Skill Align ACT BSSS Specialist Methods T Booklet 2 Practice Assessment Pack 0 - 2026 Edition
Original Skill Align Specialist Methods T practice-assessment Pack 0, Booklet 2; not an official ACT BSSS examination and not endorsed by ACT BSSS or the ACT Government.
- Paper
- Booklet 2 Showcase
- Reading
- 10 minutes
- Writing
- 65 minutes
- Assessment
- 50 marks
Suggested Skill Align Booklet 2 conditions: 10 minutes reading time and 65 minutes working time. An approved graphics calculator or CAS may be used. Offline stored notes and programs and locally supplied reference material may be used only when authorised by the administering college. Internet access and external communication are prohibited. ACT BSSS assessment is school based. The administering college may vary timing, technology, stored-note, program and reference-material conditions. No public PDF download is supplied with Pack 0.
Booklet 2
Booklet 2 assesses technology-supported probability, distributions, inference, regression and non-routine mathematical modelling. Answer all questions. Show complete working, reasoning and interpretation. State exact values unless an approximation is requested, and include units and context where applicable. Suggested Skill Align Booklet 2 conditions: 10 minutes reading time and 65 minutes working time. An approved graphics calculator or CAS may be used. Offline stored notes and programs and locally supplied reference material may be used only when authorised by the administering college. Internet access and external communication are prohibited. ACT BSSS assessment is school based. The administering college may vary timing, technology, stored-note, program and reference-material conditions.
Question 6
10 marksQuestion 7
10 marksQuestion 8
10 marksQuestion 9
10 marksQuestion 10
10 marksWorked Solutions And Marking Guide
Question 6
(a) E(X)=6.
Use the binomial mean.
(b) operatorname(Var)(X)=4.2.
Use the binomial variance.
(c) P(X=6)=binom(20)(6)(0.30)⁶(0.70)¹⁴approx0.1916.
Apply the binomial probability formula and retain the exact expression.
Detailed marking criteria
Part a (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Identifies n=20 and p=0.30.
- Uses E(X)=np.
- Obtains E(X)=20(0.30)=6.
Acceptable alternatives: For Q6(a), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part b (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Uses operatorname(Var)(X)=np(1-p).
- Substitutes 20(0.30)(0.70).
- Obtains operatorname(Var)(X)=4.2.
Acceptable alternatives: For Q6(b), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part c (4 marks)
Award all 4 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. For the final-result mark, retain the requested exact form; carry unrounded values until the final stated accuracy.
Mark-by-mark evidence:
- Selects the binomial coefficient binom(20)(6).
- Uses the success factor (0.30)^6.
- Uses the failure factor (0.70)¹⁴.
- Combines the factors and rounds the probability to 0.1916.
Acceptable alternatives: For Q6(c), award all 4 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Question 7
(a) z=2.
Subtract the mean and divide by the standard deviation.
(b) P(X>130)=0.0228.
Use the upper-tail complement.
(c) P(85<X<115)=0.6826.
Standardise both endpoints and use normal symmetry.
Detailed marking criteria
Part a (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Uses z=(x-mu) / sigma.
- Substitutes x=130, mu=100 and sigma=15.
- Obtains z=(130-100) / 15=2.
Acceptable alternatives: For Q7(a), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part b (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Recognises X>130 as Z>2.
- Uses P(Z>2)=1-P(Z<2).
- Obtains 1-0.9772=0.0228.
Acceptable alternatives: For Q7(b), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part c (4 marks)
Award all 4 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Standardises 85 and 115 to z=-1 and z=1.
- Uses symmetry to write P(-1<Z<1)=2P(Z<1)-1.
- Substitutes 2(0.8413)-1.
- Obtains 0.6826.
Acceptable alternatives: For Q7(c), award all 4 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Question 8
(a) k=1 / 8.
Set the total density area equal to one.
(b) P(X>2)=1 / 4.
Integrate the normalised density over the required interval.
(c) E(X)=4 / 3.
Integrate x f(x) over the support.
Detailed marking criteria
Part a (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Writes int_0^4k(4-x),dx=1.
- Evaluates the integral as 8k=1.
- Obtains k=1 / 8.
Acceptable alternatives: For Q8(a), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part b (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Writes P(X>2)=int_2⁴(4-x) / 8,dx.
- Uses the antiderivative x / 2-x² / 16.
- Evaluates the bounds to obtain 1 / 4.
Acceptable alternatives: For Q8(b), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part c (4 marks)
Award all 4 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Uses E(X)=int_0^4x f(x),dx.
- Substitutes f(x)=(4-x) / 8.
- Evaluates frac18int_0⁴(4x-x²),dx.
- Obtains E(X)=4 / 3.
Acceptable alternatives: For Q8(c), award all 4 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Question 9
(a) hat p=0.60.
Average the two endpoints.
(b) The margin of error is 0.06.
Use half the interval width.
(c) The interval does not establish the claim because it contains plausible values from 0.54 to 0.55, which are not above 0.55.
Compare the whole interval with the claimed threshold and use appropriately cautious confidence language.
Detailed marking criteria
Part a (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Adds the endpoints 0.54+0.66.
- Divides the sum by 2.
- Obtains the centre hat p=0.60.
Acceptable alternatives: For Q9(a), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part b (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Calculates the interval width 0.66-0.54=0.12.
- Divides the width by 2.
- Obtains a margin of error of 0.06.
Acceptable alternatives: For Q9(b), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part c (4 marks)
Award all 4 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Identifies the threshold proportion 0.55.
- Observes that the lower endpoint 0.54 is below 0.55.
- States that the interval therefore contains plausible values not exceeding 0.55.
- Concludes that the 95% interval does not establish the claim.
Acceptable alternatives: For Q9(c), award all 4 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Question 10
(a) P(0)=100 and lim_(ttoinfty)P(t)=1000.
Evaluate the model at zero and use exponential limiting behaviour.
(b) t=ln9 / 0.4approx5.49.
Set the logistic model to half its limiting value and solve.
(c) Growth accelerates before P=500, then slows; 1000 is the carrying capacity. A changed environment or population mechanism may invalidate long-term extrapolation.
Connect the S-shape to carrying capacity and qualify the model.
Detailed marking criteria
Part a (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Do not award the final-result mark when the stated mathematical form is incomplete.
Mark-by-mark evidence:
- Substitutes t=0 to obtain P(0)=1000 / (1+9)=100.
- Uses e^(-0.4t)to0 as ttoinfty.
- Obtains the limiting population 1000.
Acceptable alternatives: For Q10(a), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part b (3 marks)
Award all 3 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. For the final-result mark, carry unrounded values until the final stated accuracy.
Mark-by-mark evidence:
- Sets 1000 / (1+9e^(-0.4t))=500.
- Rearranges to e^(-0.4t)=1 / 9.
- Obtains t=ln9 / 0.4approx5.49.
Acceptable alternatives: For Q10(b), award all 3 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Part c (4 marks)
Award all 4 marks for a complete mathematically correct response to the task. The numbered statements are indicative evidence for one approach, not a compulsory calculation sequence. Award equivalent marks for a valid alternative method that establishes the requested result and justification. For incomplete responses, credit valid progress proportionately using the indicative one-mark evidence or equivalent reasoning. Apply consequential marking when a correct later method consistently uses an earlier incorrect value, unless the error materially simplifies the task. For the final-result mark, include the requested units, context or justification.
Mark-by-mark evidence:
- Identifies the initial accelerating phase of the S-shaped model.
- Identifies the slowing phase after approximately half capacity.
- Interprets 1000 as the carrying capacity.
- Explains that environmental or mechanism changes can invalidate long-term extrapolation.
Acceptable alternatives: For Q10(c), award all 4 marks for a mathematically complete alternative solution. Do not require steps belonging only to the displayed method. Retain any method, exact form, accuracy, units or justification explicitly requested in the question.
Do not credit by itself: Using an invalid relationship or leaving the requested result incomplete. Stating a conclusion without the working or justification requested by the task.
Diagnostic Checklist
| Topic | Questions | Marks | Marks Lost | Action |
|---|---|---|---|---|
| Unit 4 — Discrete random variables — Calculate a binomial expectation | Q6(a) | 3 | ___ | Q6(a): solve the task independently using a valid method. Indicative evidence for one approach: Identifies n=20 and p=0.30; Uses E(X)=np; Obtains E(X)=20(0.30)=6. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 4 — Discrete random variables — Calculate a binomial variance | Q6(b) | 3 | ___ | Q6(b): solve the task independently using a valid method. Indicative evidence for one approach: Uses operatorname(Var)(X)=np(1-p); Substitutes 20(0.30)(0.70); Obtains operatorname(Var)(X)=4.2. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 4 — Discrete random variables — Evaluate a binomial point probability | Q6(c) | 4 | ___ | Q6(c): solve the task independently using a valid method. Indicative evidence for one approach: Selects the binomial coefficient binom(20)(6); Uses the success factor (0.30)⁶; Uses the failure factor (0.70)¹⁴; Combines the factors and rounds the probability to 0.1916. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 4 — Continuous random variables and normal distribution — Standardise a normal observation | Q7(a) | 3 | ___ | Q7(a): solve the task independently using a valid method. Indicative evidence for one approach: Uses z=(x-mu) / sigma; Substitutes x=130, mu=100 and sigma=15; Obtains z=(130-100) / 15=2. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 4 — Continuous random variables and normal distribution — Convert a standard score to an upper-tail probability | Q7(b) | 3 | ___ | Q7(b): solve the task independently using a valid method. Indicative evidence for one approach: Recognises X>130 as Z>2; Uses P(Z>2)=1-P(Z<2); Obtains 1-0.9772=0.0228. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 4 — Continuous random variables and normal distribution — Find a symmetric central normal probability | Q7(c) | 4 | ___ | Q7(c): solve the task independently using a valid method. Indicative evidence for one approach: Standardises 85 and 115 to z=-1 and z=1; Uses symmetry to write P(-1<Z<1)=2P(Z<1)-1; Substitutes 2(0.8413)-1; Obtains 0.6826. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 4 — Continuous random variables and normal distribution — Normalise a continuous density | Q8(a) | 3 | ___ | Q8(a): solve the task independently using a valid method. Indicative evidence for one approach: Writes int_0^4k(4-x),dx=1; Evaluates the integral as 8k=1; Obtains k=1 / 8. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 4 — Continuous random variables and normal distribution — Integrate a density over a tail interval | Q8(b) | 3 | ___ | Q8(b): solve the task independently using a valid method. Indicative evidence for one approach: Writes P(X>2)=int_2⁴(4-x) / 8,dx; Uses the antiderivative x / 2-x² / 16; Evaluates the bounds to obtain 1 / 4. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 4 — Continuous random variables and normal distribution — Calculate the expectation of a continuous random variable | Q8(c) | 4 | ___ | Q8(c): solve the task independently using a valid method. Indicative evidence for one approach: Uses E(X)=int_0^4x f(x),dx; Substitutes f(x)=(4-x) / 8; Evaluates frac18int_0⁴(4x-x²),dx; Obtains E(X)=4 / 3. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 4 — Interval estimates for proportions — Recover a sample proportion from interval endpoints | Q9(a) | 3 | ___ | Q9(a): solve the task independently using a valid method. Indicative evidence for one approach: Adds the endpoints 0.54+0.66; Divides the sum by 2; Obtains the centre hat p=0.60. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 4 — Interval estimates for proportions — Recover a margin of error from an interval | Q9(b) | 3 | ___ | Q9(b): solve the task independently using a valid method. Indicative evidence for one approach: Calculates the interval width 0.66-0.54=0.12; Divides the width by 2; Obtains a margin of error of 0.06. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 4 — Interval estimates for proportions — Use a confidence interval to assess a threshold claim | Q9(c) | 4 | ___ | Q9(c): solve the task independently using a valid method. Indicative evidence for one approach: Identifies the threshold proportion 0.55; Observes that the lower endpoint 0.54 is below 0.55; States that the interval therefore contains plausible values not exceeding 0.55; Concludes that the 95% interval does not establish the claim. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 2 — Exponential functions — Identify initial value and carrying capacity | Q10(a) | 3 | ___ | Q10(a): solve the task independently using a valid method. Indicative evidence for one approach: Substitutes t=0 to obtain P(0)=1000 / (1+9)=100; Uses e^(-0.4t)to0 as ttoinfty; Obtains the limiting population 1000. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 3 — The logarithmic function — Solve a logistic threshold equation | Q10(b) | 3 | ___ | Q10(b): solve the task independently using a valid method. Indicative evidence for one approach: Sets 1000 / (1+9e^(-0.4t))=500; Rearranges to e^(-0.4t)=1 / 9; Obtains t=ln9 / 0.4approx5.49. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |
| Unit 2 — Exponential functions — Interpret and critique a logistic model | Q10(c) | 4 | ___ | Q10(c): solve the task independently using a valid method. Indicative evidence for one approach: Identifies the initial accelerating phase of the S-shaped model; Identifies the slowing phase after approximately half capacity; Interprets 1000 as the carrying capacity; Explains that environmental or mechanism changes can invalidate long-term extrapolation. Check that your method establishes the requested result and justification; these steps are not compulsory for an alternative method. |