21M.2.SL.TZ1.4

pestleMathematicsAISLPaper 221M· sl-4-9-normal-distribution-and-calculations, sl-4-11-expected-observed-hypotheses-chi-squared-gof-t-testsource ↗

The stopping distances for bicycles travelling at $20 {km h}^{- 1}$ are assumed to follow a normal distribution with mean $6.76 m$ and standard deviation $0.12 m$ .

Under this assumption, find, correct to four decimal places, the probability that a bicycle chosen at random travelling at $20 {km h}^{- 1}$ manages to stop

$1000$ randomly selected bicycles are tested and their stopping distances when travelling at $20 {km h}^{- 1}$ are measured.

Find, correct to four significant figures, the expected number of bicycles tested that stop between

The measured stopping distances of the $1000$ bicycles are given in the table.

It is decided to perform a $χ^{2}$ goodness of fit test at the $5 %$ level of significance to decide whether the stopping distances of bicycles travelling at $20 {km h}^{- 1}$ can be modelled by a normal distribution with mean $6.76 m$ and standard deviation $0.12 m$ .

in less than $6.5 m$ .

[2]

a.i.

in more than $7 m$ .

[1]

a.ii.

$6.5 m$ and $6.75 m$ .

[2]

b.i.

$6.75 m$ and $7 m$ .

[1]

b.ii.

State the null and alternative hypotheses.

[2]

Find the $p$ -value for the test.

[3]

State the conclusion of the test. Give a reason for your answer.

[2]

Markscheme / solution

evidence of correct probability (M1)

e.g sketch OR correct probability statement, $P (X < 6.5)$

$0.0151$ A1

[2 marks]

a.i.

$0.0228$ A1

Note: Answers should be given to $4$ decimal place.

[1 mark]

a.ii.

multiplying their probability by $1000$ (M1)

$451.7$ A1

[2 marks]

b.i.

$510.5$ A1

Note: Answers should be given to 4 sf.

[1 mark]

b.ii.

$H_{0} :$ stopping distances can be modelled by $N (6.76, 0 . 12^{2})$
$H_{1} :$ stopping distances cannot be modelled by $N (6.76, 0.12^{2})$ A1A1

Note: Award A1 for correct $H_{0}$ , including reference to the mean and standard deviation. Award A1 for the negation of their $H_{0}$ .

[2 marks]

$15.1$ or $22.8$ seen (M1)

$0.0727 (0.0726542 \dots, 7.27 %)$ A2

[3 marks]

$0.05 < 0.0727$ R1

there is insufficient evidence to reject $H_{0}$ (or “accept $H_{0}$ ”) A1

Note: Do not award R0A1.

[2 marks]

Examiners’ report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]