Microsoft Excel 2010 : Confidence Intervals and the Normal Distribution (part 3)

Using CONFIDENCE.T()

Figure 4 makes two basic changes to the information in Figure 3: It uses the sample standard deviation in cell C2 and it uses the CONFIDENCE.T() function in cell G2. These two basic changes alter the size of the resulting confidence interval.

Figure 4. Other things being equal, a confidence interval constructed using the t-distribution is wider than one constructed using the normal distribution.

Notice first that the 95% confidence interval in Figure 4 runs from 46.01 to 68.36, whereas in Figure 7.8 it runs from 46.41 to 67.97. The confidence interval in Figure 7.8 is narrower. There, you can see that there’s more area under the tails of the leptokurtic distribution than under the tails of the normal distribution. You have to go out farther from the mean of a leptokurtic distribution to capture, say, 95% of its area between its tails. Therefore, the limits of the interval are farther from the mean and the confidence interval is wider.

Because you use the t-distribution when you don’t know the population standard deviation, using CONFIDENCE.T() instead of CONFIDENCE.NORM() brings about a wider confidence interval.

The shift from the normal distribution to the t-distribution also appears in the formulas in cells G8 and G9 of Figure 4, which are:

=T.INV(F8,COUNT(A2:A17)-1)

and

=T.INV(F9,COUNT(A2:A17)-1)

Note that these cells use T.INV() instead of NORM.S.INV(), as is done in Figure 3. In addition to the probabilities in cells F8 and F9, T.INV() needs to know the degrees of freedom associated with the sample standard deviation.

Using the Data Analysis Add-in for Confidence Intervals

Excel’s Data Analysis add-in has a Descriptive Statistics tool that can be helpful when you have one or more variables to analyze. The Descriptive Statistics tool returns valuable information about a range of data, including measures of central tendency and variability, skewness and kurtosis. The tool also returns half the size of a confidence interval, just as CONFIDENCE.T() does.

To use the Descriptive Statistics tool, you must first have installed the Data Analysis add-in. Once this add-in is installed from the Office disc and made available to Excel, you’ll find it in the Analysis group on the Ribbon’s Data tab.

Once the add-in is installed and available, click Data Analysis in the Data tab’s Analysis group, and choose Descriptive Statistics from the Data Analysis list box. Click OK to get the Descriptive Statistics dialog box shown in Figure 5.

To get descriptive statistics such as the mean, skewness, count, and so on, be sure to fill the Summary Statistics check box. To get the confidence interval, fill the Confidence Level for Mean check box and enter a confidence level such as 90, 95, or 99 in the associated edit box.

If your data has a header cell and you have included it in the Input Range edit box, fill the Labels check box; this informs Excel to use that value as a label in the output and not to try to use it as an input value.

When you click OK, you get output that resembles the report shown in Figure 6.

Notice that the value in cell D16 is the same as the value in cell G2 of Figure 4. The value 11.17 is what you add and subtract from the sample mean to get the full confidence interval.

The output label for the confidence interval is mildly misleading. Using standard terminology, the confidence level is not the value you use to get the full confidence interval (here, 11.17); rather, it is the probability (or, equivalently, the area under the curve) that you choose as a measure of the precision of your estimate and the likelihood that the confidence interval is one that captures the population mean. In Figure 6, the confidence level is 95%.

Confidence Intervals and Hypothesis Testing

Both conceptually and mathematically, confidence intervals are closely related to hypothesis testing. You often test a hypothesis about a sample mean and some theoretical number, or about the difference between the means of two different samples. In cases like those you might use the normal distribution or the closely related t-distribution to make a statement such as, “The null hypothesis is rejected; the probability that the two means come from the same distribution is less than 0.05.”

That statement is in effect the same as saying, “The mean of the second sample is outside a 95% confidence interval constructed around the mean of the first sample.”