We now discuss how to come up with confidence intervals when
we try to estimate the unknown mean of some random variable,
or of some distribution, using the
sample mean as our estimator.
So here X1 up to Xn are independent, identically
distributed random variables that are drawn from a
distribution that has a certain mean theta, the
quantity that we want to estimate, and some variance
sigma squared.
Let us say that we want to construct a
95% confidence interval.
Our starting point will be the fact that the sample mean,
according to the central limit theorem, can be described
approximately using normal distributions.
And we look up at the normal table, and we observe the
following--
that if we take a standard normal random variable, then
there is probability, 97.5% of falling below this number,
1.96, which means that there is probability 2 1/2% of
falling above that number.
And by symmetry, the probability of falling below
minus 1.96 is also 2 1/2%.
This means that this middle interval here has probability
95%, and we exploit this fact as follows.
If we take the sample mean, subtract the true mean, and
then divide by the standard deviation of the sample mean,
then we obtain a random variable, which is
approximately a standard normal.
Therefore, what we have here is the probability of an
approximately standard normal random variable.
Or actually, its absolute value falling below 1.96.
This is just the event that our standard normal falls
inside this middle interval here, according to this entry
from the normal tables and the previous discussion, this
probability is going to be approximately 95%.
And now we take this statement, send this term to
the other side of the inequality, and then interpret
what it means for an absolute value to
be less than something.
And we obtain an equivalent statement.
This event here is algebraically identical to the
event that we have up there, and this provides us with the
desired confidence interval.
We think of this quantity here as the lower end of the
confidence interval.
This quantity here is the upper end of
the confidence interval.
And this statement tells us that there is probability
approximately equal to 95% that the confidence interval
constructed this way contains the true value
of the unknown parameter.
So this is how we obtain a 95% confidence interval.
If instead we wanted a 90% confidence interval, we would
proceed in more or less in the same way.
Here, we would want to have the number 0.95.
Why is that?
We want this middle interval to have probability 90%, which
means that we want to have probability 5% at
each one of the tails.
And then we look up at the normal tables, and we find
that the entry that gives us probability 95% of being below
that value is 1.645.
So if we use 1.645 in place of 1.96, we obtain a 90%
confidence interval, and similarly for other choices.
For example, if we want a 99% confidence interval.
There's only one issue that's left to discuss, and this is
the following.
In order to obtain numerical values for the endpoints of
the confidence interval, we need to know sigma, the
standard deviation of the random
variables that we are observing.
But if we do not know the value of sigma, then we may
have to do some additional work.