Problem revisited: Estimate the average GPA of the population of approximately 23000 WMU undergraduates.
Consider estimating the population average by the sample average of, say n=25 randomly selected students. Suppose this sample average, which we denote by , equals 3.05. Now chances are the true average is not equal to 3.05. For sure, the true WMU average GPA is between 1.00 and 4.00, and with high confidence between (2.50, 3.50); but what level of confidence do we have that it is between say, (2.75, 3.25) or (2.95, 3.15)? Even better, can we find an interval (a, b) which will contain with , say, 95% certainty?
Recall that if is the standard deviation for individuals, then is the standard deviation for averages (also called the SE of the sample average). More precisely, n-member averages form an approximate normal histogram with mean and standard error . This means that there is 68% likelihood that the sample average falls within of , and 95% likelihood that the sample average falls within of ; see figure below.
In statistical shorthand, we write
Of course, the distance of
is also the
so it is equally true that
in Equation ( 7.1)
results in a 68% confidence interval. In general, we can create an
interval with any desired confidence level by replacing the
multiplier with the appropriate standard normal percentile z
(also called the z-critical value).
The following table gives the appropriate critical value z for typical values of .
Theoretically speaking, has a confidence level of 95.4%. Applied statisticians usually just round off and say 95%. We can, of course, choose the more precise critical value of 1.96 to get an exact 95% confidence level. However, normality of is at best approximate anyway, so the precision seems spurious. However, the distinction is important enough to merit a reminder.