Mathematics • Year 10 • Unit 4 • Lesson 14
Bivariate Data Review in the Real World
Apply the full Lesson 14 bivariate workflow — plot, describe, fit, predict — to real Australian contexts: NRL training data, climate records, online sales, and gym performance.
1. Word problems
For each scenario, perform the requested steps in the bivariate workflow and write a one-sentence interpretation in context.
1.1 — NRL training and tackles made. A coach tracks for 6 forwards:
(training hr/week, tackles/match) = (3, 18), (4, 22), (5, 25), (6, 30), (7, 33), (8, 38).
(a) Describe the correlation.
(b) Find the line of best fit (mean point + slope from endpoints).
(c) Predict tackles made at 5.5 hr training.
(d) Comment on whether you'd predict at 12 hr training. 4 marks
1.2 — Sydney climate. Monthly average maximum temperature (°C) and rainfall (mm) for one Sydney summer:
(22, 90), (24, 95), (26, 100), (28, 80), (30, 70), (32, 50).
(a) Plot a quick scatter on the axes below.
(b) Describe the correlation — direction and shape may surprise you.
(c) Can you draw a single straight line of best fit and feel confident? Why or why not? 3 marks
|—————————————————————————————————————————————|
20 22 24 26 28 30 32 34
1.3 — Online clothing sales. A small Sydney store records weekly:
(ad spend $, sales $) = (200, 1500), (400, 2200), (600, 3000), (800, 3500), (1000, 4200), (1200, 4900).
(a) Describe the correlation.
(b) Find the equation of the line of best fit.
(c) Predict the sales when ad spend = $500 (interpolation).
(d) The owner asks: "If I spend $0 on ads, will I make $0?" Use the equation to answer and comment on whether the value is realistic. 4 marks
1.4 — Gym deadlifts. A trainer notes (weeks training, max deadlift kg) for 5 clients:
(4, 80), (8, 95), (12, 110), (16, 120), (20, 130).
(a) Describe the correlation.
(b) Predict max deadlift at 10 weeks (interpolation).
(c) Predict at 100 weeks — and discuss why the lesson misconception "trends continue forever" applies here. 3 marks
1.5 — Choosing the right tool. For each research question, state whether the appropriate first display is a scatter plot, a parallel box plot, or a bar chart. Justify in one phrase.
(a) Is there a relationship between height and shoe size in 30 Year 10s?
(b) Which of three coffee shops has the longest typical wait time?
(c) What is the most popular elective subject? 3 marks
2. Explain your thinking
This question is about justifying a workflow, not just calculating. Use full sentences. 4 marks
2.1 A Year 10 student is given a bivariate data set and immediately calculates a line of best fit before doing anything else. Using Lesson 14's recommended workflow, write a four-sentence reply that (i) names the FIRST step they should have done, (ii) explains why the order matters, (iii) gives one example where skipping the first step leads to a wrong conclusion (think non-linear data), and (iv) finishes with one rule of thumb for analysing any bivariate data set.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — NRL forwards
(a) Strong positive linear correlation.
(b) Mean: x̄ = 33/6 = 5.5; ȳ = 166/6 ≈ 27.67. m = (38 − 18)/(8 − 3) = 20/5 = 4. 27.67 = 4 × 5.5 + c → c = 27.67 − 22 = 5.67. Equation: y = 4x + 5.67.
(c) At x = 5.5: y = 4 × 5.5 + 5.67 = 22 + 5.67 ≈ 27.7 ≈ 28 tackles.
(d) At x = 12: y = 53.67. Extrapolation. Possible but suspect — 12 hr of training/week could lead to fatigue or injury, breaking the linear trend (Lesson 13 misconception).
1.2 — Sydney summer climate
(a) Plot shows 6 points rising from (22, 90) to (26, 100) then falling to (32, 50).
(b) The relationship is NOT linear — rainfall peaks at around 26 °C and then decreases. There is a clear pattern but it's curved (not monotone).
(c) No: a single straight line would be a poor fit. Lesson 12 misconception card: a strong pattern can have r ≈ 0 if it is curved. Linear methods are inappropriate here.
1.3 — Online clothing sales
(a) Strong positive linear correlation.
(b) Mean: x̄ = 4200/6 = 700; ȳ = 19300/6 ≈ 3216.67. m = (4900 − 1500)/(1200 − 200) = 3400/1000 = 3.4. 3216.67 = 3.4 × 700 + c → c = 3216.67 − 2380 = 836.67. Equation: y = 3.4x + 836.67.
(c) At x = 500: y = 3.4 × 500 + 836.67 = 1700 + 836.67 ≈ $2537.
(d) At x = 0: y = $836.67 — meaning even with zero ad spend, some baseline sales would happen (from regular customers, word of mouth, walk-ins). $0 is extrapolation; the figure is plausible because a small shop can survive without ads, but the linear model probably overstates baseline sales.
1.4 — Gym deadlifts
(a) Strong positive linear correlation, but with a slight slowing at high weeks (130 only 10 kg above 16 weeks' 120).
(b) Mean: x̄ = 60/5 = 12; ȳ = 535/5 = 107. m = (130 − 80)/(20 − 4) = 50/16 ≈ 3.125. 107 = 3.125 × 12 + c → c = 107 − 37.5 = 69.5. Equation: y = 3.125x + 69.5. At x = 10: y = 3.125 × 10 + 69.5 = 100.75 ≈ 101 kg.
(c) At x = 100: y = 3.125 × 100 + 69.5 = 382 kg — far beyond world records for human deadlifts. Lesson 13 misconception: trends don't continue forever; gains plateau as you approach physiological limits.
1.5 — Right tool
(a) Scatter plot — relationship between two numerical variables (bivariate).
(b) Parallel box plot — compare distributions across 3 groups (each group's centre + spread).
(c) Bar chart — categorical data showing counts per category.
2.1 — Explain your thinking (sample)
The first step the student should have done is plot a scatter plot — Lesson 14 says "the first step in analysing bivariate data is to draw a scatter plot". The order matters because a line of best fit is only appropriate if the points actually follow a linear trend; without plotting, you cannot tell if the data is linear, curved, or just random. For example, if the data follows a U-shape (parabola), the line of best fit will be nearly flat with r ≈ 0, hiding the very strong non-linear pattern — Lesson 12's misconception card warns about exactly this. A safe rule of thumb: always plot first, describe the correlation second, and only fit a line if the points truly look linear.
Marking: 1 mark for naming "plot the scatter plot" as the first step, 1 for explaining why order matters, 1 for a concrete example, 1 for a clear rule of thumb.