CHAPTER 10 Having Confidence in Your Results 135

Using k = 1.96 for a 95 percent confidence level (from Table 10-1), the sample

mean of 130 mg/dL, and the SD you just calculated of 8 mg/dL, you can compute

the lower and upper confidence limits around the mean using these formulas:

CLL

130

1 96

8

114 3

.

.

CLU

130

1 96

8

145 7

.

.

On the basis of your calculations, you would report your result this way: mean

glucose = 130 mg/dL (95 percent CI = 114 – 116 mg/dL).

Please note that you should not report numbers to more decimal places than their

precision warrants. In this example, the digits after the decimal point are practi-

cally meaningless, so the numbers are rounded off.

A version of the formula in the preceding section is designed to be utilized with

smaller samples, and uses k values derived from a table of critical values of the

Student t distribution. To calculate CIs this way, you need to know the number of

degrees of freedom (df). For a mean value, the df is always equal to N – 1, so in our

case, df = 25 – 1 = 24. Using a Student t table (see Chapter 24), you can find that

the Student-based k value for a 95 percent confidence level and 24 degrees of

freedom is equal to 2.06, which is a little bit larger than the normal-based k value

of 1.96. Using this k value instead of 1.96, you can calculate the 95 percent confi-

dence limits as 113.52 mg/dL and 146.48 mg/dL, which happen to round off to the

same whole numbers as the normal-based confidence limits. Generally, you don’t

have to use these more-complicated Student-based k values unless your N is quite

small (say, less than 25).

The confidence interval around

a proportion

If you were to conduct a study by enrolling and measuring a sample of 100 adult

patients with diabetes, and you found that 70 of them had their diabetes under

control, you’d estimate that 70 percent of the population of adult diabetics has

their diabetes under control. What is the 95 percent CI around that 70 percent

estimate?

There are multiple approximate formulas for CIs around an observed proportion,

which are also called binomial CIs. Let’s start by unpacking the simplest method

for calculating binomial CIs, which is based on approximating the binomial distri-

bution using a normal distribution (see Chapter 25). The N is the denominator of

the proportion, and you should only use this method when N is large (meaning at

least 50). You should also only use this method if the proportion estimate is not

very close to 0 or 1. A good rule of thumb is the proportion estimate should be

between 0.2 and 0.8.