EXM.1.AHL.TZ0.56

pestleMathematicsAIHLPaper 1EXM· sl-4-11-expected-observed-hypotheses-chi-squared-gof-t-testsource ↗

A zoologist believes that the number of eggs laid in the Spring by female birds of a certain breed follows a Poisson law. She observes 100 birds during this period and she produces the following table.

The zoologist wishes to determine whether or not a Poisson law provides a suitable model.

Calculate the mean number of eggs laid by these birds.

[2]

Write down appropriate hypotheses.

[2]

b.i.

Carry out a test at the 1% significance level, and state your conclusion.

[14]

b.ii.

Markscheme / solution

Mean $= \frac{{1 \times 19 + 2 \times 34 + \ldots + 5 \times 4}}{{100}}$ (M1)

$= 2.16$ A1 N2

[2 marks]

H₀ : Poisson law provides a suitable model A1

H₁ : Poisson law does not provide a suitable model A1

[2 marks]

b.i.

The expected frequencies are

A1A1A1A1A1A1

Note: Accept expected frequencies rounded to a minimum of three significant figures.

${\chi ^2} = \frac{{{{\left( {10 - 11.533} \right)}^2}}}{{11.533}} + \ldots + \frac{{{{\left( {4 - 6.824} \right)}^2}}}{{6.824}}$ (M1)(A2)

$= 5.35$ (accept 5.33 and 5.34) A2

$v = 4$ (6 cells − 2 restrictions) A1

Note: If candidates have combined rows allow FT on their value of $v$ .

Critical value ${\chi ^2} = 13.277$

Because 5.35 < 13.277, the Poisson law does provide a suitable model. R1 N0

[14 marks]

b.ii.

Examiners’ report

[N/A]

b.i.

[N/A]

b.ii.