Confidence Interval for a Mean
It is certain that the estimated value will not be exactly equal to the true parameter value. We want to
have some idea of the precision of the estimate. Hence, rather than simply calculating a point estimate
of the parameter value, we find a confidence interval estimate.
Defn: A confidence interval estimate of a parameter consists of an interval of numbers obtained from
a point estimate of the parameter, together with a percentage that specifies how confident we are that
the true parameter value lies in the interval. This percentage is called the confidence level, or
confidence coefficient.
Thus, there are three quantities that must be specified: 1) the point estimate of the parameter, 2) the
width of the interval (which is usually centered at the value of the point estimate), and 3) the
confidence level.
Confidence Interval for a Mean:
If we are sampling from a population which has a normal distribution, or if our sample size is large
enough (n  30), then we may find the confidence interval as follows. We know that, in either of the
X  X
situations (normality or large sample size), the random variable T 
has either a t distribution
 sX 


 n
with d.f. = n – 1 (in the first case) or an approximate t distribution with d.f. = n – 1 (in the second
case). Hence, for a given percentage , we may make the following probability statement:




X  X


P  t n 1,  
 t n 1,    1   .
 sX 
2
2






n




We can rearrange the quantities in parentheses to obtain an equivalent probability statement by
s
multiplying throughout the inequality by the quantity X . We obtain
n

s
s 
P  t n 1,  X  X    t  X   1   .
n 1,
2
n
n
2

We want to isolate the parameter, , in the middle of the inequality, so we subtract X throughout,
obtaining
s
s 

P  X  t n 1,  X      X  t n 1,  X   1   .
2
2
n
n

We are almost done. We do not want - in the center, but rather , so we multiply throughout by –1,
obtaining
s
s 

P X  t n 1,  X    X  t n 1,  X   1   .
2
2
n
n

Then the endpoints of our confidence interval are X  t n 1, 
2
sX
n
and X  t n 1, 
2
sX
. The confidence
n
level is 1 - .
Intervals constructed in the above manner are called confidence intervals for the corresponding
parameter (in this case, a population mean, ). We are usually interested in one of several specific
values of the confidence level, either 90%, or 95% or 99%.
We interpret a confidence interval as follows. A confidence interval with confidence level (1 )100% is an interval obtained from sample data by a method such that, (1 - )100% of all intervals
obtained by this method would, in fact, contain the true value of the parameter, and 100% of all
intervals so obtained would not contain the true value of the parameter.