Normal Distribution | CIE AS Maths: Probability & Statistics 1 Topic Questions 2020

1a

1 mark

A continuous random variable can take any value within a given range. Many naturally occurring continuous quantities can be modelled using the Normal Distribution, for example the height of human beings; the mass of new born puppies or the distribution of all A Level maths exam results.

Give a different example of a quantity that could be modelled using the normal distribution.

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1b

3 marks

The graph of the normal distribution has a characteristic bell shape that is symmetrical about the mean, $μ$ . If $X$ has a normal distribution with a mean, $μ$ , and variance, $σ^{2}$ , then it can be written as $X ~ N (μ, σ^{2}) .$

For $X ~ N (μ, σ^{2})$ , state:

(i)

P (X < μ)

(ii)

P (X \geq μ)

(iii)

P (X = μ)

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1c

1 mark

Using your answers to part (b), or otherwise, explain why there is no difference between $\geq$ and $>$ or $\leq$ and $<$ when calculating normal probabilities.

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1a

5 marks

The test scores, $X$ , of a group of RAF recruits in an aptitude test are modelled as a normal distribution with $X ~ N (210, 27 . 8^{2})$ .

(i)

Find the values of

a

and

b

such that

P (X < a) = 0.25

and

P (X > b) = 0.25

.

(ii)

Hence find the interquartile range of the scores.

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1b

3 marks

Those who score in the top 30% on the test move on to the next stage of training.

One of the recruits, Amelia, achieves a score of 231. Determine whether Amelia will move on to the next stage of training.

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1a

2 marks

For the random variable, $X ~ N (32, 9)$ ,

(i)

Write down the mean and standard deviation.

(ii)

Draw a sketch of the graph and clearly label the mean.

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1b

2 marks

(i)

With the help of your diagram, explain how you should know, without carrying out any calculations, that

P (X \leq 34) > 0.5 .

(ii)

With the help of your diagram, explain how you should know, without carrying out any calculations, that

P (X \geq 39) < 0.5 .

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1c

6 marks

Use the formula $Z = \frac{X - μ}{σ}$ to find appropriate -values, to three decimal places, and then use the table of $z$ -values to calculate, to three decimal places,

(i)

P (X \leq 34)

(ii)

P (X \geq 39)

(iii)

P (34 \leq X \leq 39) .

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1a

4 marks

The test scores, $X$ , of a group of Royal Navy recruits in an aptitude test are modelled as a normal distribution with $X ~ N (520, 89 . 9^{2})$ .

Find the interquartile range of the scores.

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1b

5 marks

Those who get a score over 670 are offered a role within the submarine service. Only those who get a score over 750 are offered a leadership role within the submarine service.

Given that Mervyn, one of the recruits, has been offered a role within the submarine service, find the probability that he is offered a leadership role.

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CIE AS Maths: Probability & Statistics 1

Topic Questions

3.3 Normal Distribution