The Normal Distribution
More Complicated Examples

The following examples illustrate some of the many uses and applications of the normal distribution.

Worked Examples

1

A machine produces bolts which are N(4, 0.09), where measurements are in mm. Bolts are measured accurately and any which are smaller than 3.5 mm or bigger than 4.4 mm are rejected. Out of a batch of 500 bolts how many would be acceptable?

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P(X < 4.4) = Φ⎡⎤ ≈ Φ(1.33) = 0.90824

P(X < 3.5) = Φ⎡⎤ ≈ Φ(−1.67) = 0.04746

Hence

P(3.5 < X < 4.4) ≈ 0.90824 − 0.04746 = 0.86078

The number of acceptable items is therefore

0.86078 × 500 = 430.39 ≈ 430 (to nearest whole number).
2

IQ tests are measured on a scale which is N(100, 225). A woman wants to form an 'Eggheads Society' which only admits people with the top 1% of IQ scores. What would she have to set as the cut-off point in the test to allow this to happen?

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From tables you need to find z such that Φ(z) = 0.99.

This is most easily carried out using a 'percentage points of the normal distribution' table, which gives the values directly.

Now Φ−1(0.99) = 2.3263

which is an alternative way of saying that

Φ(2.3263) = 0.99

(Check this using the usual tables.)

This means that

\frac{x - 100}{\sqrt{225}} = 2.3263

Hence

x= 100 + 2.3263 × 15
= 134.8945 ≈ 134.9
3

A manufacturer does not know the mean and SD of the diameters of ball bearings he is producing. However, a sieving system rejects all bearings larger than 2.4 cm and those under 1.8 cm in diameter. Out of 1000 ball bearings 8% are rejected as too small and 5.5% as too big. What is the mean and standard deviation of the ball bearings produced?

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Assume a normal distribution of

Φ−1(1 − 0.08) = 1.4

so 1.8 is 1.4 standard deviations below mean.

Also Φ−1(1 − 0.055) = 1.6

so 2.4 is 1.6 standard deviations above the mean.

This can be written as two simultaneous equations and solved:

μ + 1.6σ = 2.4

μ − 1.4σ = 1.8

Subtracting,

3.0σ= 0.6
⇒ σ= 0.2

Using the first equation,

μ + (1.6 × 0.2) = 2.4
⇒ μ = 2.4 − (1.6 × 0.2)
⇒ μ = 2.08

So diameters are distributed N(2.08, 0.22).

Exercises

Bags of sugar are sold as 1 kg. To ensure bags are not sold underweight the machine is set to put a mean weight of 1004 g in each bag. The manufacturer claims that the process works to a standard deviation of 2.4. What proportion of bags are underweight?

%

Parts for a machine are acceptable within the 'tolerance' limits of 20.5 to 20.6 mm. From previous tests it is known that the machine produces parts to

N(20.56, (0.02)2).

Out of a batch of 1000 parts how many would be expected to be rejected?

Buoyancy aids in watersports are tested by adding increasing weights until they sink. A club has two sets of buoyancy aids. One set is two years old, and should support weights according to N(6.0, 0.64) kg; the other set is five years old and should support weights of N(4.5, 1.0) kg. All the aids are tested and any which are unable to support at least 5 kg are thrown out.

(a)

If there are 24 two-year-old aids, how many are still usable?

(b)

If there are 32 five-year-old aids how many are still usable?

About

Sacks of potatoes are packed by an automatic loader with mean weight 114 lb. In a test it was found that 10% of bags were over 116 lb. Use this to find the SD of the process. If the machine is now adjusted to a mean weight of 113 lb, what % are now over 116 lb if the SD remains unaltered?

,   %

In a soap making process it was found that 6% of bars produced weighed less than 90.50 g and 4% weighed more than 100.25 g.

(a)

Find the mean and the SD of the process.

,  
(b)

What % of the bars would you expect to weigh less than 88 g?

%

A light bulb manufacturer finds that 5% of his bulbs last more than 500 hours. An improvement in the process meant that the mean lifetime was increased by 50 hours. In a new test, 20% of bulbs now lasted longer than 500 hours.

Find the mean and standard deviation of the original process.

,  

The masses of plums from a certain orchard have mean 24 g and standard deviation 5 g. The plums are graded small, medium or large. All plums over 28 g in mass are regarded as large and the rest equally divided between small and medium.

Assuming a normal distribution find:

(a)

the proportion of plums graded large;

(b)

the upper limit of the masses of the plums in the small grade.

g

The weights of pieces of home made fudge are normally distributed with mean 34 g and standard deviation 5 g.

(a)

What is the probability that a piece selected at random weighs more than 40 g?

(b)

For some purposes it is necessary to grade the pieces as small, medium or large. It is decided to grade all pieces weighing over 40 g as large and to grade the heavier half of the remainder as medium. The rest will be graded as small. What is the upper limit of the small grade?

g

Yuk Ping belongs to an athletics club. In javelin throwing competitions her throws are normally distributed with mean 41.0 m and standard deviation 2.0 m.

(a)

What is the probability of her throwing between 40 m and 46 m?

(b)

What distance will be exceeded by 60% of her throws?

m

Gwen belongs to the same club. In competitions 85% of her javelin throws exceed 35 m and 70% exceed 37.5 m. Her throws are normally distributed.

(c)

Find the mean and standard deviation of Gwen's throws, each correct to two significant figures.

,