Mathematics Standard • Year 12 • Ratios and rates • Lesson 5
Reversing Estimates and Energy, Mastery Challenge
Run the capture-recapture formula backwards, adjust an estimate when the population itself changes, size a solar system against a household's use, and diagnose the two errors that cost the most marks in this topic.
1. Find the missing count
You are given the estimate and asked for one of the inputs. 4 marks
Q1.1 A capture-recapture study estimated a population of 480. 60 animals had been marked, and 72 were caught in the second sample. How many of that second sample were marked?
2. When the population changes
Part (b) asks you to repair an assumption, not just repeat the calculation. 4 marks
Q2.1 200 birds are banded. Later, 150 birds are caught and 12 carry a band.
(a) Estimate the population. (b) Suppose a survey shows that 30 of the banded birds had died before the second sample was taken. Give a better estimate, and explain why it changes in the direction it does.
3. Sizing a solar system
Three linked parts. Carry your answers forward. 4 marks
Q3.1 A household uses 28 kWh of electricity a day. It installs a 5.5 kW solar system that generates at full output for an average of 4.5 hours a day.
(a) How much energy does the system generate each day? (b) What is the daily shortfall that must still be bought from the grid? (c) At $0.32 per kWh, what does that shortfall cost over 30 days?
4. Find the error
Identify the mistake, name it, and give the correct answer. 4 marks
Q4.1 A student writes: "A 2000 W heater runs for 5 hours and electricity costs $0.30 per kWh, so the cost is 2000 × 5 × 0.30 = $3000." Explain what has gone wrong, state by what factor the answer is out, and give the correct cost.
5. A flawed study
Name the assumption and give the direction of the error. 3 marks
Q5.1 A researcher tags fish in one small bay of a very large lake, then takes the second sample from that same bay. Explain which assumption this breaks and whether the resulting estimate of the lake's fish population is too high or too low.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1, The missing count
Start from N = (n1 × n2) ÷ m2, so 480 = (60 × 72) ÷ m2 [1]. That gives 480 = 4320 ÷ m2 [1], so m2 = 4320 ÷ 480 [1] = 9 animals [1]. Check: (60 × 72) ÷ 9 = 4320 ÷ 9 = 480.
Q2.1, When the population changes
(a) N = (200 × 150) ÷ 12 = 30 000 ÷ 12 = 2500 birds [2].
(b) If 30 of the banded birds died, only 170 banded birds were actually alive in the population when the second sample was taken, so n1 should be 170, not 200. N = (170 × 150) ÷ 12 = 25 500 ÷ 12 = 2125 birds [1].
The estimate falls, and it should. Finding 12 banded birds out of 150 is more impressive when only 170 banded birds exist than when 200 do, so that same catch now implies a smaller population [1].
Q3.1, Sizing a solar system
(a) Energy = power × time = 5.5 × 4.5 = 24.75 kWh a day [1].
(b) Shortfall = 28 − 24.75 = 3.25 kWh a day [1].
(c) Over 30 days that is 3.25 × 30 = 97.5 kWh [1], costing 97.5 × 0.32 = $31.20 [1]. Check: 31.20 ÷ 0.32 = 97.5 kWh.
Q4.1, Find the error
The power was never converted from watts to kilowatts [1]. The price is given per kilowatt hour, so the power must be in kilowatts before it is multiplied by anything. Because a kilowatt is 1000 watts, the answer is out by a factor of 1000 [1].
Correctly: 2000 W = 2 kW, so the energy is 2 × 5 = 10 kWh [1], and the cost is 10 × 0.30 = $3.00 [1]. A sense check settles it at a glance: no household heater costs three thousand dollars to run for an afternoon.
Q5.1, A flawed study
This breaks the assumption that the marked individuals mix evenly back through the whole population [1]. Sampling the same small bay twice means the tagged fish are concentrated exactly where the researcher is looking, so far more of them turn up in the second sample than their share of the lake would justify. That makes m2 much too large [1].
Since m2 is the denominator of N = (n1 × n2) ÷ m2, an inflated m2 makes N too small, so the estimate is too low: the study would badly understate how many fish the lake holds [1]. Note that the arithmetic here is not wrong at all. The sampling design is.