Goodness of fit: binomial and Poisson models

Poisson model

If λ is not given, estimate it by the sample mean: λ̂ = Σfx/Σf. For the calls data, λ̂ = 1.92.

Expected frequency for x = N × P(X = x). The last cell must be 'x or more' so that the expected frequencies total N: E(≥ 4) = N - (sum of the others).

ν = cells (after combining) - 1 - 1, the extra 1 because λ was estimated.

Calls to a help line in 100 one-minute intervals

Binomial model

With n trials per observation, estimate p by p̂ = x̄/n if p is not given. For the seeds, x̄ = 2.5875 so p̂ = 0.6469.

Expected frequencies: 80 × P(X = x) for X ~ B(4, p̂) = 1.24, 9.11, 25.05, 30.59, 14.01. The first is below 5, so combine x = 0 and x = 1.

80 trays, each with 4 seeds

Degrees of freedom: the common slips

• Parameter given (for example 'test whether B(4, 0.5) fits'): ν = cells - 1.
• Parameter estimated from the data: ν = cells - 2.
• Count cells after combining.

Examiners report candidates using 6 - 1 instead of 6 - 2 when p had been estimated, which can change the conclusion.

Worked example

Test at the 5% level whether a Poisson distribution is a suitable model for the calls data.

1. H0: a Poisson distribution is a suitable model; H1: it is not.
2. λ̂ = 1.92. E for 0, 1, 2, 3, ≥ 4: 14.66, 28.15, 27.02, 17.29, 12.87 (x = 4 and 5 combined into '≥ 4').
3. X² = 0.2276
4. ν = 5 - 1 - 1 = 3; critical value 7.815; 0.228 < 7.815.

Answer: Do not reject H0: the Poisson distribution is a suitable model for the number of calls per minute.

Check yourself

1. Using the calls data, estimate λ for a Poisson model.
   Show answer

   1.92. Σfx = 192, Σf = 100, λ̂ = 1.92.

2. A Poisson model with λ = 1.92 is fitted to 100 intervals. Find the expected frequency for x = 2. Give your answer to 2 decimal places.
   Show answer

   27.02. 100 × e^-1.92 × 1.92²/2! = 27.0226.

3. For the calls data, with x = 4 and 5 combined into '≥ 4' and expected frequencies 14.66, 28.15, 27.02, 17.29, 12.87, calculate X². Give your answer to 3 significant figures.
   Show answer

   0.228. Σ(O - E)²/E = 0.2276.

4. A Poisson model with λ estimated from the data is tested using 5 cells (after combining). How many degrees of freedom are there?
   Show answer

   3. 5 - 1 - 1 = 3: one for the total and one for the estimated λ.

5. Using the seed data, estimate p for a binomial B(4, p) model. Give your answer to 3 significant figures.
   Show answer

   0.647. x̄ = 207/80 = 2.5875, so p̂ = x̄/4 = 0.6469.

6. Test whether B(4, 0.5) fits the seed data. The expected frequencies are 5, 20, 30, 20, 5. Calculate X². Give your answer to 3 significant figures.
   Show answer

   34.1. 9/5 + 121/20 + 36/30 + 100/20 + 100/5 = 34.05.

7. For the B(4, 0.5) model, X² = 34.05 with 5 cells and p given. What is the conclusion at the 5% level?
   • Do not reject H0, since 34.05 < 9.488
   • Reject H0, since 34.05 > 9.488: B(4, 0.5) is not a suitable model
   • Reject H0, using ν = 3 and critical value 7.815, but only because p was estimated
   • No conclusion is possible because one expected frequency is exactly 5
   Show answer

   Reject H0, since 34.05 > 9.488: B(4, 0.5) is not a suitable model. p is given, so ν = 5 - 1 = 4 and the critical value is 9.488.

8. A binomial model with p estimated from the data has 5 cells; after combining the first two there are 4 cells. How many degrees of freedom are there?
   • 2
   • 3
   • 4
   • 1
   Show answer

   2. 4 cells - 1 (total) - 1 (estimated p) = 2.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S3
• Pearson Edexcel IAL Statistics S3 (WST03) examiners' report, June 2025
• Pearson Edexcel Mathematical Formulae and Statistical Tables

Last updated 8 November 2026. Spotted a mistake? Tell us.