To use the foregoing theory to construct a confidence interval for the universe mean researchers must first decide what level of confidence they require. Roughly, a level of confidence is the probability that one’s confidence interval (based on a given sample) will in fact encompass the universe mean.
The higher the confidence level the more likely that the confidence interval will in fact be correct. It is commonly chosen to be 80, 90, 95, or some higher percent, depending on how confident the analyst wishes to be about the location of the universe mean. For illustration, it is assumed that a 99.7 percent confidence interval estimate of the mean is wanted. In terms of the foregoing discussion of the sampling distribution of the mean, 99.7 percent confidence (practical certainty) implies that Z = 3. (for 95.4 percent, Z = 2,. For values associated with other levels of confidence, see Table.
After the confidence level has been directed, the analyst performs certain calculations (illustrated below) based on the available information. This will consist of the size of the simple random sample used (n) the mean of the sample (always symbolized by x) and information about the universe standard deviation (denoted by σ). The course of the calculations depends on whether the universe standard deviation, σ, is known (possibly from pervious work), or whether it is not known and must be estimated from the sample. The simplest case is when σ is known.
If the Universe Standard Deviation (σ) is known:
Using the income universe of Figure for illustration, assume it is known from pervious experience that the universe standard deviation is σ=$40,000. An investigator wishes to construct a 99.7 percent confidence interval (Z=3) for the unknown universe mean (M) using a maple mean (x) based on a sample size of n = 400.
The applicable theory is that for samples of n= 400, the sample mean (x) will have a normally shaped sampling distribution. That distribution of sample means will have a mean value of M (the mean of the income universe) and a standard error of:
σ x = σ / √n = $40,000 /√400 = $2,000
This implies that 99.7 percent of sample means, based on n=400, will be located within an interval of:
Interval = ± 3 standard errors = ± 3σ x
= ± 3 ($2,000) = ± $ 6,000
Around the universe mean of M
That is, before the sample was selected, the probability was 99.7 percent that the interval
(M – 3 σx) to (M + 3 σx), or (M ± 3 σ x)
Would include the mean of a sample of n = 400 households
After the sample is selected, a particular specific value for x will be obtained. The investigator than asserts that the obtained sample mean is one of those sample means that does in fact lie within Z = ± 3 standard errors of the universe mean. If this is true and by definition it will be true 99.7 percent of the time then the 99.7 percent confidence interval.
x ± 3 σ x
will include (or cover) the universe mean. M. In this particular example, σ x = $2,000, so the 99.7 percent confidence interval for the mean, M, of the income universe would be:
x ± $6,000
Thus, the investigator would be 99.7 percent confident that the obtained sample mean would be within $6,000 of the actual mean income M.
Other Confidence Levels and sample sizes:
Using similar reasoning, an investigator would be 90 percent confident that the interval x ± 1.64 σ x would include M. For example, with n = 400 as earlier, the analyst would be 90 percent confident that the interval
x ± (1.64) ($2,000) = x ± $3,280
will include the universe mean.
As a final example of confidence interval reasoning, to be 95.4 percent (Z=2) confident of the location of the universe mean, M, for the income distribution, a researcher with a simple random sample of n= 900 would construct the confidence interval:
x ± 2 σ x = x ± (2) $40,000 / √ 900 = x ± $2,667