IB Maths AI HL AI 4.3 Probability and distributions questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

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Question 1

Paper 2Medium5 marks

At a parcel centre, $X$ is the excess handling time, in minutes, for a randomly selected parcel. Figure 1 shows the probability distribution of $X$

(a)

Calculate $\mathrm{E}(X)$ and $\mathrm{Var}(X)$

[3]

(b)

For a parcel with excess handling time $X$ the additional handling charge $Y$ in AUD, is modelled by $Y=3X+2$ Hence determine $\mathrm{E}(Y)$ and $\mathrm{Var}(Y)$

[2]

A snack company models the mass, $X$ g, of a packet as normally distributed with mean $503$ g and standard deviation $4.8$ g. Packets with mass below $494$ g do not meet the lower specification. The company wants at least $94\%$ of packets to meet the specification $494\leq X\leq U$ where $U$ is the upper specification.

Use your GDC for normal probability and inverse normal calculations.

(a)

Show that $P(X<494)=0.0304$

[2]

(b)

Using your result from part (a), determine the smallest value of $U$ to the nearest $0.1$ g, that meets the company’s requirement. A manager proposes setting $U$ to the 95th percentile of the distribution. Assess whether this proposal meets the requirement.

[4]

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