## Exam-Style Questions on Normal Distribution## Problems on Normal Distribution adapted from questions set in previous exams. |

## 1. | IB Standard |

A random variable \(X\) is normally distributed with \(\mu=250\) and \(\sigma=25\).

Find the interquartile range of \(X\).

## 2. | IB Studies |

The heights of palm trees along a beach are normally distributed with a mean of 4.55m and a standard deviation of 0.37m.

(a) Find the probability that a randomly chosen tree has a height greater than 4.55m.

(b) Find the probability that a randomly chosen tree will be within 2 standard deviations of the mean. Give your answer as an integer percentage.

(c) Use your graphic display calculator to calculate the probability that a randomly chosen tree will have a height greater than 4m.

(d) The probability that a particular tree is less than \(x\) metres tall is 0.75. Find the value of \(x\).

## 3. | IB Studies |

A group of students sat a Biology examination and a Computer Studies examination. The students' marks in the Biology examination are normally distributed with a mean of 70 and a standard deviation of 9.

(a) Draw a diagram that shows this information.

(b) Find the probability that a randomly chosen student who sat the Biology examination scored at most 70 marks.

Eric scored 82 marks in the Biology examination.

(c) Find the probability that a randomly chosen student who sat the Biology examination scored more than Eric.

The students' marks in the Computer Studies examination are normally distributed with a mean of 68 and a standard deviation of 11. Eric also scored 82 marks in the Computer Studies examination.

(d) Find the probability that a randomly chosen candidate who sat the Computer Studies examination scored less than Eric.

(e) Determine whether Eric's Computer Studies mark, compared to the other students, is better than his mark in Biology. Give a reason for your answer.

To obtain a grade A a student must be in the top 12% of the students who sat the Computer Studies examination.

(f) Find the minimum possible mark to obtain a grade A. Give your answer correct to the nearest integer.

## 4. | IB Standard |

The weights of players in a sports league are normally distributed with a mean of 75.2 kg, (correct to three significant figures). It is known that 75% of the players have weights between 67 kg and 80 kg. The probability that a player weighs less than 67 kg is 0.05.

(a) Find the probability that a player weighs more than 80 kg.

(b) Write down the standardized value, z, for 67 kg.

(c) Hence, find the standard deviation of weights.

To take part in a tournament, a player's weight must be within 1.5 standard deviations of the mean.

(d) Find the set of all possible weights of players that take part in the tournament.

(e) A player is selected at random. Find the probability that the player takes part in the tournament.

Of the players in the league, 22% are women. Of the women, 60% take part in the tournament.

(f) Given that a player selected at random takes part in the tournament, find the probability that the selected player is a woman.

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