The Normal Distribution
Transformation of Normal P.D.Fs

The method needed to transform any normal variable to the standardised variable is illustrated in the example below.

Worked Examples

1

Eggs laid by a particular hen are known to have lengths normally distributed, with mean 6 cm and standard deviation 1.4 cm. What is the probability of:

(a)

finding an egg bigger than 8 cm in length;

Show me

The number of SDs that 8 is above the mean is given by

z = = = 1.429

but Φ(1.43) = 0.92364 (from tables)

so P(X > 8) = 1 − 0.92364 = 0.07646.
(b)

finding an egg smaller than 5 cm in length?

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z = = −0.7143

but Φ(0.7143) = 0.7625 (from tables using interpolation),

so P(X < 5) = 1 − 0.7625 = 0.2375.

Note that in order to find the probability you need to establish whether you need the area greater than a half or less than a half. Drawing a diagram will help.

When a variable X follows a normal distribution, with mean m and variance σ2, this is denoted by

X ~ N(μ, σ2)

So in the last example, X ~ N(6, 1.42).

To use normal tables, the transformation

Z =

is used. This ensures that Z has mean 0 and standard deviation 1, and the tables are then valid.

The tallest person in recorded history for whom there is irrefutable evidence of height is Robert Wadlow (1918 - 1940), an American who was born and raised in Illinois. He reached the height of 2.72 m (8 ft 11.1 in)

Using the UK data on heights in the Exercise in section 6.1, the z value for Robert Wadlow's height is

z = ≈ 11.5

So his height is 11.5 SDs above the mean. The most accurate tables show that 6 SDs is only exceeded with a probability of 10−10, so it is extremely unlikely that a taller person will ever appear in our lifetime!

2

If X ~ N(4, 9), find

Now Z = = .
(a)

P(X > 6)

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Hence

P(X > 6)= 1 − P(X < 6)
= 1 − Φ()
= 1 − Φ(0.67)
= 1 − 0.74857
= 0.25143
(b)

P(X > 1)

Show me
P(X > 1)= P(X < 7)   (by symmetry)
= Φ()
= Φ(1)
= 0.84134

Exercises

If X ~ N(200, 625), find

(a)
P(X > 250)
(b)
P(175 < X < 225)
(c)
P(X < 275)

If X ~ N(6, 4), find

(a)
P(X > 8)
(b)
P(4 < X < 8)
(c)
P(5 < X < 9)

If X ~ N(−10, 36), find

(a)
P(X < 0)
(b)
P(−12 < X < −8)
(c)
P(−15 < X < 0)

Components in a personal stereo are normally distributed with a mean life of 2400 hours with SD 300 hours. It is estimated that the average user listens for about 1000 hours in one year. What is the probability that a component lasts for more than three years.

The maximum flow of a river in Africa during the 'rainy season' was recorded over a number of years and found to be distributed

N(6300, 19002) m3s−1

For the banks to burst a flow of 8700 m3s−1 is required. What is the probability of this happening in a particular year?

(to 4 d.p.)

IQs are designed to be N(100, 225). To join Mensa an IQ of 138 is required. What percentage of the population are eligible to join?

A psychologist claims that any child with an IQ of 150 + is 'gifted'. How many 'gifted' children would you expect to find in a school of 1800 pupils?

%;

Rainfall in a particular area has been found to be N(850, 1002) mm over the years. What is the probability of rainfall exceeding 1000 mm?

(to 3 d.p.)

In a verbal reasoning test on different ethnic groups, one group was found to have scores distributed N(98.42, 15.312). Those with a score less than 80 were deemed to be in need of help. What percentage of the overall group were in need of help?

% (to 1 d.p.)