Mini-Lesson

HL Statistics

This HL-only mini-lesson covers the extra statistics HL demands: the Poisson distribution, the χ² goodness-of-fit and independence tests, the t-test for comparing means, and the long-run behaviour of Markov chains (steady states) — with careful hypothesis-test interpretation.

AI HL flavour: you choose the right test, run it on your GDC, and state a conclusion in context with the significance level.

Work through each screen, answer the questions as you go (some are wordy, some are calculations) and collect ⭐ stars. Every number on the calculation screens has been re-derived and checked. Press Start when you are ready.

Poisson

The Poisson distribution

The Poisson distribution models the number of random events in a fixed interval, with mean (and variance) λ:

P(X = r) = e^−λ λ^r ⁄ r!mean = variance = λ

Worked example

λ = 3 calls per hour.

P(X = 2) = e⁻³ × 3² ⁄ 2! = 0.2240

P(X ≤ 1) = e⁻³(1 + 3) = 0.1991

Calculate

Your turn — Poisson

1For X ~ Poisson(λ = 3), find P(X = 2) to 4 decimal places using e^(−λ) λ^r ⁄ r!.

Hint: e^(−3) × 3² ⁄ 2!.

Quick check

When Poisson?

?The Poisson distribution is appropriate when events:

χ² tests

Goodness-of-fit & independence

Both χ² tests use χ²calc = Σ (O − E)² ⁄ E and compare to a critical value.

Independence (contingency table): df = (r − 1)(c − 1).
Goodness-of-fit: df = (categories − 1) − (parameters estimated).

Worked example

2×3 independence table: χ²calc = 16.67, df = 2, critical 5.991.

16.67 > 5.991 ⇒ reject H₀ (the variables are associated).

Calculate

Your turn — χ² statistic

2A 2 × 3 contingency table gives Σ(O − E)²⁄E = 16.67 (checked). For a 3 × 3 table, state the degrees of freedom (r − 1)(c − 1).

Hint: (3 − 1)(3 − 1).

t-test

Comparing two means

A two-sample t-test compares the means of two groups. State H₀: μ₁ = μ₂ and H₁ (one- or two-tailed), run the test on your GDC, and read the p-value.

Decision rule: if p < significance level (e.g. 0.05), reject H₀ — there is evidence the means differ. Otherwise there is insufficient evidence.

Calculate

Your turn — t-test decision

3A two-sample t-test at the 5% level returns p = 0.03. Enter 1 if you reject H₀ (means differ), or 0 if you do not.

Hint: reject H₀ when p is less than 0.05.

Quick check

Reading a p-value

?A t-test gives p = 0.20 at the 5% significance level. You should:

Markov chains

Steady-state distributions

A Markov chain with transition matrix T settles to a steady state s with Ts = s and components summing to 1.

Worked example

T = [[0.8, 0.3], [0.2, 0.7]].

Solve: 0.8s₁ + 0.3s₂ = s₁ with s₁ + s₂ = 1 ⇒ s₁ = 0.3⁄0.5 = 0.6, s₂ = 0.4

Calculate

Your turn — steady state

4For T = [[0.8, 0.3], [0.2, 0.7]], the steady state (s₁, s₂) satisfies Ts = s and s₁ + s₂ = 1. Find s₁ to 1 decimal place.

Hint: 0.2 s₁ = 0.3 s₂ and s₁ + s₂ = 1 ⇒ s₁ = 0.3 ⁄ 0.5.

Poisson tail

Cumulative Poisson

Cumulative Poisson probabilities add the individual terms (GDC poissoncdf).

Worked example

λ = 3: P(X ≤ 1) = e⁻³(1 + 3) = 0.1991

Calculate

Your turn — cumulative Poisson

5For X ~ Poisson(λ = 3), find P(X ≤ 1) to 4 decimal places.

Hint: e^(−3)(1 + 3).

Sort it

Choose the test / model

Tap a scenario, then the correct statistical tool.

📞 Poisson

🧪 χ² test

⚖️ t-test

Quick check

Poisson property

?A key property of the Poisson distribution is that its mean and variance are:

Match it

Match the tool to its use

Tap an item on the left, then its match on the right.

Item

Match

Choosing a test

Which test, which distribution?

Match the tool to the data: counts of rare events → Poisson; two categorical variables → χ² independence; comparing two group means → t-test; fit to a claimed distribution → χ² goodness-of-fit.

Always state H₀ and H₁, the significance level, and interpret the p-value in context.

Quick check

One- vs two-tailed

?A researcher tests only whether a new method gives a higher mean than the old one. This calls for a:

Recap

The big ideas to take away

Poisson: e^(−λ)λ^r ⁄ r!; mean = variance = λ; random events at rate λ

χ² tests: independence df=(r−1)(c−1); goodness-of-fit df=(k−1)−params

t-test: compare two means; reject H₀ if p < significance level

Markov: steady state Ts = s, components sum to 1

Interpretation: always state H₀/H₁ and conclude in context

GDC: poissonpdf/cdf, χ²-test, 2-Samp t-test

You have worked through the whole topic. Press Finish to see your score.

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Mini-lesson complete!

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