Scatterplots and numerical association

Temperature explains ice cream sales, and it comes first. That is the explanatory variable followed by the response.

(C) has a real explanatory and response pair in the wrong order: fertiliser explains growth, so growth is the response. 21.0% chose it, which is what happens when you miss the words “followed by”. (A) and (B) pair variables with no clear direction. 73.7% of students chose (D).

Older vehicles have had more time to travel, so age explains distance. Age is explanatory and goes on the horizontal axis.

Each wrong option gets one half right. (C) gives distance the right role and the wrong axis, and 14.4% chose it. (D) gives it the right axis and the wrong role. (A) swaps the roles. 68.3% of students chose (B).

a) Distance is the response, so it goes up the side and sleep goes across. Sleep runs from 6 to 9.5 hours, so steps of 1 hour with half-hour points between gridlines work. Distance runs from 4 to 8 km.

1 mark · correctly identifies the axis for the response variable (this mark may be implied by subsequent working)

1 mark · formats scatterplot with appropriate scaling and labelling for both axes

1 mark · accurately plots all points

The sleep values are not in order in the table. Plot them as pairs, one pair at a time, and the half-hour values (7.5, 6.5, 9.5, 8.5) are where accuracy is lost.

b) Reading left to right, the points fall. More sleep goes with less distance.

1 mark · identifies direction (accept similar statements, e.g. downward)

For reference, $r = -0.78$. The question did not ask for strength. The 2023 report listed identifying the direction of an association as done well.

a) Age explains the number of stands, so age goes across. You choose everything else. The marking guide used 0 to 100 in 10s and 0 to 25 in 5s. The broken-scale layout in the figure above is the one the 2024 subject report showed as effective, and it spreads the points further.

1 mark · correctly identifies the explanatory and response variables (this mark may be implied by subsequent working)

1 mark · accurately plots points (FT allowed)

1 mark · formats scatterplot with appropriate scaling and labelling of axes

b) The points sit along a straight line with no bend.

1 mark · states form as linear

“Negative” and “strong” are true ($r = -0.99$), but they are direction and strength. The question asked for form, so only “linear” earns the mark.

The points fall from left to right, so the direction is negative. They stay in a fairly narrow band all the way down, so the association is strong. The points I read off the plot give $r \approx -0.95$.

Do not mistake a little scatter for weakness. Strong does not mean perfect. The 2020 subject report has no multiple choice results.

(C) falls from left to right in a tight band. (A) is a perfect positive association, (B) shows no association and (D) is strong positive. 95.4% of students chose (C).

(D) rises from left to right, so it is positive, but the points are loosely spread around that trend, so it is weak. (C) is strong positive, (B) falls, so it is negative, and (A) shows no clear trend at all.

84.8% of students chose (D). The most common wrong answer was (A), at 10.9%, which mistakes no association for a weak one.

a) Temperature explains pie sales, so temperature goes across. The guide used 0 to 40 °C in 5s across and started the vertical axis at 18 pies, in 2s, with no 0 written.

1 mark · correctly identifies the explanatory and response variables (this mark may be implied by subsequent working)

1 mark · accurately plots points

1 mark · formats scatterplot with appropriate scaling and labelling of axes

b) The points fall from left to right and stay close to a straight line.

1 mark · describes association as strong

1 mark · describes association as negative

The guide’s sample says “The association is a strong, negative, linear relationship.” For reference, $r = -0.905$. The one point at (31, 25) might tempt you to say moderate, but the other seven sit almost on a line.

a) $g$ goes across and $h$ goes up. Label the axes yourself, because the printed grid had numbers but no titles.

1 mark · correctly formats the Cartesian plane with $g$ along the $x$-axis and $h$ along the $y$-axis

1 mark · correctly plots all the data points (accurate plotting accepted even if the axes are back-to-front)

b) The points follow a curve, so the form is non-linear. They sit close to that curve, so it is strong.

1 mark · identifies form (non-linear, accept equivalent such as quadratic or not linear)

1 mark · identifies strength (strong, accept equivalent strengths)

This is the trap question of the topic. Put this data into your calculator and you get $r = -0.04$, which would suggest no association. That only says no straight line fits. The curved pattern itself is very tight. Strength describes how closely the points follow their pattern, whatever its shape, and $r$ only measures straight-line patterns.

$r$ is positive, so the direction is positive. Its size is small, so of the two strengths offered, weak is the only fit.

On the convention above, 0.23 sits just under the 0.25 line, where I would say no linear association. The options do not include that, so choose the closest. 84.0% of students chose (A).

$r = 0.92$ is strong. The model is a straight line, and $r$ measures linear association, so the form is linear.

The slope of 4.6 and the intercept of $-35$ say nothing about strength or form. 77.4% of students chose (B), and 13.6% chose (D).

$r = -0.989$ is negative and very close to $-1$, so the points lie almost exactly on a straight line. That makes it negative, linear and strong. Only (B) has no wrong word. 92.8% of students chose it.

Pick $r$, not $R^2$, from the box: $r = 0.986$. That is close to 1, so the association is strong.

1 mark · correctly identifies the value of Pearson’s correlation coefficient

1 mark · correctly describes the strength as strong (this mark may only be awarded if a coefficient value is identified, accept equivalent statement, e.g. better than moderate)

Read that note again. A correct word with no value of $r$ beside it earned nothing. Parts b) and c) ask about a confounding variable, which is in the linear models and causation guide.

Step 1 · $r$ for each dataset, from the calculator, with fish caught as $y$ both times:

$$\begin{aligned} r_{t,f} &= 0.31 \\ r_{p,f} &= 0.81 \end{aligned}$$

1 mark · correctly calculates correlation coefficient for each dataset (accept equivalent decimal values, the guide shows 0.3 and 0.8)

Step 2 · Compare. $0.81 > 0.31$, so the number of people fishing, $p$, is the explanatory variable for the stronger linear association. Jiro’s claim has the better support. On the convention in this part, 0.81 is strong and 0.31 is weak.

1 mark · identifies explanatory variable for stronger linear association (this mark may be implied by subsequent working)

Step 3 · Least-squares line for $f$ against $p$ (the method is in guide 3):

$$f = -129.84 + 10.893p$$

1 mark · determines least-squares line equation for dataset with stronger linear association

Step 4 · Predict. The 25 °C belongs to the weaker model, so it is not used.

$$\begin{aligned} f &= -129.84 + 10.893 \times 50 \\ &= 414.8 \approx 415 \text{ fish} \end{aligned}$$

1 mark · substitutes value for relevant explanatory variable (this mark may be implied by subsequent working)

1 mark · predicts number of fish caught

The guide’s sample rounds the coefficients first, to $f = -130 + 11p$, and predicts 420 fish. It accepts equivalent whole numbers based on valid working, so 415 from full precision and 420 from the rounded line both score. Keep full precision yourself.

The 2024 sheet gave $r = \dfrac{1}{n-1}\sum\left(\dfrac{x_i - \bar{x}}{s_x}\right)\left(\dfrac{y_i - \bar{y}}{s_y}\right)$, so

$$r = \frac{-4.5}{10 - 1} = -0.50$$

(B) divides by 10 instead of 9, and 38.9% chose it. 33.7% of students chose (A).

The stem fixes $y$ as the response, so the explained variation is the variation in $y$. (A) and (D) talk about the variation in $x$, and (C) talks about predicted outcomes. (B) is the keyed answer. In your own response, write the full sentence: 36% of the variation in $y$ is explained by the linear association with $x$.

Only 24.6% of students chose (B), the lowest result on that paper. 35.2% chose (A), which swaps the roles of $x$ and $y$, and 29.4% chose (C).

$$0.997 \times 100\% = 99.7\%$$

1 mark · correctly states the percentage

99.7% of the variation in average weekly earnings is explained by the linear relationship with the year. Parts b) and c), on the slope and a prediction, are in guide 3.

a)

$$\begin{aligned} y &= 2.3 \times 12 + 31.4 \\ &= 59 \text{ fish} \end{aligned}$$

1 mark · correctly calculates 59

b) Use each value separately, as the question says. $r = 0.688$ is a moderate positive linear association. $R^2 = 0.473$ means 47.3% of the variation in the number of fish caught is explained by the linear association with hours spent fishing, so more than half is down to other factors. The prediction of 59 fish may be reasonable, but with only a moderate association it should be treated with caution.

1 mark · correctly describes the strength as either moderate or strong (accept moderately positive or strongly positive)

1 mark · correctly describes the meaning of the coefficient of determination

1 mark · evaluates the reasonableness of the solution (FT allowed, accept alternative statements)

Check that the two values agree: $0.688^2 = 0.473$. The 2020 report listed the difference between $r$ and $R^2$ as something to strengthen, so give each its own sentence.

Step 1 · Plot the data. People go across, water usage goes up, as printed.

1 mark · correctly constructs a scatterplot

Step 2 · $r$ from the calculator, people as $x$ and water as $y$:

$$r = 0.886$$

1 mark · correctly determines the correlation coefficient (accept equivalent decimal values, e.g. 0.89, 0.8864)

Step 3 · Interpret it. On its own, 0.886 says a strong positive linear association. The guide’s sample says “a very strong positive relationship”.

1 mark · interprets the value of the correlation coefficient

Step 4 · Evaluate. The plot curves upward. From 1 to 4 people the usage climbs slowly (130 L to about 300 L), then from 5 to 7 people it climbs steeply to 1340 L. The association is not linear, and $r$ measures linear association only, so $r$ should not be used here, despite its high value.

1 mark · correctly identifies that the scatterplot is not linear

1 mark · correctly identifies that the correlation coefficient should not be used

Two of the five marks need the plot. Evaluate means weigh it up and give a verdict, so end with a clear statement that $r$ is not appropriate, and why.

Step 1 · Scale the given axes. Customers run from 6 to 22 and checkouts from 1 to 5, so with six divisions use 0 to 30 in 5s across and 0 to 6 in 1s up.

1 mark · correctly formats scatterplot with appropriate scaling shown on axes

Step 2 · Plot all five points.

1 mark · accurately plots points (FT allowed)

Step 3 · Explain. The points for 8 am, 10 am, 12 pm and 2 pm lie on a straight line: every 4 extra customers adds one checkout. The 4 pm point is an anomaly. At 14 customers the pattern gives 3 open checkouts.

1 mark · provides mathematical reasoning or working to support the answer (accept alternative working, e.g. indicating on the scatterplot where the plotted point should be repositioned)

Step 4 · Answer the question asked. It asks for additional checkouts, which is a difference.

$$3 - 1 = 2$$

1 mark · determines number of additional checkouts that should have been open at 4 pm

Writing 3 answers a question that was not asked. The 2025 report listed constructing a scatterplot, with or without provided axis scaling, as done well.