Mathematics Methods 3&4: Exploring Probability Distributions

This Investigation Consists of the Take-Home Part Only to Be Completed Out of Class

You are to produce a digital and/or handwritten Investigation Report that shows how you have applied general inquiry skills, course-related knowledge and skills, and modelling skills. The planning, research, execution and communication of the findings of your investigation will be included in your Investigation Report. Including snips from websites and photos of your ClassPad screen in your report to show your research, analysis and calculations is recommended.

Evaluation of your investigation report will be based on the criteria described in the rubric on page 3 which, amongst other things, considers the Mathematical Thinking Process as follows.

The Mathematical Thinking Process

The mathematical thinking process is a problem-solving process that consists of a sequence of steps that guide the approach to finding a solution to a problem. One description of the mathematical thinking process includes the following steps:

• clarifying and interpreting the task, identifying the key information and any relevant assumptions
• choosing the course-related mathematics which could help to investigate, model or solve the problem
• using existing mathematical knowledge and strategies to explore or model the problem to obtain a set of observations or a solution
• interpreting the results in relation to the original mathematical situation or context
• communicating the generalisation of the findings or the solution to the problem.

120 results from a probabilistic experiment were collected and are shown in the following table.
The model of this probabilistic experiment is known to be one of continuous Uniform, Binomial or Normal distribution.

Your task is to:

• use a random number generator website to randomly select 20 of the 120 numbers of the table above.
• enter your randomly selected sample of 20 numbers into list1 of the Statistics Application of your ClassPad and analyse the data with the aim of deciding, with reasons , which type of distribution (continuous uniform, binomial or normal) would be best suited to model the data and which distribution would be the least suited to model the data.

Your investigation should consider the following for your sample of 20 numbers

• the shape of distribution
• the mean, standard deviation and the parameters for the uniform (a and b) and binomial (n and p) distributions
• the distribution of probability from the mean

Note:

• For a continuous Uniform distribution defined for a ≤ x ≤ b , the parameters a and b .
• For a Binomial distribution defined for 0 ≤ x ≤ n the parameters n and p .

Investigation Assessment Criteria

Your investigation report will be assessed via the following criteria.

1. Introduction, Assumptions and Planning of the Investigation

Clearly defines the purpose of the investigation, states any assumptions made, explains the process to obtain a random sample, identifies key information, and outlines a well-structured plan to complete the investigation.

2. Application and Analysis of Mathematical Modelling 

Evidence of competence and sophistication in the application of mathematical knowledge and models to complete the investigation and the analysis of the success of the various mathematical models used.
Discussion and interpretation of progression and results, with supporting examples obtained from mathematical calculations/representations (graphs/tables/diagrams etc).
Communication of results obtained, including the appropriate use of mathematical conventions, symbols and terminology.

3. Reflection, Evaluation, and Conclusion 

Critically assess the entire investigation process, identifying both strengths and areas for improvement, evaluate the success and practicality of the various models used, summarise key findings and any relationships discovered and reflect on what you have learned mathematically (or otherwise) through the completion of this investigation.

Include references/bibliography listing websites and artificial intelligence used.

Summary of Assessment Requirements

The assessment required students to complete a digital and/or handwritten Investigation Report demonstrating their ability to apply general inquiry skills, course-related knowledge, and modelling techniques. The investigation was to be supported by planning, research, execution, and effective communication of findings. Key requirements included:

• Applying the Mathematical Thinking Process:

  • Clarifying and interpreting the task.
  • Identifying key information and assumptions.
  • Selecting suitable mathematical models.
  • Using knowledge and strategies to explore solutions.
  • Interpreting and communicating results.

• Conducting a probabilistic experiment by:

  • Using a random number generator to select 20 values from 120 collected results.
  • Analysing the data using ClassPad’s Statistics Application.
  • Determining which distribution (continuous uniform, binomial, or normal) best suited the data and which was least suited.
• Investigating through:
  • Shape of the distribution.
  • Mean, standard deviation, and parameters for uniform (a, b) and binomial (n, p).
  • Distribution of probability around the mean.

• Meeting Assessment Criteria:

  1. Introduction, Assumptions, and Planning.
  2. Application and Analysis of Mathematical Modelling.
  3. Reflection, Evaluation, and Conclusion.

Step-by-Step Guidance by Academic Mentor

The academic mentor guided the student through the assessment in the following structured approach:

1. Introduction, Assumptions and Planning

• The mentor explained the purpose of the investigation, ensuring the student clearly defined the task and stated assumptions.
• Guidance was provided on how to use the random number generator and document the sampling process.
• Together, they outlined a plan for analysis—starting with data collection, followed by model application, and ending with result interpretation.

2. Application and Analysis of Mathematical Modelling

• The mentor helped the student enter the sample into ClassPad and explore its statistical functions.
• Step-by-step, they calculated the mean, standard deviation, and distribution parameters.
• The student was guided to compare different distributions (uniform, binomial, normal) by examining graphs, histograms, and probability behaviour.
• The mentor ensured that the student used mathematical conventions, diagrams, and tables to present results clearly.

3. Reflection, Evaluation and Conclusion

• The mentor encouraged the student to evaluate the suitability of each distribution.
• The student reflected on the strengths and limitations of models and discussed why one distribution fit better than others.
• Finally, the student summarised findings, drew generalisations, and linked the results back to the original mathematical context.

Final Outcome and Learning Objectives Covered

• Outcome:
  The student successfully produced an investigation report that demonstrated logical inquiry, effective use of mathematical modelling, and clear communication of results. The report identified the most suitable and least suitable distributions, supported by statistical evidence.
• Learning Objectives Achieved:
  • Developed skills in data sampling and statistical analysis.
  • Applied modelling techniques to real data.
  • Strengthened ability to compare and evaluate mathematical models.
  • Enhanced critical thinking, problem-solving, and reflection skills.

  • Practiced academic presentation using graphs, tables, and structured reporting.

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