Mathematical Methods — Question 16

QCAA 2020, Paper 2 · 4 marks

Q16 · 2020 · Technology-activeComplex familiar4 marks

QUESTION 16 (4 marks)

Bottles of soft drink should contain a volume with a mean of 591 mL, but some variation is expected. Any bottle at or below the 20th percentile of the volume distribution is rejected. A percentile is a measure in statistics that shows the values below which a given percentage of observations occur. Thirty-five per cent of the bottles contain 593 mL or more of soft drink. Assuming the volumes are normally distributed, determine the smallest volume (in mL) that will be accepted.

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WORKED SOLUTION

QCAA guide · typeset solution

4 marks

ANSWER

587 mL

Worked solution

Given

\mu=591

, the

z

-score associated with the 65th percentile is

0.38532

. Thus

0.38532=\frac{593-591}{\sigma},

giving

\sigma=5.1905

. The

z

-score associated with the 20% rejection region is

-0.841621

. Therefore

-0.841621=\frac{x-591}{5.1905},

x=587\;\mathrm{mL}

correctly determines the

z

-score associated with the 65th percentile

[1 mark]

determines

\sigma

[1 mark]

correctly determines the

z

-score associated with the 20% rejection region

[1 mark]

determines the smallest volume

[1 mark]

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Mathematical Methods — Question 16

QUESTION 16 (4 marks)

QCAA guide · typeset solution

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