Predict x from y

Choose the right regression line

In 4 of 20 recent exams
Paper 1 or 2
1 to 3 marks

How it is set in the exam

Paper 2 (calculator) mostly (6 of 7 logged parts), 1–3 marks a part; once on Paper 1 (no calculator) with both lines given. Typical (4 of 20 exam versions). It sits inside a longer regression question, next to parts that predict $y$ or find the means, so the whole point is to notice which variable is being estimated.

Spot it in the exam

Exam phrases to spot (4)

“estimate $x$ when $y$ is …”
“the regression line of $x$ on $y$ is …”
“find the regression line of $x$ on $y$ ”
“explain why the line $y = ax + b$ should not be used to estimate $x$ ”

In disguise (4)

both lines are given in the stem, each with its own subject: pick the one whose subject is the variable asked for. This holds for an estimate of $y$ too
the table's first row is the variable to be estimated: what matters is which variable is asked for, not which row it is in or what letter it has
the given value of $y$ comes from an earlier part or from a model given as a formula: evaluate that first
"show that the estimate is $72$ , to the nearest whole number": the unrounded value must appear before the rounded one

Variants (3)

given: both lines (choose one), or the data (swap the lists on the GDC)
asked: an estimate, a "show that" to a rounded value, the reason the other line is wrong
the input value comes from an earlier part: a given model evaluated first

Looks like it, but isn't (2)

both lines are given and the means are asked for: The mean point
the data, or only the line of $y$ on $x$ , are given and $y$ is estimated from $x$ : Regression on the GDC

Worked example

Both lines given: choose by the subject

Paper 1 (no calculator)

Question

On a sample of winter days, the daily gas use $G$ m³ of a house and the outside temperature $T$ °C were recorded. The temperatures were between $-6$ °C and $14$ °C, and the gas use between $4$ m³ and $16$ m³. The regression line of $G$ on $T$ is $G = 12 - 0.5T$ , and the regression line of $T$ on $G$ is $T = 20 - 1.6G$ .
(a) On one day the house used $7$ m³ of gas. Estimate the outside temperature that day. [2]
(b) Explain why the line $G = 12 - 0.5T$ was not used in (a). [1]
(c) Estimate the gas used on a day when the temperature is $-2$ °C. [2]

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Which method?

$y$ from $x$
You are given
the data
Use
the line of $y$ on $x$ : the $x$ -values as the $x$ -list
$x$ from $y$
You are given
the data
Use
the line of $x$ on $y$ : swap the lists, to get $x = cy + d$
either
You are given
both lines
Use
the line whose subject is the variable you estimate

You estimate	You are given	Use
$y$ from $x$	the data	the line of $y$ on $x$ : the $x$ -values as the $x$ -list
$x$ from $y$	the data	the line of $x$ on $y$ : swap the lists, to get $x = cy + d$
either	both lines	the line whose subject is the variable you estimate

Never rearrange the line of $y$ on $x$ to estimate $x$ from a value of $y$ : rearranged, it is still the line of $y$ on $x$ , not the line of $x$ on $y$ (the two are the same only when $r = \pm 1$ ). Solving a line for its other variable is right only when nothing is being estimated: finding $\bar{x}$ from $\bar{y}$ (both lines pass through $(\bar{x}, \bar{y})$ , so both give the same $\bar{x}$ : The mean point), or finding where the line's own prediction reaches a value ("when does the line predict …").

Steps

Name the variable to be estimated.
It must be the subject of the line you use.
Get that line:
from the two given lines, or on the GDC with the lists swapped (linear regression). Write it as $x = cy + d$ with the question's letters.
Get the input value:
from the question, an earlier part, or a model given as a formula (define a function and evaluate it). Keep it unrounded (keep full-accuracy values).
Substitute
into the line, written out in full. Check that the value you substitute is inside the range of the data.
"Show that" with a rounded target:
write the unrounded value, then the rounded one. "Why not the other line?": the line of $y$ on $x$ is made for estimating $y$ ; estimating $x$ needs the line of $x$ on $y$ .

Common mistakes

Rearranging the line of $y$ on $x$
to estimate $x$ : when the lines are given, and at best follow-through marks on . In Example 2.2 it gives $69.7$ instead of $72.0$ .
Choosing by the letters in the type's name.
"Predict x from y" is only a name: what decides the line is which variable is estimated. If a question calls the speed $y$ and the flow $x$ , estimating the speed needs the line of $y$ on $x$ .
Not swapping the lists
on the GDC, so the "line of $x$ on $y$ " is really the line of $y$ on $x$ .

More mistakes (3)

Only the rounded value
in a "show that": $72$ without $71.992\ldots$ loses the accuracy mark.
A value outside the data
an estimate from a value of $y$ beyond the recorded range is an extrapolation, and a follow-through answer outside the range has lost the accuracy mark.
The wrong $t$
in a model feeding the estimate: 9 pm is $t = 3$ when $t$ counts hours after 6 pm.

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