Recall that if X has a normal distribution with mean m and standard deviation σ, then the transformation Z = (X – μ)/σ leads to a random variable Z with a standard normal distribution.

*Z*** DISTRIBUTION OBTAINED WHEN
STANDARD DEVIATION IS KNOWN**

Recall that if *X*
has a normal distribution with mean *m* and standard deviation σ, then the transformation *Z* = (*X*
– *μ*)/*σ* leads to a random variable *Z* with a standard normal distribution.
We can do the same for the sample mean *X*.
Assume *n* is large so that the sample
mean has an approximate normal distribution. Now, let us pretend for the
mo-ment that the distribution of the sample mean is exactly normal. This is
reasonable since it is approximately so. Then define the standard or normal *Z* score as follows:

Then *Z*
would have a standard normal distribution because has a normal distribution with mean *μ* = *μ* and standard deviation *σ*/√*n*.

Because in practice we rarely know *σ*, we can approximate σ by the
sample estimate,

For large sample sizes, it is acceptable to use *S* in place of *σ*; under these conditions, the
standard normal approximation still works. So we use the following formula for
the approximate *Z* score for large
sample sizes:

However, in small samples such as *n* < 20, even if the observations are
normally distributed, using Formula 7.2 does not give a good approximation to
the normal distribution. In a famous paper under the pen name Student, William
S. Gosset found the distribution for the statistic in Formula 7.2 and it is now
called the Stu-dent’s *t* statistic;
the distribution is called the Student’s *t*
distribution with *n* – 1 de-grees of
freedom. This is the subject of the next section.

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