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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