Learn with Amar
Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Compute a goodness-of-fit statistic from expected counts.
- Justify the conclusion "χ²=0.8, with two degrees of freedom for this fixed-probability model" using the stated assumptions.
Before you start
Categorical counts and squared differences.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A model predicts three equally likely categories. In thirty independent trials the counts are 8,12,10. Compute Pearson's chi-square statistic.
Why this math matters
Compute a goodness-of-fit statistic from expected counts. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Trials are independent with the stated fixed category probabilities under the null model.
- Expected counts are ten each, supporting the usual large-sample chi-square approximation if a tail probability is later used.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Compare observed categorical counts with a specified model
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A model predicts three equally likely categories. In thirty independent trials the counts are 8,12,10. Compute Pearson's chi-square statistic.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Expected counts are 10,10,10
Multiply each category probability one third by the total thirty.
Work through the mathematics
χ²=(8−10)²/10+(12−10)²/10+(10−10)²/10
Each squared discrepancy is scaled by its expected count.
Check the conclusion
χ²=0.8, with two degrees of freedom for this fixed-probability model
The counts sum to a fixed total, leaving one fewer free category discrepancy; this statistic alone is not a posterior probability.
The result
χ²=0.8, with two degrees of freedom for this fixed-probability model
The counts sum to a fixed total, leaving one fewer free category discrepancy; this statistic alone is not a posterior probability.
Common mistakes to catch
- Estimated model parameters can change the degrees of freedom.
- Observed counts, expected counts, and category probabilities have different roles.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Would changing the category order alter the statistic?
Show a hint
The same three terms are still summed.
Reveal answer and explanation
No
Reordering labels does not change the discrepancy total.
Practice 2
Why not interpret a small statistic as proof the model is true?
Show a hint
Many models can be compatible with one sample.
Reveal answer and explanation
It only indicates limited discrepancy under this measure
Lack of strong evidence against a model does not establish exact truth.
Take the idea with you
Compute and label expected counts before using a categorical model-checking statistic.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
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