21M.2.SL.TZ2.4

pestleMathematicsAISLPaper 221M· sl-4-9-normal-distribution-and-calculations, sl-4-8-binomial-distributionsource ↗

It is known that the weights of male Persian cats are normally distributed with mean $6.1 kg$ and variance $0 . 5^{2} {kg}^{2}$ .

A group of $80$ male Persian cats are drawn from this population.

Sketch a diagram showing the above information.

[2]

Find the proportion of male Persian cats weighing between $5.5 kg$ and $6.5 kg$ .

[2]

Determine the expected number of cats in this group that have a weight of less than $5.3 kg$ .

[3]

It is found that $12$ of the cats weigh more than $x kg$ . Estimate the value of $x$ .

[3]

Ten of the cats are chosen at random. Find the probability that exactly one of them weighs over $6.25 kg$ .

[4]

Markscheme / solution

A1A1

Note: Award A1 for a normal curve with mean labelled $6.1$ or $μ$ , A1 for indication of SD $(0.5) :$ marks on horizontal axis at $5.6$ and/or $6.6$ OR $μ - 0.5$ and/or $μ + 0.5$ on the correct side and approximately correct position.

[2 marks]

$X ~ N (6.1, 0 . 5^{2})$

$P (5.5 < X < 6.5)$ OR labelled sketch of region (M1)

$= 0.673 (0.673074 \dots)$ A1

[2 marks]

$(P (X < 5.3) =) 0.0547992 \dots$ (A1)

$0.0547992 \dots \times 80$ (M1)

$= 4.38 (4.38393 \dots)$ A1

[3 marks]

$0.15$ OR $0.85$ (A1)

$P (X > x) = 0.15$ OR $P (X < x) = 0.85$ OR labelled sketch of region (M1)

$6.62 (6.61821 \dots)$ A1

[3 marks]

$(P (X > 6.25) =) 0.382088 \dots$ (A1)

recognition of binomial (M1)

e.g. $B (10, 0.382088 \dots)$

$0.0502 (0.0501768 \dots)$ A2

[4 marks]

Examiners’ report

[N/A]