059. Confidence Intervals for the Population Mean – Large Samples
One of the most common uses for confidence intervals is to estimate the population mean. Many typical business situations require an estimate of 
. A producer may wail to estimate the mean monthly level of output for his firm; a marketing representative might be concerned about a drop in mean quarterly sales; a management director might be interested in the mean wage level for hourly employees. There is an almost infinite number of situations calling for an estimate of the unknown population mean.
Example 6.1. Consider the producer just mentioned, who wishes to construct a confidence interval for his mean monthly output. Recall that a confidence interval consists of an upper confidence limit (UCL) and a lower confidence limit (LCL). Using the sample mean 
as a point estimator, the producer will add a certain amount to it to get tie UCL, and subtract the same amount to get the LCL. The question is, how much should be added and subtracted? The answer depends on how precise the producer wishes to be. Assume he wants a 95 percent interval. Figure 8-3 illustrates the producer’s objective. He must identify an LCL and a UCL that will encompass 95 percent of all the possible values for p, along the axis. If the population standard deviation n is known, the limits on this confidence interval (C. I.) can be found by when past experience and familiarity with the population may reveal its variance but the mean remains a mystery. In any event, in the likely circumstance the population standard deviation is unknown, we simply substitute the sample standard deviation, S, and obtain the interval using
C. I. for = 
± Zs
. (6.1)
Returning to the producer’s efforts to construct the 95 percent interval for mean output, assume he takes a sample of N = 100 and calculates a sample mean of = 112 tons. Past experience has shown that 
= 50 tons. Then,
C. I. for, = ± Z= 112 ± (Z)
The producer still needs a value for Z, which will be taken from the Z-table. Thus, Z-value is 1.96. The producer can now complete his answer.
C. I. for, = ± Z= 112 ± (Z)
= 112 ± (1,96)
102.2 < < 121.8 tons.
The producer can draw two inferences about the population from his sample, each based on one of the two interpretations of a confidence interval discussed above:
1. He can be 95 percent confident that the mean daily output lies between 102.2 and 121.8 tons.
2. Ninety-five percent of all confidence intervals formed in this manner will include the true value for .
With reference to this second interpretation, consider this question: If the producer would repeat the experiment, would he get the same answer for the interval? No, because he would likely get a different value for . However, he can be certain that 95 percent of all the confidence intervals he might construct in this manner would include .
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