Explain it
A classmate reads y = 25 + 15x and writes down that the slope is 25 because b is the second letter and comes last.
Discussion prompt
In two sentences or fewer, correct them.
Hint: Ask which position the letter occupies in each convention.
Answer:
In the statistics form the constant comes first and is called a, so 25 is the intercept and 15 — the coefficient of x — is the slope b.
The algebra form y = mx + b does put b last, which is exactly the clash: trust the positions, since the constant term is always the intercept and the coefficient of x is always the slope.
Section
Section 4
Concept
The book asks for the intercept and slope to be interpreted using complete sentences. The intercept is the value of y when x is zero; the slope is how much y changes for a one-unit increase in x.
an interpretation — A sentence in the problem's own units, naming what the number is rather than restating the arithmetic. At the start of the session she charges a one-time fee of 25 dollars is an interpretation; a equals 25 is not.
\[ a = y \text{ at } x = 0; \qquad b = \Delta y \text{ per unit } \Delta x \]
This looks like a small exercise and is the section's most durable content. Section 12.3 gives exactly the same instruction about a fitted regression line — for a one-point increase in the score on the third exam, the final exam score increases by 4.83 points, on average — and the only new word in that sentence is on average. Everything else is the habit built here.
Figure (svg): How to interpret a slope and intercept in context
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, p. 620 — Example 12.4's interpretations
Picture it
Five steps, ending in two sentences.
Figure (svg): How to interpret a slope and intercept in context
Naming the units before writing the sentence is what keeps it honest. A slope reported without units is just a number, and a number alone cannot be checked against the situation for plausibility.
Worked example
Try It 12.4: Ethan charges 25 dollars per visit plus 20 dollars per hour, so y = 25 + 20x.
\[ y = 25 + 20x \]
Units of x
Why: What varies.
Units of y
Why: What results.
The intercept sentence
Why: At x = 0.
\[ 25\text{ dollars to arrive} \]
The slope sentence
Why: Per one more hour.
\[ 20\text{ dollars more} \]
Figure (svg): The solution to Worked example interpreting both constants shown as a ladder of expressions, one row per legal move
\[ a = 25 \text{ dollars}; \qquad b = 20 \text{ dollars per hour} \]
Verify: confirm the sentences by testing them against a value
Why: Three hours should give 25 plus three lots of 20, which is 85 — and the interpretation predicts exactly that: 25 dollars to arrive, then 20 dollars an hour for three hours. If a sentence cannot be used to reconstruct a value this way, it has restated the arithmetic rather than interpreted it.
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, p. 620
Two truths and a lie
All three concern interpretation.
Eliminate the wrong options
Two are true. Knock those out and keep the false one.
Survives elimination: B
Why: The survivor is false. When x = 0 falls outside the data or describes an impossible case — a height of zero, a year of zero — the intercept places the line correctly without describing anything real.
Worked example
A line fitted to adult heights and weights has intercept -220 pounds.
\[ a = -220 \]
Read it literally
Why: y at x = 0.
Ask if that occurs
Why: No adult has it.
Ask if it is an error
Why: No.
State the limit
Why: Position, not meaning.
Figure (svg): The solution to Worked example when the intercept is not meaningful shown as a ladder of expressions, one row per legal move
\[ a \text{ is a placement, not always a quantity} \]
Verify: confirm this does not undermine the model
Why: The line can predict weight accurately across the range of adult heights while its intercept describes an impossible case, because the intercept sits far outside the data. Svetlana's intercept was interpretable precisely because zero hours is a real possibility; that is the test to apply, and section 12.5 formalises it as the distinction between interpolation and extrapolation.
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, pp. 619-620
Error analysis
Which are genuine interpretations?
Annotate
On: \( \begin{aligned} &(1)\; \text{a equals } 25 \text{ and b equals } 15 \\ &(2)\; \text{each hour tutored adds } \$15 \\ &(3)\; \text{she earns } \$15 \text{ in total} \\ &(4)\; \text{with no hours tutored she still collects } \$25 \end{aligned} \)
Error (3) is the common one, and it becomes more tempting in section 12.3 where the slope is 4.83 exam points. A slope is always a rate of change, never a level.
Estimation
The tax business charges y = 31.50 + 32x.
Predict first
Roughly what does a job taking 4 hours cost?
Correct: About 160 dollars.
Why: Four hours at 32 dollars is 128, dollars plus the one-time 31.50 dollars, giving 159.50 dollars. Forgetting the fixed charge would give 128 dollars — which is the second option and exactly the error of treating the intercept as though it were not there.
Faded example
For y = 50 + 20x, Emma's cost with x students.
Fill in the blanks
\text0 50 \text___ ___ \text___
Why: The 50 dollars is a per-class fee, so it is owed even for a class nobody signs up for — which makes this intercept describe something that really can happen.
Prediction
Commit before reasoning.
Predict first
Section 12.3 interprets a fitted slope as: the final exam score increases by 4.83 points for a one-point increase in the third exam, ON AVERAGE. What do those two words add?
Correct: That it is a trend, not a guarantee for each student.
Why: The lines in this section are exact — Svetlana really does earn exactly 15 dollars more per hour. A fitted line only describes the average behaviour of points scattered around it, so individual students depart from it. Those two words are the entire difference between an exact relationship and a statistical one.
Section
Section 5
Concept
All four of the section's situations have the same structure: a fixed amount that does not depend on x, plus a rate multiplied by x. Recognising the structure is faster than deriving the equation each time.
the fixed-plus-rate shape — A one-time charge, session fee or baseline becomes a; a per-hour, per-student or per-unit rate becomes b.
\[ y = \underbrace{\text{fixed}}_{a} + \underbrace{\text{rate}}_{b} \times x \]
The introduction to the chapter names this pattern in the wild: the amount you pay a repair person for labor is often determined by an initial amount plus an hourly fee. What makes the rest of the chapter necessary is that most real pairs of variables do not follow an exact rule like this one — so a line has to be FITTED to scattered points rather than read off a price list.
Figure (svg): A table of four cost situations with their linear equations, intercepts and slopes
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, pp. 617-620 — the chapter introduction, and the section's examples
Picture it
Each with its equation and the meaning of both constants.
Figure (svg): A table of four cost situations with their linear equations, intercepts and slopes
Every row is fixed-plus-rate, and every slope's units are y-units per x-unit. The next section starts from data that follows no such exact rule, which is where the chapter's real work begins.
Worked example
A gym charges a joining fee of 40 dollars plus 28 dollars a month.
\[ \$40 \text{ once}, \; \$28 \text{ monthly} \]
Find the fixed part
Why: Charged once.
\[ a = 40 \]
Find the rate
Why: Per month.
\[ b = 28 \]
Name x
Why: What multiplies the rate.
Assemble
Why: Intercept first.
\[ y = 40 + 28 x \]
Figure (svg): The solution to Worked example writing an equation from a description shown as a ladder of expressions, one row per legal move
\[ y = 40 + 28x \]
Verify: confirm by checking a year
Why: Twelve months should cost 40 plus twelve lots of 28, which is 40 plus 336, or 376 — and the equation gives exactly that. Testing a value large enough that the two terms differ in size catches a swapped a and b: the reversed equation would give 28 plus 480, or 508, which is obviously not the right total.
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, p. 619
Faded example
A printer charges a setup fee of 15 dollars plus 0.40 dollars per page.
Fill in the blanks
y = 15 + 0.40x
Why: Fixed first, rate second — and the slope's units here are dollars per page, so a 200-page job costs 15 plus 80, or 95 dollars.
Worked example
Age and vocabulary size for four children, from section 12.2's data.
\[ (3, 655), (4, 1098), (6, 2463), (7, 3195) \]
Try a line through the first two
Why: Slope 443.
\[ y = -674 + 443 x \]
Test the third point
Why: At x = 6.
\[ \text{predicts } 1984,\text{ not } 2463 \]
Try another pair
Why: Different line.
Conclude
Why: No exact line.
Figure (svg): The solution to Worked example what this section cannot do shown as a ladder of expressions, one row per legal move
\[ \text{no } (a,b) \text{ satisfies all four} \]
Verify: confirm this is the chapter's actual subject
Why: Section 12.3 opens by saying data rarely fit a straight line exactly and that usually you must be satisfied with rough predictions. Everything in this section assumed an exact rule — a price list — and the remaining five sections are about what to do when no such rule exists: choose the line that misses by as little as possible, then ask whether it is worth using at all.
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, pp. 617-620
Trap
\[ y = 50 + 20x \;\Rightarrow\; \text{Emma pays } \$20 \text{ for the class} \]
Read the slope as an amount
Why: It is a dollar figure.
\[ \$20 \text{ is per STUDENT} \]
A class of twelve costs 50 plus 240, which is 290 dollars — not 20 dollars and not 70 dollars.
\[ y = 50 + 20(12) = 290 \]
Multiply the rate by the number of units before adding the fixed part
Why: That is what the slope being a rate means.
Attaching units to the slope prevents this entirely. Twenty dollars PER STUDENT cannot be mistaken for a total, whereas a bare 20 easily can — which is why the interpretation sentences the book asks for are a practical safeguard rather than an exercise in phrasing.
Sorting
Each describes a pair of quantities.
Sort into buckets
Sort by whether an exact line describes the relationship.
The first bucket is this section; the second is the rest of the chapter. Everything from section 12.2 onward concerns pairs of measured quantities, which never fall exactly on a line.
Two truths and a lie
All three concern the shape.
Eliminate the wrong options
Two are true. Knock those out and keep the false one.
Survives elimination: B
Why: The survivor is false and is why the chapter continues. Real bivariate data scatters, so the rest of chapter 12 is about choosing a line that fits well rather than one that fits exactly.
Prediction
Commit before reasoning.
Predict first
Given scattered points, what will section 12.3 choose the line to minimise?
Correct: The sum of squared vertical distances.
Why: That quantity is the sum of squared errors, and the line minimising it is called the least-squares line. Squaring makes every miss count positively regardless of direction — the same reasoning that made chapter 11's statistics sums of squares.
Comparison
Fill the blanks. The line is the same; the notation is not.
Comparison matrix
| Algebra | Statistics | |
|---|---|---|
| Written | y = mx + b | y = a + bx |
| Slope is called | m | b |
| Intercept is called | b | a |
| What is asked of them | graph and solve | interpret in context |
The third row is the collision: b names the intercept on one side and the slope on the other. The fourth row is the genuinely new demand, and it is the one the rest of the chapter builds on.
Pattern
Five steps, and the last two are the ones this chapter cares about.
Check any equation by substituting one convenient value; a swapped intercept and slope shows up immediately.
OpenStax Introductory Business Statistics 2e, §13.3 Linear Equations §13.3 Linear Equations
Check
The convention.
Check your understanding
In the equation y = 8 + 3x, what are a and b?
Answer: A
Why: The constant term is the intercept a, and the coefficient of x is the slope b.
Check
The graph.
Check your understanding
Which lines can be written in the form y = a + bx?
Answer: A
Why: The book says any line that is not vertical can be described by this equation. A vertical line has one x for many y values, which no choice of a and b produces.
Check
Interpretation.
Check your understanding
A repair cost is y = 25 + 20x with x in hours. What does the 20 mean?
Answer: A
Why: The slope is a rate of change: dollars per hour, so each extra hour raises the total by 20 dollars.
Real world
A software company advertises: our plan costs 99 dollars plus 7 dollars per user per month. A finance team writes the monthly cost as y = 7 + 99x, computes the cost for 40 users as 3,967, dollars and budgets accordingly.
Discussion prompt
Find the error, give the right figure, and say what would have caught it.
Hint: Ask which number is charged once and which is charged per user.
Answer:
The intercept and slope have been swapped. The 99 dollars is charged once per month regardless of headcount, so it is the intercept a; the 7 dollars is charged per user, so it is the slope b. The correct equation is y = 99 + 7x.
\[ y = 99 + 7x \;\Rightarrow\; y(40) = 99 + 280 = 379 \]
The real monthly cost is 379, dollars not 3,967 dollars — the error overstates the budget by more than a factor of ten, and it grows with headcount. At 200 users the correct figure is 1,499 dollars while the swapped equation gives 19,807 dollars.
Two checks would have caught it. Substituting a small value is the fastest: one user should cost 99 dollars plus 7, dollars which is 106, dollars and the swapped equation gives 106 dollars too — so a single user is exactly the case that does NOT distinguish them. Trying two users separates them at once, giving 113 dollars correctly against 205 dollars. That is worth knowing generally: test a value where the two terms differ in size, not where they happen to coincide.
The second check is units. The slope's units must be dollars per user, and 99 dollars per user is plainly not what the advertisement says. Attaching units to both constants before computing anything makes the swap visible without any arithmetic at all — which is the practical reason the book insists on interpreting both numbers in complete sentences.
Commit first
Answer, then rate your confidence honestly.
Predict first
In the statistics form y = a + bx, what does b represent?
Correct: The slope.
\[ y = \underbrace{a}_{\text{intercept}} + \underbrace{b}_{\text{slope}}\,x \]
Why: Statistics writes the intercept first, so a is the intercept and b is the slope — the reverse of algebra's y = mx + b, where b names the constant. Every formula in chapter 12 follows the statistics convention, so reading the positions rather than the letters is what keeps the rest of the chapter straight.
Explain it
They wrote a gym's cost as y = 28 + 40x, where the gym charges 40 dollars to join and 28 dollars a month.
Discussion prompt
In two sentences or fewer, correct them.
Hint: Ask which amount is paid every month.
Answer:
The 40 dollars joining fee is paid once, so it is the intercept a, and the 28 dollars is paid per month, so it multiplies x as the slope.
The equation is y = 40 + 28x — check it at two months, which should be 40 plus 56, or 96 dollars.
Exit ticket
Name the weakest spot before you close the deck.
Predict first
Which of these would you least want handed to you cold?
Correct: Whichever you picked is tonight's ten minutes, and each has a one-line fix.
Why: For the first, the constant term is a and the coefficient of x is b. For the second, ask which quantity depends on which. For the third, fixed amount first, per-unit rate second. For the fourth, name the units of x and y before writing anything. Do five problems of your chosen kind rather than twenty mixed ones.
Connect it up
Paper. Ten minutes — this is a short section.
Draw it
At the top, write y = mx + b on the left and y = a + bx on the right, and draw an arrow between the two b's with the words SAME LETTER, DIFFERENT JOB. Underneath, write the statistics form once more and label the constant term as the intercept and the coefficient of x as the slope. In the middle of the page, draw the three cases side by side — a line sloping up for b greater than zero, a horizontal line for b equal to zero, and a line sloping down for b less than zero — and write beneath them that a only slides the line vertically. Below that, make a four-row table of the section's situations: tax returns at y = 31.50 + 32x, tutoring at y = 25 + 15x, hang-gliding at y = 50 + 20x, and repairs at y = 25 + 20x, with a column saying what a means and a column giving the units of b. At the bottom, write out the two interpretation sentences for the tutoring line in full — one about the 25 dollars fee at zero hours, one about the 15 dollars per hour — and note that section 12.3 will ask for the same two sentences about a fitted line, adding only the words on average.
Check your table by confirming every b has units of the form dollars per something, and that the something is the unit of x in that row. Check your three graphs by covering the labels and asking whether the sign of b alone would let you redraw each one.
Recap
Five things, and the last one is what the chapter is built on.
| If you see | Then |
|---|---|
| A one-time or fixed amount | It is a, the intercept |
| An amount stated per unit | It is b, the slope |
| The constant term of an equation | That is a, whatever the letters suggest |
| b greater than zero | The line slopes upward |
| b equal to zero | A horizontal line: y does not depend on x |
| b less than zero | The line slopes downward |
| A slope to interpret | State it as y-units per one x-unit |
| x = 0 outside the data | The intercept places the line but may mean nothing |
Section 12.2 drops the assumption that made this section easy. Real pairs of measured quantities scatter rather than falling on a line, so the first task is simply to look at them — which is what a scatter plot is for.
OpenStax Introductory Statistics 2e, §12.1 Linear Equations §12.1, pp. 617-620 — everything on these slides traces back here
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