IB DP Mathematics: Applications and Interpretation – Data collection, non-linear regression, hypothesis tests and Markov chains (HL) Study Guide

This study guide teaches one HL unit of IB Diploma Programme Mathematics: Applications and Interpretation. It is aligned to the IB Mathematics: applications and interpretation guide, first assessment 2021, and covers syllabus sections 4.12, 4.13, 4.18 and 4.19. All of this content is HL only (AHL). It follows the IB guide for first assessment 2021, which remains the examined syllabus until the new course is first assessed in May 2029, so it applies to the May and November 2026, 2027 and 2028 HL sessions.

At HL, Papers 1, 2 and 3 all require a GDC. You set problems up mathematically, calculate with technology and interpret in context.

Revise from the revision notes and test yourself with the practice questions. The IB DP Maths AI course hub and the printable syllabus checklist show where this unit sits. SL groundwork is in the statistics and probability guide.

What this unit covers

Section	What you must be able to do	Level
4.12	Design valid surveys; choose relevant variables and data; choose χ² categories with expected frequencies over 5; set degrees of freedom when parameters are estimated; explain reliability (test-retest, parallel forms) and validity (content, criterion-related)	HL only
4.13	Fit linear, quadratic, cubic, exponential, power and sine models with technology; use SSres and R² to judge fit	HL only
4.18	Use critical values and regions; test a normal mean (z or t), a binomial proportion, a Poisson mean and ρ = 0; find Type I and Type II error probabilities	HL only
4.19	Use transition matrices and diagrams, sₙ = Tⁿs₀, regular Markov chains and steady states	HL only

4.12 Designing valid data collection

A biased question pushes people towards one answer (“Don’t you agree the canteen is too expensive?”). An unbiased question is neutral (“How would you rate canteen prices?”). Personal questions (income, health) may cause refusals or dishonest answers, so ask them only if needed.

Unstructured questions are open (“What do you think of the bus service?”). They give rich answers that are hard to analyse. Structured questions give consistent answer choices, such as a five-point scale or ranges that do not overlap. Precise questions fix the time frame and units: “How many hours did you spend on homework last week?” not “Do you do a lot of homework?”

You must also pick the relevant variables from a large data set and choose data that answers the question.

Reliability and validity

• Reliability: the method gives consistent results when repeated.
  • Test-retest: give the same test to the same group at two different times and compare the results.
  • Parallel forms: give two equivalent versions of the test to the same group and compare the results.
• Validity: the method measures what it claims to measure.
  • Content validity: the questions cover every part of what is being measured.
  • Criterion-related validity: the results agree with an established measure (the criterion).

χ² goodness of fit with estimated parameters

For a χ² goodness of fit test, choose categories so that every expected frequency is greater than 5. Merge neighbouring categories if needed, and say why.

Degrees of freedom = (number of categories after merging) − 1 − (number of parameters estimated from the data).

Estimating a Poisson mean or a binomial p costs 1; estimating a normal μ and σ costs 2.

Worked example 1

A café counts customers arriving in 120 five-minute intervals.

Customers	0	1	2	3	4	5	6	7
Frequency	22	31	27	20	10	6	3	1

Test at the 5% level whether a Poisson model fits.

H₀: the data follow a Poisson distribution
H₁: the data do not follow a Poisson distribution
Sample mean = 240/120 = 2, so use Po(2)
Expected: 0: 16.24  1: 32.48  2: 32.48  3: 21.65  4: 10.83
          5: 120 × P(X = 5) = 4.33,  ≥6: 120 × P(X ≥ 6) = 1.99

The expected frequencies for 5 and for ≥ 6 are below 5, so merge them into “≥ 5”: expected 6.32, observed 10. There are now 6 categories and 1 parameter was estimated, so df = 6 − 1 − 1 = 4.

Technology gives χ² = 5.37 and p = 0.251. Since 0.251 > 0.05, do not reject H₀. There is not enough evidence that the Poisson model is unsuitable.

4.13 Non-linear regression

Your GDC fits least squares curves: linear, quadratic, cubic, exponential, power and sine models can all appear in examinations.

• The residual for a point is the observed y minus the value the model predicts.
• SSres, the sum of square residuals, measures fit. Smaller is better.
• The coefficient of determination R² is the proportion of the variability in y accounted for by the model. It lies between 0 and 1.
• For a linear model, R² = r², where r is Pearson’s product moment correlation coefficient.

R² alone is not a good way to choose between models. Adding terms (quadratic to cubic) never lowers R². Also ask whether the model makes sense in context and outside the data range.

Worked example 2 (SSres by hand)

Data: (1, 5), (2, 9), (3, 14), (4, 22). Compare model A: y = x² + x + 3 and model B: y = 3.2 × 1.6ˣ.

Model A predictions: 5, 9, 15, 23      residuals: 0, 0, −1, −1
SSres(A) = 0 + 0 + 1 + 1 = 2
Model B predictions: 5.12, 8.192, 13.107, 20.972
residuals: −0.12, 0.808, 0.893, 1.028
SSres(B) = 2.52 (3 s.f.)

Model A fits these four points better because its SSres is smaller.

Worked example 3 (technology)

The area y m² of an algae patch after x weeks:

x	1	2	3	4	5	6	7
y	3.1	4.0	5.6	8.2	11.5	16.0	21.9

• Linear: y = 3.08x − 2.29, R² = 0.930.
• Quadratic: y = 0.487x² − 0.813x + 3.56, R² = 0.999.

The quadratic explains 99.9% of the variation in area, against 93.0% for the line. It predicts 28.2 m² at week 8, a small extrapolation. A cubic gives R² even closer to 1, but that alone does not make it better.

4.18 Hypothesis testing

The critical value is the boundary of the critical region: the values of the test statistic that lead you to reject H₀.

• Type I error: rejecting H₀ when it is true.
• Type II error: accepting H₀ when it is false.

For a normal test with known σ, P(Type I) equals the significance level. For binomial and Poisson tests the variable is discrete. The critical region is chosen so that P(Type I) is as large as possible while staying below the significance level.

Which test?

Situation	Test
Normal mean, σ known	z-test using X̄ ~ N(μ, σ²/n)
Normal mean, σ unknown (any n)	t-test
Matched pairs	t-test on the differences (a single sample)
Proportion	binomial, one-tailed only
Poisson mean	Poisson, one-tailed only
Correlation, bivariate normal	test ρ = 0 with technology

You will not be asked to calculate critical regions for t-tests.

Worked example 4 (normal, σ known)

Bottles are filled with volumes N(μ, 4²) ml. The label claims μ = 500. A sample of 25 has x̄ = 498.3. Test at 5% whether μ < 500.

H₀: μ = 500    H₁: μ < 500
Under H₀, X̄ ~ N(500, 4²/25), so the standard deviation of X̄ is 0.8
z = (498.3 − 500)/0.8 = −2.125,  p = 0.0168

Since 0.0168 < 0.05, reject H₀. There is evidence that the mean volume is below 500 ml.

Critical region: z < −1.645, which gives x̄ < 500 − 1.645 × 0.8 = 498.68.

If in fact μ = 498, then P(Type II) = P(X̄ ≥ 498.684 | X̄ ~ N(498, 0.8²)) = 0.196.

Worked example 5 (matched pairs)

Eight runners’ scores before and after a training plan:

Before	62	55	71	48	66	59	74	52
After	66	58	70	55	71	60	79	57

Let d = after − before: 4, 3, −1, 7, 5, 1, 5, 5. H₀: μ_d = 0, H₁: μ_d > 0. σ is unknown, so use a one-sample t-test on d. Technology gives t = 4.01 and p = 0.00258 < 0.05. Reject H₀: there is evidence that scores improved.

Worked example 6 (binomial, with errors)

It is claimed that 30% of seeds fail. After a new treatment, 20 seeds are sown. Test at 5% whether the proportion has fallen.

H₀: p = 0.3    H₁: p < 0.3    X ~ B(20, 0.3) under H₀
P(X ≤ 2) = 0.0355 < 0.05     P(X ≤ 3) = 0.107 > 0.05
Critical region: X ≤ 2       P(Type I) = 0.0355

If the true proportion is 0.15, P(Type II) = P(X ≥ 3 | X ~ B(20, 0.15)) = 0.595.

Worked example 7 (Poisson)

A junction averaged 3.5 accidents per month. In the 8 months after a redesign there were 19. Test at 5% whether the rate has fallen.

Under H₀ the total is Y ~ Po(8 × 3.5) = Po(28). P(Y ≤ 19) = 0.0478 < 0.05, so reject H₀. The critical region is Y ≤ 19, because P(Y ≤ 20) = 0.0727 is too big.

Testing ρ = 0

For bivariate normal data, H₀: ρ = 0 and H₁: ρ > 0, ρ < 0 or ρ ≠ 0. The data will be given. Your GDC returns r and the p-value to compare with the significance level.

4.19 Markov chains

A transition matrix T holds the probabilities of moving between states. In IB notation Tᵢⱼ is the probability of moving from state j to state i, so each column sums to 1. A transition diagram shows the same information as arrows between states.

If s₀ is the initial state matrix (a column), then after n transitions sₙ = Tⁿs₀.

A Markov chain is regular if some power of T has all entries positive. A regular chain has a steady state s, with Ts = s and entries summing to 1. You find it by working out Tⁿs₀ for large n, or by solving the linear equations. The steady state is the eigenvector for eigenvalue 1, scaled so its entries add to 1. Questions will say when an exact answer from equations is required.

Worked example 8

Each week a gym member either attends (A) or does not (N). An attender attends the next week with probability 0.8. A non-attender attends with probability 0.3.

        from A  from N
T = [   0.8     0.3  ]  to A
    [   0.2     0.7  ]  to N

s₀ = (0.5, 0.5)ᵀ
s₁ = T s₀ = (0.55, 0.45)ᵀ
s₂ = T² s₀ = (0.575, 0.425)ᵀ

For the exact steady state, solve 0.8a + 0.3b = a with a + b = 1. This gives 0.2a = 0.3b, so a = 3/5 and b = 2/5. In the long run, 60% of members attend in a given week. Check: T¹⁰s₀ = (0.5999, 0.4001)ᵀ.

Using your GDC

• χ² goodness of fit: enter observed and expected lists, and change df yourself when you estimate parameters.
• Regression: quote parameters to 3 s.f. but predict with the stored full values.
• Tests: write down the hypotheses, the distribution and the p-value, not just “GDC”.

Common errors

• Merging because an observed frequency is small. The expected frequencies must exceed 5.
• Forgetting to subtract estimated parameters from df.
• Mixing up reliability (consistent) with validity (accurate).
• Comparing models by R² alone, or extrapolating a polynomial far outside the data.
• Using z when σ is unknown, even for large n.
• Writing a two-tailed binomial or Poisson test.
• Giving a critical region with P(Type I) just over the significance level.
• Building T with rows summing to 1, then multiplying in the wrong order.

Where to go next

Condense this with the revision notes, then try the practice questions. See also the exam preparation guide, syllabus guide and subject guide.

Official syllabus

International Baccalaureate Organization, Diploma Programme, Mathematics: applications and interpretation guide, first assessment 2021, syllabus sections AHL 4.12, 4.13, 4.18 and 4.19.