Introduction to Interval Estimation

From the discussion on the 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 is a matter of common-sense that we have to be extremely lucky to have a sample which has a mean \overline X exactly equal to the population mean \mu , otherwise it will be a little higher or a little lower.

A point estimate of any parameter, may be theoretically unbiased, might be misleading. We, therefore, try to determine two values, instead of one point estimate, within which the true value of the parameter is expected to fall. We can also attach a certain degree true values of a parameter are called confidence limits (lower and upper limit confidence limits) and the two together is called a confidence interval. The word confidence refers to the 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 infect 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 as possible one would prefer an interval with the least 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.