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# Estimating the Population Mean Using Intervals

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.

Selected areas under histogram for

In statistical shorthand, we write

where is read as the probability that ".

Of course, the distance of from is also the distance of from , so it is equally true that

 (7.1)

Recall the question posed earlier: Can we find an interval which we know contains   with 95% confidence?'' Answer: .

Example 1: Suppose that the SD of student GPA is .30. If a random sample of n=25 students has average GPA equal to 3.05, then we are 95% confident that is contained within or (2.93, 3.17).

Replacing by 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).

 (7.2)

where z is determined from as follows:

The following table gives the appropriate critical value z for typical values of .

 (7.3)

Example 1 (con't.):
a. Construct a 99% confidence interval for WMU GPA.
b. Construct a 75% confidence interval for WMU GPA.
c. Construct a 95.4% confidence interval for WMU GPA.
d. Construct a 95.0% confidence interval for WMU GPA.

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.

Next: t-based Confidence Interval for Up: Confidence Intervals Previous: Confidence Intervals

2003-09-08