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Normal Distribution

The standard normal distribution is a normal distribution with a mean of zero and a standard deviation of one, crucial for computing areas under the normal curve through numerical integration. Standardization is the process of converting any normal distribution to the standard normal distribution using a specific formula. An example illustrates that the probability of a dishwasher lasting less than one year is 0.0062, indicating a low failure rate and supporting a twelve-month guarantee.

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6 views2 pages

Normal Distribution

The standard normal distribution is a normal distribution with a mean of zero and a standard deviation of one, crucial for computing areas under the normal curve through numerical integration. Standardization is the process of converting any normal distribution to the standard normal distribution using a specific formula. An example illustrates that the probability of a dishwasher lasting less than one year is 0.0062, indicating a low failure rate and supporting a twelve-month guarantee.

Uploaded by

sheeeerazahmad
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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THE STANDARD NORMAL DISTRIBUTION

A normal distribution whose mean is zero and whose standard deviation is 1 is known as
the standard normal distribution
This distribution has a very important role in computing areas under the normal curve.
The reason is that the mathematical equation of the normal distribution is so complicated
that it is not possible to find areas under the normal curve by ordinary integration. Areas
under the normal curve have to be found by the more advanced method of numerical
integration. The point to be noted is that areas under the normal curve have been
computed for that particular normal distribution whose mean is zero and whose standard
deviation is equal to 1, i.e. the standard normal distribution.
In any problem involving the normal distribution, the generally established procedure is
that the normal distribution under consideration is converted to the standard normal
distribution. This process is called standardization.
The formula for converting N (μ, σ) to N (0, 1) is:

THE PROCESS OF STANDARDIZATION


The standardization formula is:

If X is N (μ, σ), then Z is N (0, 1). In other words, the standardization formula given
above converts our normal distribution to the one whose mean is 0 and whose standard
deviation is equal to 1.
We illustrate this concept with the help of an interesting example:

EXAMPLE
The length of life for an automatic dishwasher is approximately normally distributed with
a mean life of 3.5 years and a standard deviation of 1.0 years. If this type of dishwasher is
guaranteed for 12 months, what fraction of the sales will require replacement?
SOLUTION
Since 12 months equal one year, hence we need to compute the fraction or proportion of
dishwashers that will cease to function before a time-span of one year. In other words, we
need to find the probability that a dishwasher fails before one year.

This means that the probability of a dishwasher lasting less than a year is 0.0062 i.e.
0.62% --- even less than 1%.Hence, the owner of the factory should be quite happy with
the decision of placing a twelve-month guarantee on the dishwasher!

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