DP IB Maths: AA SL

Topic Questions

4.6 Normal Distributions

1a
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2 marks

The random variable,X  is seen on the following diagram which shows the distribution of heights, in cm, of adult women in the UK:

ib1a-ai-sl-4-6-ib-maths-medium

The distribution of heights follows a normal distribution, with a mean of 162 cm and a standard deviation of 6.3 cm.

On the diagram above, shade in the region representing Popen parentheses X greater than 155 close parentheses

1b
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4 marks
(i)
Find the probability that a randomly selected woman has a height of more than 155cm.

(ii)
Use your answer from part (b)(i) to find the probability that a randomly selected woman has a height of more than 169cm.
1c
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3 marks

Suggest a range of heights within which the height of approximately

(i)
68%

(ii)
95%

(iii)
99.7%

of adult women in the UK will fall.

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2a
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3 marks

For the random variable X tilde N left parenthesis 23 comma 4 squared right parenthesis find the following probabilities:

(i)

P left parenthesis X less than 20 right parenthesis

(ii)

P left parenthesis X greater or equal than 29 right parenthesis

(iii)
P left parenthesis 20 less or equal than X less than 29 right parenthesis
2b
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3 marks

For the random variable Y tilde N left parenthesis 100 comma 225 right parenthesis find the following probabilities:

(i)

P left parenthesis Y less or equal than 90 right parenthesis

(ii)

P left parenthesis Y greater than 140 right parenthesis

(iii)
P left parenthesis 85 less or equal than Y less or equal than 115 right parenthesis

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3a
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2 marks

The weight, W g, of a chocolate bar produced by a certain manufacturer is modelled as W tilde N left parenthesis 200 comma space 1.75 squared right parenthesis.

Find:

(i)

P left parenthesis W less than 195 right parenthesis

(ii)
P left parenthesis W greater than 203 right parenthesis
3b
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2 marks

Heledd buys a pack containing 12 of the chocolate bars.  It may be assumed that the 12 bars in the pack represent a random sample.

Find the probability that all of the bars in the pack have a weight of at least 195 g.

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4a
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4 marks

The random variable X tilde N left parenthesis 330 comma 10 squared right parenthesis

Find the value of a, to 2 decimal places, such that:

(i)

P left parenthesis X less than a right parenthesis equals 0.25

(ii)

P left parenthesis X greater than a right parenthesis equals 0.25

(iii)
P left parenthesis 315 less or equal than X less or equal than a right parenthesis equals 0.5
4b
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2 marks

The random variable Y tilde N left parenthesis 10 comma 10 right parenthesis.

Find the value of b and the value of c, each to 2 decimal places, such that:

(i)

P left parenthesis Y less than b right parenthesis equals 0.4

(ii)
P left parenthesis Y greater than c right parenthesis equals 0.25
4c
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2 marks

Use a sketch of the distribution of Y to explain why P left parenthesis b less or equal than Y less or equal than c right parenthesis equals 0.35.

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5a
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3 marks

The test scores,X , of a group of RAF recruits in an aptitude test are modelled as a normal distribution with X tilde N left parenthesis 210 comma 27.8 squared right parenthesis.

(i)
Find the values of a and b such that  P left parenthesis X less than a right parenthesis equals 0.25  and  P left parenthesis X greater than b right parenthesis equals 0.25.

(ii)
Hence find the interquartile range of the scores.
5b
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2 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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6a
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4 marks

For the standard normal distribution Z tilde N left parenthesis 0 comma 1 squared right parenthesis, find:

(i)

P left parenthesis Z less than 1.5 right parenthesis

(ii)

P left parenthesis Z greater than negative 0.8 right parenthesis

(iii)
P left parenthesis negative 2.1 less than Z less than negative 0.3 right parenthesis
6b
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3 marks

The random variable X tilde N left parenthesis 2 comma 0.1 squared right parenthesis.

By using the coding relationship between and , re-express the probabilities from parts (a) (i), (ii) and (iii) in the forms P left parenthesis X less than a right parenthesis,   P left parenthesis X greater than b right parenthesis and  P left parenthesis c less than X less than d right parenthesis  respectively, where a comma b comma c, and d are constants to be found.

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7a
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3 marks

The table below shows the percentage points of the normal distribution.  The values z in the table are those which a random variable space Z tilde N left parenthesis 0 comma 1 right parenthesis space space exceeds with probability p.

p z p z
0.5000 0.0000 0.0500
1.6449
0.4000 0.2533 0.0250
1.9600
0.3000 0.5244 0.0100
2.3263
0.2000 0.8416 0.0050
2.5758
0.1500 1.0365 0.0010
3.0902
0.1000 1.2816 0.0005
3.2905

(i)
Use the percentage points table for the standard normal distribution to find the value of z for which P left parenthesis Z greater than z right parenthesis equals 0.2 .

(ii)
Use your answer to part (a)(i) along with the properties of the normal distribution to work out the values of a and b spacefor which  P left parenthesis Z less than a right parenthesis equals 0.2  and P left parenthesis Z less than b right parenthesis equals 0.8 .

7b
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2 marks

The weights, W kg, of coconuts grown on the Coconutty As They Come coconut plantation are modelled as a normal distribution with mean 1.25 kg and standard deviation 0.38 kg.  The plantation only considers coconuts to be exportable if their weight falls into the 20% to 80% interpercentile range.

Use your answer to part (a)(ii) to find the range of possible weights, to the nearest 0.01 kg, for an exportable coconut.

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8a
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2 marks

A machine is used to fill cans of a particular brand of soft drink.  The volume, V ml, of soft drink in the cans is normally distributed with mean 330 ml and standard deviation space sigma ml.  Given that 15% of the cans contain more than 333.4 ml of soft drink, find:

the value of space sigma

8b
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1 mark

P left parenthesis 320 less or equal than V less or equal than 340 right parenthesis.

8c
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3 marks

Six cans of the soft drink are chosen at random.

Find the probability that all of the cans contain less than 329 ml of soft drink.

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9a
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2 marks

The random variable X tilde N left parenthesis mu comma sigma squared right parenthesis.  It is known that  P left parenthesis X greater than 36.88 right parenthesis equals 0.025  and  P left parenthesis X less than 27.16 right parenthesis equals 0.1

Find the values of a and b for which  P left parenthesis Z greater than a right parenthesis equals 0.025  and P left parenthesis Z less than b right parenthesis equals 0.1 ,  where Z is the standard normal variable.  Give your answers correct to 4 decimal places.

9b
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2 marks

Use your answers to part (a), along with the relationship between Z and X , to show that the following simultaneous equations must be true:

mu plus 1.96 sigma equals 36.88

mu minus 1.2816 sigma equals 27.16

9c
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2 marks

By solving the simultaneous equations in (b), determine the values of mu and sigma.  Give your answers correct to 2 decimal places.

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10
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5 marks

The ages, A in years, that Liverpool players have made their debuts over the past 20 years are normally distributed with a mean of 22.5 years and a standard deviation of sigma years.

Given that 10% of Liverpool players make their debuts before turning 20 years old, find:

(i)
the value of sigma ,

(ii)
the probability that a randomly selected player made his debut before his 18th birthday.

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