Introduction to Interval Estimation

From the discussion on point estimation we know that \overline X is the best possible estimator of the population mean \mu , which is a fixed, usually unknown parameter. It would be extremely lucky to have a sample which has a mean \overline X exactly equal to the population mean \mu , so in most cases it will be a little higher or a little lower.

 

A point estimate of any parameter might be misleading. We, therefore, try to determine two values within which the true value of the parameter is expected to fall instead of one point estimate. We can also attach a certain degree of true values of a parameter which are called confidence limits (lower and upper limit confidence limits), and the two together is called a confidence interval. The word 'confidence' refers to probability. We can determine an interval estimate of any parameter to any degree of confidence, but usually it is estimated with 90, 95 or 99 percent confidence. Thus, if we determine a 95 percent confidence interval estimate, we understand that the probability that the interval contains the true parameter is 0.95, or in other words, out of 100 possible intervals 95 of the intervals are certain to contain the true parameter.

 

Remembering that Z is a standard normal variate consulting the normal table.

P (-1.96 < Z < 1.96) = 0.95
P (-1.80 < Z < 2.19) = 0.95       
P (-2.36 < Z < 1.74) = 0.95       
P (-2.69 < Z < 1.68) = 0.95       
……………………………
……………………………       

We can in fact construct thousands of such intervals each having a probability equal to 0.95. Let us now look at the range (length) of each of these intervals:

R1 = 1.96 – (– 1.96) = 3.92
R2 = 2.19 – (– 1.80) = 3.99 
R3 = 1.74 – (– 2.36) = 4.10 
R4 = 1.68 – (– 2.69) = 4.37

In general we can write

P\left( {a < Z < b} \right) = 0.95

with a range equal to \left( {b - a} \right). To get an interval as precise as possible one would prefer an interval with the lowest value of \left( {b - a} \right). Hence, in the above mentioned four statements, the first interval is more precise than the other three. In the forthcoming discussion we will try to construct confidence intervals with the least possible length.