Mathematics Applications & Interpretation (AI) IA Exemplar: Female… | RevisionDojo

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IB Mathematics Applications & Interpretation (AI) SL Internal Assessment Example

my aim with this paper is to find if the observed number of female government officials is equivalent to the expected number of female officials in Congress and representative of the female population in the United States of America.SL

Official IB Result

11/20

General feedback

11/20

No overall summary is available for this report.

6.1·Weakness

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The introductory motivation is engaging, but some of the political background stretches beyond what is needed to support the mathematical investigation. Trimming a few of these broader comments would make the exploration more concise and keep the focus on the statistical question.

6.2·Suggestion

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The student could show even more independence by briefly justifying why the expected distribution is split equally across women and men in each chamber. A short rationale here would make the modelling choice feel more deliberate and personally reasoned.

6.3·Strength

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The student demonstrates initiative by sourcing 2020 census data and using a real population figure rather than relying on a vague estimate. That choice gives the investigation a more authentic evidence base and shows the student is making independent methodological decisions.

6.4·Strength

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The division into clearly labeled trials and the use of appendices for tables make the exploration easy to navigate. This structure supports the reader by separating the core argument from supporting calculations.

6.5·Weakness

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This reflective comment is more motivational than analytical. The student should move beyond broad statements about hope and progress by asking whether the model itself is the best way to measure change, or whether the expected values should be adjusted to reflect population proportions.

6.6·Weakness

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The method for Trial Two repeats the Trial One setup almost verbatim. The student could save space and improve conciseness by stating that the same hypotheses and procedure are being reused, then only listing the changed observed values and new result.

6.7·Suggestion

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Including a brief note explaining why a second trial was necessary would make the personal approach even stronger. The student could connect the updated data to a new civic question, which would show purposeful initiative rather than just repetition.

6.8·Question

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What would change in the conclusion if the expected distribution were based on the female share of the population rather than a 50/50 split? Considering that alternative model would deepen the analysis of whether the conclusion is driven by representation or by the choice of expectation.

6.9·Weakness

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The conclusion is directionally reasonable, but it remains broad. The student should tie the final claim more directly to the statistical evidence by naming the test outcome, the implication of the large chi-square value, and what that says about the size of the gap between observed and expected counts.

6.10·Suggestion

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A stronger conclusion would briefly acknowledge one methodological limitation and one implication. For example, the student could note that the choice of expected proportions affects the interpretation, then state how that shapes confidence in the conclusion about underrepresentation.

Criteria A: Presentation

3/4

Criteria Strands

A.1Coherence and logical development

Good

A.2Organization and structure

Good

A.3Conciseness and relevance

Moderate

Criteria Feedback

Your exploration is clearly structured with a logical progression from context to method, analysis, and conclusion.
Your headings, tables, and appendices make the work easy to navigate.
You keep the investigation mostly focused on the topic, and the key statistical steps are easy to locate.
You maintain a readable format that supports the reader throughout the investigation.

Some sections repeat the same ideas and definitions, which makes the work longer than necessary.
The context sometimes expands into broader commentary that is only loosely connected to the mathematics.
The decision-making section could be more streamlined and explicit.
A few sections would benefit from tighter transitions so the argument feels more seamless.

1.1·Strength

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The exploration is clearly structured into purpose, method, two trials, and conclusion. This makes the line of argument easy to follow and gives the mathematical process a logical progression from question to evidence to interpretation.

1.2·Weakness

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The transition from context to method could be tighter: the discussion of why a chi-square goodness-of-fit test is appropriate is useful, but it is repeated later with very similar wording. Streamlining this explanation once would improve conciseness without losing meaning.

1.3·Weakness

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The interpretation line still confuses the test statistic with the significance level. The student should separate

χ^{2}

, the p-value, and the chosen significance level

α = 0.05

, then state the decision using the correct comparison. This is important for the logical development of the argument.

Criteria B: Mathematical Communication

2/4

Criteria Strands

B.1Mathematical language and notation

Good

B.2Multiple representations

Moderate

B.3Clarity and consistency

Moderate

Criteria Feedback

You use appropriate statistical notation such as hypotheses, chi-square notation, degrees of freedom, and observed/expected tables.
Your calculations are presented in a format that is generally easy to read and follow.
You consistently reuse the same mathematical structure across both trials, which helps the reader track the process.
Your definition of degrees of freedom is shown clearly.

You sometimes use statistical terms imprecisely, especially when discussing chi-square, p-values, and significance.
The communication would be clearer if you separated the test statistic from the decision rule.
There are a few wording and labeling issues, including a heading typo.
The work would benefit from more varied mathematical representations, not just tables and formulas.

2.1·Weakness

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The explanation of p-values is still not accurate enough: the student says the p-value determines the significance of the test statistic, but later treats the test statistic itself as the significance level. The terminology needs to be kept distinct so the inference is mathematically valid.

2.2·Strength

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The student communicates the degrees-of-freedom step clearly and uses the formula correctly:

df = (4 - 1) (2 - 1) = 3

. This explicit working helps the reader follow the statistical procedure and shows good mathematical communication.

2.3·Suggestion

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It would strengthen communication to add a short sentence showing the full decision rule after the formula, for example: compare the calculated p-value with

α = 0.05

, or compare

χ^{2}

with the critical value for

df = 3

. That would make the inference chain complete.

2.4·Strength

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The student continues to present the formula and variable meanings in a readable way, which supports clarity for the reader. Repeating the notation consistently between trials helps make the method easy to track.

2.5·Weakness

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The heading typo "Trail Two" weakens precision in a section where terminology should be exact. Correcting it to "Trial Two" would improve consistency and avoid distracting the reader from the mathematics.

Criteria C: Personal Engagement

2/3

Criteria Strands

C.1Independent thinking

Good

C.2Personal approach

Excellent

C.3Creativity and initiative

Good

Criteria Feedback

You chose a topic that clearly matters to you and that gives the investigation a real purpose.
You show independent thinking by shaping the question around a concrete issue rather than using a generic example.
You took initiative by gathering your own data and extending the investigation to a second trial.
Your personal interest gives the work a sense of direction and ownership.

The mathematical approach stays fairly conventional once the topic is chosen.
There is limited exploration of alternative modelling choices or deeper statistical questions.
The work would feel even more original if you justified and tested different expected distributions.
Some opportunities for creative statistical investigation were not developed.

3.1·Strength

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The student shows clear independent thinking by choosing a real issue connected to lived experience and civic concern. Framing the investigation around women in Congress gives the mathematics a genuine purpose, which strengthens personal engagement and makes the research question feel self-directed rather than assigned.

Criteria D: Reflection

1/3

Criteria Strands

D.1Depth of reflection

Good

D.2Critical analysis

Poor

D.3Connection to understanding

Good

Criteria Feedback

You do include some reflection on what the results suggest over time.
You acknowledge at least one limitation in your data or approach.
You make an effort to connect the statistical outcome back to the real-world issue you are studying.

Your reflection does not yet engage deeply with the statistical choices behind the model.
The discussion of what the result means is often broad rather than critically analytical.
You need to reflect more carefully on how the expected distribution affects the conclusion.
The interpretation would be stronger if you discussed the mechanics of the test rather than only the general social meaning.

4.1·Strength

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The student shows some awareness of methodological planning by explaining that the data must be reorganized before the second test. That indicates an understanding that statistical analysis depends on how the information is structured, not just on the final answer.

4.2·Weakness

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The student says the p-value "continued to be .05," but this is not supported by the calculation shown and appears to conflate the significance level with the result of the test. Reflection should explicitly use the computed p-value and explain what it means in context.

4.3·Strength

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The comparison between Trial 1 and Trial 2 shows that the student is thinking about change over time rather than treating the second calculation as isolated. Recognising that the statistic decreased is a useful first step toward deeper reflection on whether representation is improving.

Criteria E: Use of Mathematics

3/6

Criteria Strands

E.1Relevance and level

Moderate

E.2Accuracy and correctness

Moderate

E.3Knowledge and understanding

Moderate

Criteria Feedback

You apply the chi-square goodness-of-fit method appropriately to a relevant real-world question.
Your chi-square calculations are mostly correct and clearly set out.
You show a good understanding of observed versus expected frequencies and the degrees-of-freedom calculation.
The mathematics you use is appropriate for the level of the course and relevant to the investigation.

Your interpretation of the hypothesis test is not fully correct, especially when discussing chi-square, p-values, and significance.
You do not clearly justify why the chosen expected distribution is the right model for the question.
The final inference would be stronger if you used the correct decision rule throughout.
There is no deeper statistical extension beyond the standard test procedure.

5.1·Weakness

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The note that raw data must be compiled in a specific way is helpful, but the analysis does not yet explain why those expected values are appropriate for a goodness-of-fit model. Without that justification, the model risks looking imposed rather than derived from the research question.

5.2·Suggestion

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The student could strengthen the method by adding a short line explaining how the expected frequency 50/50 split was chosen. Since the investigation is about representation, linking expected counts to the modelling assumption would make the mathematics more defensible.

5.3·Weakness

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The inferential step is still incorrect:

χ^{2}

should not be described as the significance level, and it should not be compared directly with 0.05. The student needs to compute or cite the p-value for

df = 3

, or compare

χ^{2}

to the appropriate critical value, before rejecting or failing to reject

H_{0}

5.4·Strength

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The calculation of

χ^{2}

is set out clearly and the observed and expected values are substituted correctly. This shows the student understands the structure of the goodness-of-fit formula and can apply it to a four-cell dataset.

5.5·Question

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How would the chi-square result change if the expected counts were based on the 2024 female population proportion rather than equal halves? Exploring that would help test whether the conclusion depends on the modelling choice or on the data itself.

5.6·Weakness

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The statement "102 > 0.05" is not a valid statistical decision rule. The correct comparison is either

p < 0.05

χ^{2}

greater than the critical value. This is the main correctness issue left in the investigation.

Loading scores...

Official IB Result

11/20

General feedback

11/20

No overall summary is available for this report.

6.1·Weakness

Page 3• Click to view

6.2·Suggestion

Page 5• Click to view

6.3·Strength

Page 5• Click to view

6.4·Strength

Page 6• Click to view

6.5·Weakness

Page 9• Click to view

6.6·Weakness

Page 9• Click to view

6.7·Suggestion

Page 9• Click to view

6.8·Question

Page 11• Click to view

6.9·Weakness

Page 12• Click to view

6.10·Suggestion

Page 12• Click to view

Criteria A: Presentation

3/4

Criteria Strands

A.1Coherence and logical development

Good

A.2Organization and structure

Good

A.3Conciseness and relevance

Moderate

Criteria Feedback

Your exploration is clearly structured with a logical progression from context to method, analysis, and conclusion.
Your headings, tables, and appendices make the work easy to navigate.
You keep the investigation mostly focused on the topic, and the key statistical steps are easy to locate.
You maintain a readable format that supports the reader throughout the investigation.

Some sections repeat the same ideas and definitions, which makes the work longer than necessary.
The context sometimes expands into broader commentary that is only loosely connected to the mathematics.
The decision-making section could be more streamlined and explicit.
A few sections would benefit from tighter transitions so the argument feels more seamless.

1.1·Strength

Page 4• Click to view

1.2·Weakness

Page 5• Click to view

1.3·Weakness

Page 8• Click to view

The interpretation line still confuses the test statistic with the significance level. The student should separate

χ^{2}

, the p-value, and the chosen significance level

α = 0.05

, then state the decision using the correct comparison. This is important for the logical development of the argument.

Criteria B: Mathematical Communication

2/4

Criteria Strands

B.1Mathematical language and notation

Good

B.2Multiple representations

Moderate

B.3Clarity and consistency

Moderate

Criteria Feedback

You use appropriate statistical notation such as hypotheses, chi-square notation, degrees of freedom, and observed/expected tables.
Your calculations are presented in a format that is generally easy to read and follow.
You consistently reuse the same mathematical structure across both trials, which helps the reader track the process.
Your definition of degrees of freedom is shown clearly.

You sometimes use statistical terms imprecisely, especially when discussing chi-square, p-values, and significance.
The communication would be clearer if you separated the test statistic from the decision rule.
There are a few wording and labeling issues, including a heading typo.
The work would benefit from more varied mathematical representations, not just tables and formulas.

2.1·Weakness

Page 7• Click to view

2.2·Strength

Page 7• Click to view

The student communicates the degrees-of-freedom step clearly and uses the formula correctly:

df = (4 - 1) (2 - 1) = 3

. This explicit working helps the reader follow the statistical procedure and shows good mathematical communication.

2.3·Suggestion

Page 8• Click to view

It would strengthen communication to add a short sentence showing the full decision rule after the formula, for example: compare the calculated p-value with

α = 0.05

, or compare

χ^{2}

with the critical value for

df = 3

. That would make the inference chain complete.

2.4·Strength

Page 10• Click to view

2.5·Weakness

Page 11• Click to view

Criteria C: Personal Engagement

2/3

Criteria Strands

C.1Independent thinking

Good

C.2Personal approach

Excellent

C.3Creativity and initiative

Good

Criteria Feedback

You chose a topic that clearly matters to you and that gives the investigation a real purpose.
You show independent thinking by shaping the question around a concrete issue rather than using a generic example.
You took initiative by gathering your own data and extending the investigation to a second trial.
Your personal interest gives the work a sense of direction and ownership.

The mathematical approach stays fairly conventional once the topic is chosen.
There is limited exploration of alternative modelling choices or deeper statistical questions.
The work would feel even more original if you justified and tested different expected distributions.
Some opportunities for creative statistical investigation were not developed.

3.1·Strength

Page 3• Click to view

Criteria D: Reflection

1/3

Criteria Strands

D.1Depth of reflection

Good

D.2Critical analysis

Poor

D.3Connection to understanding

Good

Criteria Feedback

You do include some reflection on what the results suggest over time.
You acknowledge at least one limitation in your data or approach.
You make an effort to connect the statistical outcome back to the real-world issue you are studying.

Your reflection does not yet engage deeply with the statistical choices behind the model.
The discussion of what the result means is often broad rather than critically analytical.
You need to reflect more carefully on how the expected distribution affects the conclusion.
The interpretation would be stronger if you discussed the mechanics of the test rather than only the general social meaning.

4.1·Strength

Page 9• Click to view

4.2·Weakness

Page 11• Click to view

4.3·Strength

Page 11• Click to view

Criteria E: Use of Mathematics

3/6

Criteria Strands

E.1Relevance and level

Moderate

E.2Accuracy and correctness

Moderate

E.3Knowledge and understanding

Moderate

Criteria Feedback

You apply the chi-square goodness-of-fit method appropriately to a relevant real-world question.
Your chi-square calculations are mostly correct and clearly set out.
You show a good understanding of observed versus expected frequencies and the degrees-of-freedom calculation.
The mathematics you use is appropriate for the level of the course and relevant to the investigation.

Your interpretation of the hypothesis test is not fully correct, especially when discussing chi-square, p-values, and significance.
You do not clearly justify why the chosen expected distribution is the right model for the question.
The final inference would be stronger if you used the correct decision rule throughout.
There is no deeper statistical extension beyond the standard test procedure.

5.1·Weakness

Page 6• Click to view

5.2·Suggestion

Page 7• Click to view

5.3·Weakness

Page 8• Click to view

The inferential step is still incorrect:

χ^{2}

should not be described as the significance level, and it should not be compared directly with 0.05. The student needs to compute or cite the p-value for

df = 3

, or compare

χ^{2}

to the appropriate critical value, before rejecting or failing to reject

H_{0}

5.4·Strength

Page 8• Click to view

The calculation of

χ^{2}

5.5·Question

Page 10• Click to view

5.6·Weakness

Page 11• Click to view

The statement "102 > 0.05" is not a valid statistical decision rule. The correct comparison is either

p < 0.05

χ^{2}

greater than the critical value. This is the main correctness issue left in the investigation.