## Exam-Style Question on Normal Distribution## A mathematics exam-style question with a worked solution that can be revealed gradually |

Question id: 575. This question is similar to one that appeared on an IB AA Standard paper in 2021. The use of a calculator is allowed.

The random variable X follows a normal distribution with mean \(\mu\) and standard deviation \(\sigma\).

(a) Find \( P(\mu - 1.25\sigma \lt X \lt \mu + 1.25\sigma) \).

The pineapples grown on a farm in Thailand have weights, in grams, that are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Pineapples are categorised as tiny, regular, super or giant, according to their weight. The following table shows the probability a pineapple grown on the farm is classified tiny, regular, super or giant.

Size | Tiny | Regular | Super | Giant |
---|---|---|---|---|

Probability | 0.08 | 0.72 | 0.16 | 0.04 |

The maximum weight of a tiny pineapple is 895 grams.

The minimum weight of a giant pineapple is 1804 grams.

(b) Find the values of \(\mu\) and \(\sigma\).

One season a wholesaler purchased all the regular, super and giant pineapples from the farm.

Find the probability that a pineapple chosen at random from this purchase is categorized as

(c) regular.

(d) super.

(e) giant.

The wholesaler sells the pineapples at the following prices:

Size | Regular | Super | Giant |
---|---|---|---|

Price | 40 ฿ | 60 ฿ | 80 ฿ |

The wholesaler pays the farm 8000 ฿ for the pineapples and assumes it will then sell them in exactly the same proportion as purchased from the farm.

(f) Find the minimum number of pineapples that must be sold so that the net profit for the supermarket is at least 4000 ฿.

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