Regression on the GDC

The line, r, a prediction and what they mean

In 12 of 20 recent exams
2% of the marks
Paper 2, calculator
1 to 3 marks

How it is set in the exam

Paper 2 (calculator) only, 1–3 marks a part (usually 2), in a Section A question of 4–7 marks; once inside a Section B question. Very common (12 of 20 exam versions, 32 logged parts). A question usually runs line → $r$ → prediction, sometimes with an interpretation or a reliability mark at the end.

Spot it in the exam

Exam phrases to spot (6)

“find the equation of the regression line of $y$ on $x$ ”
“the regression line has equation $y = ax + b$ ; find $a$ and $b$ ”
“write down the value of $r$ ”
“use your equation to estimate …”
“interpret the gradient in context”
“explain why the estimate may be unreliable”

In disguise (5)

the variables have names instead of $x$ and $y$ : "the regression line of $T$ on $h$ " predicts $T$ from $h$ , so $h$ goes in the $x$ -list
the $x$ -values are years: code them as years since the start, or use the year itself (Example 1.3)
a change in the predictor with no starting value ("she adds 10 km a week to her training"): the change in $y$ is $a$ times it, and $b$ is not needed
"what does the value of $a$ represent?": the gradient in context
a second model (a percentage growth) to be compared with the line's prediction (Example 1.4)

Variants (4)

given: the data (the usual case, on the GDC), or the line itself in the stem, with only a prediction, a change or an interpretation asked (Your turn 2)
asked: the line, $r$ , a prediction, the change for an increase in $x$ , the gradient (or intercept) in context, the reliability of a prediction, a comparison with another model
$x$ = a year, coded as the year or as years since the start; any consistent coding is accepted
the prediction uses a value from an earlier part (follow-through from the student's own line)

Looks like it, but isn't (5)

$x$ is estimated from a value of $y$ : Predict x from y
both regression lines are given and either variable is estimated: Predict x from y (choose the line by its subject)
both regression lines are given and the means are asked for: The mean point
a value is missing from the table and the line is given: Work backwards from the mean or median, with the mean from the line first (The mean point)
a line through two exact points, with no data set: Equation of a straight line

Worked example

The line, r and an estimate

Paper 2 (calculator)

Question

Eight weather stations in a mountain region are at altitudes $h$ metres. The mean July temperature at each is $T$ °C.

$h$ (m)	$T$ (°C)
$150$	$23.4$
$320$	$23.1$
$480$	$21.2$
$650$	$21.0$
$870$	$19.6$
$1040$	$18.1$
$1230$	$17.5$
$1410$	$16.2$

$h$ (m)	$150$	$320$	$480$	$650$	$870$	$1040$	$1230$	$1410$
$T$ (°C)	$23.4$	$23.1$	$21.2$	$21.0$	$19.6$	$18.1$	$17.5$	$16.2$

The regression line of $T$ on $h$ is $T = ah + b$ .
(a) Find $a$ and $b$ . [2]
(b) Write down Pearson's product-moment correlation coefficient, $r$ , for these data. [1]
(c) Estimate the mean July temperature at a station $900$ m above sea level. [2]

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Write your working on paper or on screen. Paper Zero marks it step by step, like the worked example, and shows where each mark went. Free, with no account.

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See it move: Drag a point, the line follows

The line of $T$ on $h$ : $a = -0.0058337\ldots$ and $b = 24.497\ldots$ , so $T = -0.00583h + 24.5$ . It passes through the mean point $(\bar{h}, \bar{T}) = (768.75, 20.01)$ and makes the squares of the dashed gaps as small as they can be.
$r = -0.99183\ldots \approx -0.992$ : strong negative correlation. $a$ and $r$ always have the same sign.
At $h = 900$ : $T = -0.0058337\ldots \times 900 + 24.497\ldots = 19.246\ldots \approx 19.2$ °C. 900 m is inside the data, 150 to 1410 m: an interpolation.
(a) and (b) Three significant figures, with the sign: $a = -0.00583$ (not $-0.006$ ), $b = 24.5$ and $r = -0.992$ . The temperature falls as the altitude rises, so both are negative.
(c) Predict with the unrounded line: $19.246\ldots \approx 19.2$ °C, between the stations either side, 19.6 at 870 m and 18.1 at 1040 m. The rounded line gives $-0.00583 \times 900 + 24.5 = 19.253$ , which rounds to $19.3$ .

Steps

Decide which variable is predicted:
that one is $y$ . Enter the other as the $x$ -list and run a linear regression (linear regression). Code years first if the question says how.
Write the line
with the question's variables and each coefficient to 3 s.f., labelled ( $a = \ldots$ , $b = \ldots$ ), and $r$ to 3 s.f. with its sign. Sign check: $a$ and $r$ always have the same sign.
Predict:
substitute into the unrounded line, written out in full with its inputs, then the value (keep full-accuracy values). Check that the value you substitute is inside the range of the data. A count may be given as a whole number or to 3 s.f.
Interpret:
$a$ is the change in $y$ per one-unit increase in $x$ , in context, with units and a direction. A change in $x$ changes $y$ by $a$ times that change: an increase of $10$ in $x$ changes $y$ by $10a$ , whatever the starting value.
Reliability:
name the reason (How far to trust a prediction): extrapolation (with the value that is outside the data), weak correlation, or the wrong line.

Common mistakes

The lists the wrong way round.
The predicted variable is the $y$ -list. Swapped lists give the line of $x$ on $y$ , a different line.
Data entered wrongly,
or a gap in the years closed up: for the coefficients and $r$ , even when the rounded values happen to be close.
Dropping the sign of $r$
, or giving $r^2$ instead of $r$ .

More mistakes (8)

Rounding a small gradient to 3 decimal places
$-0.006$ instead of $-0.00583$ . Three significant figures, for $r$ too: $0.98$ for $0.981$ scores .
$a$ and $b$ unlabelled or swapped
one mark lost. Write $a = \ldots$ , $b = \ldots$ , or the whole equation with the question's letters.
An expression when the equation is asked for
$-0.176d + 33.1$ is not an equation; write $t = -0.176d + 33.1$ . (When the question asks for $a$ and $b$ , labelled values are enough.)
A prediction with only the final number.
The () is for substituting into your line, and the follows through from it: write the substitution out in full, even if you doubt (a). A bare wrong value scores nothing.
Predicting with rounded coefficients.
It can change the third significant figure (Example 1.1), and with years as $x$ it ruins the answer (Example 1.3). Use the stored line.
An interpretation with no context
" $a$ is the gradient" or " $y$ goes up by $0.176$ ". Say what changes, by how much, per one unit of what, and the direction ("decreases", not "changes").
A reliability answer with no reason
"the model may not be accurate" scores nothing. Name extrapolation and the value outside the data.
Writing "correlation does not imply causation"
when the question asks about something else: .

Easy to confuse with

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