SPM.1.AHL.TZ0.16

pestleMathematicsAIHLPaper 1SPM· ahl-4-18-t-and-z-test-type-i-and-ii-errorssource ↗

The number of fish that can be caught in one hour from a particular lake can be modelled by a Poisson distribution.

The owner of the lake, Emily, states in her advertising that the average number of fish caught in an hour is three.

Tom, a keen fisherman, is not convinced and thinks it is less than three. He decides to set up the following test. Tom will fish for one hour and if he catches fewer than two fish he will reject Emily’s claim.

State a suitable null and alternative hypotheses for Tom’s test.

[1]

Find the probability of a Type I error.

[2]

The average number of fish caught in an hour is actually 2.5.

Find the probability of a Type II error.

[3]

Markscheme / solution

${{\text{H}}_0}\,{\text{:}}\,m = 3$ , ${{\text{H}}_1}\,{\text{:}}\,m < 3$ A1

Note: Accept equivalent statements in words.

[1 mark]

(let $X$ be the number of fish caught)

${\text{P}}\left( {X \leqslant 1\left| {m = 3} \right.} \right) = 0.199$ M1A1

[2 marks]

${\text{P}}\left( {X \geqslant 2\left| {m = 2.5} \right.} \right)\,\,\,\,\left( { = 1 - {\text{P}}\left( {X \leqslant 1\left| {m = 2.5} \right.} \right)} \right)$ M1A1

Note: Award M1 for using $m$ = 2.5 to evaluate a probability, award A1 for also having $X$ ≥ 2 .

= 0.713 A1

[3 marks]

Examiners’ report

[N/A]