8.1. Population Mean: Large Sample Case (
)
General confidence interval:
Definition of 
:
Let
Z be the standard normal random variable. Then,
As 
,
.
Derivation of 
confidence
interval:
As the sample size is large,
.
Thus,
. Thus,
There is an
approximate 
chance that the
population mean 
will fall
between
and 
, i.e., 
,
falls in the
interval 
with a chance
close to 
.
confidence
interval:
Suppose
the sample size is large.
As 
is known,
is a
confidence
interval estimate of the population mean .
As is unknown,
is a
confidence
interval estimate of the population mean ,
where
is the standard
error of the estimator 
.
Note:
95%
confidence interval: Suppose the sample size is large.
As is known,
is a 95% confidence interval estimate of the
population mean .
As is unknown,
is a 95% confidence
interval estimate of the population mean , where 
is
the sample variance.
Example 1:
In
the survey conducted by CJW, Inc., a mail-order firm, the satisfaction scores
(1~100) of
100
customers (n=100) are obtained. Suppose 
is known. Also,
.
Since
, 
, and 
, thus
is a
95% confidence interval estimate of the population mean .
As 
,
is a
90% confidence interval estimate of the population mean .
Note: In the CJW, Inc. example, the 95% confidence interval is wider than
the 90%
confidence interval. Intuitively, if we want to make
sure that we will make less mistakes,
we should speak vaguely (wider confidence interval).
For instance, if we want to get
a 100% confidence interval (for sure), the interval 
would make us not
make
any mistake.
Note: The length of the confidence interval is 
or 
. Therefore, a larger
sample size 
will provide a
narrow interval and a greater precision.
JavaStatSoft:
Confidence
interval:
Statistics
-> Estimation -> One Sample -> Mean -> Z Interval