22M.1.AHL.TZ2.12

pestleMathematicsAIHLPaper 122M· sl-4-9-normal-distribution-and-calculations, sl-4-6-combined-mutually-exclusive-conditional-independence-prob-diagramssource ↗

The sex of cuttlefish is difficult to determine visually, so it is often found by weighing the cuttlefish.

The weights of adult male cuttlefish are known to be normally distributed with mean $10 kg$ and standard deviation $0.5 kg$ .

The weights of adult female cuttlefish are known to be normally distributed with mean $12 kg$ and standard deviation $1 kg$ .

A zoologist uses the null hypothesis that in the absence of information a cuttlefish is male.

If the weight is found to be above $11.5 kg$ the cuttlefish is classified as female.

$90 %$ of adult cuttlefish are male.

Find the probability of making a Type I error when weighing a male cuttlefish.

[2]

Find the probability of making a Type II error when weighing a female cuttlefish.

[2]

Find the probability of making an error using the zoologist’s method.

[2]

Markscheme / solution

$P$ (Type I error) $= P$ (stating female when male)

$= P (W_{M a l e} > 11.5)$ (M1)

$= 0.00135 (0.00134996 \dots)$ A1

[2 marks]

$P$ (Type II error) $= P$ (stating male when female)

$= P (W_{F e m a l e} < 11.5)$ (M1)

$= 0.309 (0.308537 \dots)$ A1

[2 marks]

attempt to use the total probability (M1)

$P$ (error) $= 0.9 \times 0.00134996 \dots + 0.1 \times 0.308537 \dots$

$= 0.0321 (0.0320687 \dots)$ A1

[2 marks]

Examiners’ report

This was a straightforward problem on Type I and Type II errors which some candidates answered successfully in a couple of lines but many candidates were unable to do the correct calculations.

[N/A]