Understanding t-Tests

Overview

This 60-minute lesson is designed for Year 13 Maths students (aged 17-18) focusing on t-tests in the Statistics element of the A-level curriculum, specifically aligned with the National Curriculum for England and the AQA Mathematics specifications that underpin the Year 13 statistical inference content.

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National Curriculum Alignment

• Mathematics Programmes of Study (KS5) - A-Level

  • Statistical inference (AQA Specification, broadly referenced):
    • Understanding and performing hypothesis tests (including one-sample and two-sample t-tests)
    • Interpreting the results of inferential tests in context
    • Calculating and using test statistics and critical regions

• Learning objectives from national framework:

  • Apply t-tests to real data with understanding of assumptions
  • Analyse output from statistical tests
  • Use mathematical reasoning to interpret statistical inference results
  • Develop problem-solving and exam technique skills through targeted practice

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Learning Objectives by the end of the lesson

Students will be able to:

1. Explain the purpose and assumptions of one-sample and two-sample t-tests.
2. Calculate t-test statistics using sample data.
3. Carry out hypothesis tests using appropriate critical values and significance levels.
4. Interpret the results of t-tests in real-world contexts.
5. Demonstrate exam technique by tackling a variety of problem-solving questions.
6. Collaborate effectively to discuss solutions and clarify misunderstandings.

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Resources Required

• Whiteboard and markers
• Projector/slides with worked examples
• Student worksheets (exam-style questions tailored for t-tests)
• Statistical tables (t-distribution critical values)
• Calculators or spreadsheet software (optional for calculations)
• Timer for activities

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Lesson Structure (60 minutes)

Introduction (5 minutes)

• Starter question: "When might you want to compare the means of two sets of data? How do you determine if the difference is significant?"
• Review briefly what a t-test is and when it is used compared to other tests (e.g., z-test).
• Outline learning objectives.

Direct Instruction (15 minutes)

• Explain t-tests:

  • Types: one-sample, two-sample (independent), and paired t-tests
  • Assumptions: normality, independence, unknown variance
  • Setting null (H0) and alternative (H1) hypotheses, specifying one-tailed or two-tailed tests

• Step-by-step calculation of t-statistic

• Interpretation of results with reference to critical values from t distribution tables

• Clarify how to choose degrees of freedom and significance levels (typically 5% for A-level)

• Present a worked example on the board/projector: calculate and interpret a one-sample t-test

Guided Practice (15 minutes)

• Distribute a worksheet with varied exam-style questions:

  • Question 1: One-sample t-test on sample mean
  • Question 2: Two-sample independent t-test assuming equal variance
  • Question 3: Two-tailed test interpretation

• Students work individually or in pairs (maximising opportunity for peer discussion)

• Teacher circulates, offers support, and prompts students to justify each step explicitly

• Emphasis on writing conclusions in context, not just numerical answers

Group Discussion and Deepening Understanding (10 minutes)

• Review worksheet questions with the whole class.

• Select different students/groups to explain their answers and reasoning.

• Discuss common pitfalls:

  • Interpreting significance incorrectly
  • Confusing types of t-tests
  • Errors in degrees of freedom calculation

• Use a “challenge question” for extension: hypothesis test for difference of means with unequal variances (Welch’s t-test), briefly explaining context without going into full calculation (to challenge deeper thinking)

Exam Practice (10 minutes)

• Provide a mini mock exam question designed to replicate AQA A-level mark scheme style.
• Students complete the question under timed conditions on their own.
• Encourage showing full working, including stating hypotheses, calculating test statistic, critical region, and concluding.

Plenary and Assessment (5 minutes)

• Quick "exit slip": Students write down one thing they found easy and one point they wish to clarify about t-tests.
• Teacher addresses misconceptions identified from exit slips.
• Set homework: Extended worksheet of additional exam questions or online past paper questions (teacher to provide).

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Assessment and Feedback

• Formative assessment through worksheet completion and group discussions.
• Summative assessment through timed exam practice.
• Verbal immediate feedback during activities.
• Written feedback on homework to track progress.

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Differentiation and Inclusion

• Provide scaffolding prompts on worksheets for lower attainers (e.g., step-by-step guidance, formula reminders)
• Challenge higher attainers with extended questions involving paired t-tests or Welch’s test.
• Use visual aids to support learners with difficulty in abstract concepts.
• Ensure all students access statistical tables to aid calculation.

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Extension Ideas (for interested students)

• Discuss assumptions and limitations of t-tests using real datasets.
• Explore use of software (e.g., Excel, GraphPad) for t-tests.
• Investigate confidence intervals corresponding to hypothesis tests.
• Link to other inferential tests in the curriculum (chi-square, ANOVA).

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By the end of this lesson, students will be confident applying t-tests confidently, understanding their implications, and improving problem-solving skills required for succesful exam performance on this critical statistical inference topic.