22M.1.SL.TZ2.10

pestleMathematicsAISLPaper 122M· sl-4-9-normal-distribution-and-calculationssource ↗

The masses of Fuji apples are normally distributed with a mean of $163 g$ and a standard deviation of $6.83 g$ .

When Fuji apples are picked, they are classified as small, medium, large or extra large depending on their mass. Large apples have a mass of between $172 g$ and $183 g$ .

Approximately $68 %$ of Fuji apples have a mass within the medium-sized category, which is between $k$ and $172 g$ .

Determine the probability that a Fuji apple selected at random will be a large apple.

[2]

Find the value of $k$ .

[3]

Markscheme / solution

sketch of normal curve with shaded region to the right of the mean and correct values (M1)

$0.0921 (0.0920950 \dots)$ A1

[2 marks]

EITHER

$(P (x < 172))$

$0.906200 \dots$ (A1)

$(0.906200 \dots - 0.68)$

$0.226200 \dots$ (A1)

$(P (163 < x < 172))$

$0.406200 \dots$ (A1)

$0.5 - (0.68 - 0.406200 \dots)$ OR $0.5 + (0.68 - 0.406200 \dots)$

$0.226200 \dots$ OR $0.773799 \dots$ (A1)

(A1)(A1)

Note: Award A1 for a normal distribution curve with a vertical line on each side of the mean and a correct probability of either $0.406$ or $0.274$ or $0.906$ shown, A1 for a probability of $0.226$ seen.

THEN

$(k =) 158 g (157.867 \dots g)$ A1

[3 marks]

Examiners’ report

A straightforward calculation of probability for a normal distribution was well done.

The majority of candidates used 0.68 as the area to the left of k. Very few candidates knew how to approach the question when the probability given was not the complete area to the left or right of k.