# Exam-Style Question on Normal Distribution

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 698. This question is similar to one that appeared on an IB AA Higher paper in 2021. The use of a calculator is allowed.

The travelling times by train, $$T$$ minutes, between two towns can be modelled by a normal distribution with a mean of 55 minutes and a standard deviation of $$\sigma$$ minutes.

(a) Given that 8% of the times are longer than 68 minutes, find the value of $$\sigma$$.

(b) Find the probability that a randomly selected train journey will have a duration of more than 75 minutes.

(c) Given that a train journey between the two towns takes longer than an hour, find the probability that it takes less than 70 minutes.

On a particular day, there are 42 trains scheduled to travel between these two towns.

(d) Find the expected number of train journeys that will have a duration of more than an hour.

(e) Find the probability that more than 10 of the journeys on this particular day will have a duration of more than one hour.

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