19N.1.SL.TZ0.T_12

pestleMathematicsAISLPaper 119N· sl-4-9-normal-distribution-and-calculationssource ↗

The Malthouse Charity Run is a $5$ kilometre race. The time taken for each runner to complete the race was recorded. The data was found to be normally distributed with a mean time of $28$ minutes and a standard deviation of $5$ minutes.

A runner who completed the race is chosen at random.

Write down the probability that the runner completed the race in more than $28$ minutes.

[1]

Calculate the probability that the runner completed the race in less than $26$ minutes.

[2]

It is known that $20 %$ of the runners took more than $28$ minutes and less than $k$ minutes to complete the race.

Find the value of $k$ .

[3]

Markscheme / solution

$0.5 (\frac{1}{2}, 50 %)$ (A1) (C1)

[1 mark]

$P (X \leq 26)$ (M1)

Note: Award (M1) for a correct mathematical statement.

OR
Award (M1) for a diagram that shows the value $26$ labelled to the left of the mean and the correct shaded region.

$3.45 (0.344578 \dots, 34.5 %)$ (A1) (C2)

[2 marks]

$0.7$ OR $0.3$ (seen) (A1)

Note: Award (A1) for $0.7$ or $0.3$ seen.

$P (time < 7) = 0.7$ OR $P (time > k) = 0 .3$ (M1)

Note: Award (M1) for a correct mathematical statement.
OR
Award (M1) for a diagram that shows $k$ greater than the mean and shading in the region below $k$ , above $k$ , or between $k$ and the mean.

$(k =) 30.6 (30.6220 \dots)$ (minutes) (A1) (C3)

Note: Accept “ $30$ minutes and $37$ seconds” or (from $3$ sf $k$ value) “ $30$ minutes and $36$ seconds”.

[3 marks]

Examiners’ report

[N/A]