BUS 461 Decision Modeling & Analysis Wk 3 Assignment

8-3 CONFIDENCE INTERVAL FOR A MEAN

We now come to the main topic of this chapter: using properties of sampling distributions to construct confidence intervals. We assume that data have been generated by some random mechanism, either by observing a random sample from some population or by performing a randomized experiment. The goal is to infer the values of one or more population parameters such as the mean, the standard deviation, or a proportion from sample data. For each such parameter, you use the data to calculate a point estimate, which can be considered a best guess for the unknown parameter. You then calculate a confidence interval around the point estimate to measure its accuracy.

We begin by deriving a confidence interval for a population mean μ, and we discuss its interpretation. Although the particular details pertain to a specific parameter, the mean, the same ideas carry over to other parameters as well, as will be described in later sections. As usual, the sample ¯¯¯X is used as the point estimate of μ.

To obtain a confidence interval for μ, you first specify a confidence level, usually 90%, 95%, or 99%. You then use the sampling distribution of the point estimate to determine the multiple of the standard error (SE) to go out on either side of the point estimate to achieve the given confidence level. If the confidence level is 95%, the value used most frequently in applications, the multiple is approximately 2. More precisely, it is a t-value. That is, a typical confidence interval for μ is of the form in Equation (8.4), where SE(¯¯¯X)=s/√n.

Confidence Interval for Population Mean

To obtain the correct t-multiple, let α be one minus the confidence level (expressed as a decimal). For example, if the confidence level is 90%, then α = 0.10. Then the appropriate t-multiple is the value that cuts off probability α/2 in each tail of the t distribution with n − 1 degrees of freedom. For example, if n= 30 and the confidence level is 95%, cell B25 of Figure 8.2 indicates that the correct t-value is 2.045. The corresponding 95% confidence interval for μ is then

¯¯¯X±2.045(s/√n)

If the confidence level is instead 90%, the appropriate t-value is 1.699 (change the probability in cell B24 to 0.10 to see this), and the resulting 90% confidence interval is

¯¯¯X±1.699(s/√n)

If the confidence level is 99%, the appropriate t-value is 2.756 (change the probability in cell B24 to 0.01 to see this), and the resulting 99% confidence interval is

¯¯¯X±2.756(s/√n)

Note that as the confidence level increases, the length of the confidence interval also increases. Because narrow confidence intervals are desirable, this presents a trade-off. You can either have less confidence and a narrow interval, or you can have more confidence and a wide interval. However, you can also take a larger sample. As n increases, the standard error s/√n tends to decrease, so the length of the confidence interval tends to decrease for any confidence level. (Why won’t it decrease for sure? The larger sample might result in a larger value of s that could offset the increase in n.)

Confidence interval lengths increase when you ask for higher confidence levels, but they tend to decrease when you use larger sample sizes.

Example 8.1 illustrates confidence interval estimation for a population mean. Starting in this edition of the book, we illustrate this in two ways: with Excel-only formulas and with StatTools. The advantages of the Excel-only method are that no add-in is required, and it shows exactly how the calculations are performed. The advantage of StatTools is that it is much faster, requiring only that you fill in a dialog box. You might want to try the Excel-only method at first and then move to StatTools when you are more comfortable with the procedure, but this is totally up to you. These comments apply to the other confidence intervals in this chapter as well.