If
data is normally distributed, we can calculate a standard normal
distribution
A normal distribution is:
The standard normal distribution is:
If m = 68, s^{2}
= 100, and the 87^{th} observation is X_{87} = 70
The observation is standardized by
Form
a confidence interval
Usually set a = 5%
(or 0.05). It is okay to have an a
= 10% or a = 1%
If a = 5%, then
For two sided confidence intervals, we usually put a/2
in each tail
Thus, z_{a}_{/2}
= z_{0.025} = 1.96 for a standard normal
Example
= 68, which is an unbiased estimate for the
population parameter, m
The standard deviation is s
= 10 and a = 0.05
We would expect 95% of the data to fall between
[48.4, 87.6]
Standard
Errors – use one sample to determine variability of population
parameter, m
We have the following distribution
Take a random sample
n = 90,
= 110, and s^{2}
= 81
We are assuming we know the variance now; usually this
is unknown too!
We calculate the standard error (SE)
Form a 95% Confidence Interval
There is a 95% chance that the true population mean
lies between [108.1, 111.9]
We
assume we know s^{2}
However, we have to estimate s^{2}
too
We switch the distribution to a tdistribution
The tdistribution is shorter with fatter tails
Uses degrees of freedom
df = n – 1
The one is we estimated the variance, so we lose one
piece of information
As the degrees of freedom approaches infinity, the
tdistribution collapses onto the normal distribution
As the sample size becomes larger, the standard error
becomes smaller. The confidence intervals become smaller too!
