8.1 A confidence interval for a single population mean using the  (Page 2/7)

Another way to approach confidence intervals is through the use of something called the Error Bound. The Error Bound gets its name from the recognition that it provides the boundary of the interval derived from the standard error of the sampling distribution. In the equations above it is seen that the interval is simply the estimated mean, sample mean, plus or minus something. That something is the Error Bound and is driven by the probability we desire to maintain in our estimate, Zα, times the standard deviation of the sampling distribution. The Error Bound for a mean is given the name, Error Bound Mean, or EBM.

Calculating the confidence interval: an alternative approach

To construct a confidence interval for a single unknown population mean μ , where the population standard deviation is known , we need $\stackrel{-}{x}$ as an estimate for μ and we need the margin of error. Here, the margin of error ( EBM ) is called the error bound for a population mean (abbreviated EBM ). The sample mean $\stackrel{-}{x}$ is the point estimate of the unknown population mean μ .

The confidence interval estimate will have the form:

(point estimate - error bound, point estimate + error bound) or, in symbols,( $\stackrel{-}{x}–EBM,\stackrel{-}{x}\text{+}EBM$ )

The mathematical formula for this confidence interval is:

The margin of error ( EBM ) depends on the confidence level (abbreviated CL ). The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. However, it is more accurate to state that the confidence level is the percent of confidence intervals that contain the true population parameter when repeated samples are taken. Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because that person wants to be reasonably certain of his or her conclusions.

There is another probability called alpha ( α ). α is related to the confidence level, CL . α is the probability that the interval does not contain the unknown population parameter.
Mathematically, 1 - α = CL .

A confidence interval for a population mean with a known standard deviation is based on the fact that the sampling distribution of the sample means follow an approximately normal distribution. Suppose that our sample has a mean of $\stackrel{-}{x}$ = 10, and we have constructed the 90% confidence interval (5, 15) where EBM = 5.

To get a 90% confidence interval, we must include the central 90% of the probability of the normal distribution. If we include the central 90%, we leave out a total of α = 10% in both tails, or 5% in each tail, of the normal distribution.

To capture the central 90%, we must go out 1.645 standard deviations on either side of the calculated sample mean. The value 1.645 is the z -score from a standard normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail.

It is important that the standard deviation used must be appropriate for the parameter we are estimating, so in this section we need to use the standard deviation that applies to the sampling distribution for means which we studied with the Central Limit Theorem and is, $\frac{\sigma }{\sqrt{n}}$ . The fraction $\frac{\sigma }{\sqrt{n}}$ , is commonly called the "standard error of the mean" in order to distinguish clearly this standard deviation from the population standard deviation σ and the standard deviation for a sample, s .