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 distribution 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.

Using this method, you first calculate the SE of the proportion using this formula:

where p stands for proportion. Next, you use the normal-based formulas in the earlier section “Before

you begin: Formulas for confidence limits in large samples” to calculate the ME and the confidence

limits.

Using the numbers from the sample of 100 adult diabetics (of whom 70 have their diabetes under

control), you have

and

. Using those numbers, the SE for the proportion is

or 0.046. From Table 10-1, k is 1.96 for 95 percent confidence limits. So for the

confidence limits,

and

. If you calculate these out,

you get a 95 percent CI of 0.61 to 0.79 (around the original estimate of 0.7). To express these fractions

as percentages, you report your result this way: “The percentage of adult diabetics in the sample

whose diabetes was under control was 70 percent (95 percent CI = 61 – 79 percent).”

The confidence interval around an event count or rate

Suppose that you learned that at a large hospital, there were 36 incidents of patients having a serious

fall resulting in injury in the last three months. If that’s the only incident report data you have to go on,

then your best estimate of the monthly serious fall rate is simply the observed count (N), divided by the

length of time (T) during which the N counts were observed: 36/3, or 12.0 serious falls per month.

What is the 95 percent CI around that estimate?

There are many approximate formulas for the CIs around an observed event count or rate, which is

also called a Poisson CI. The simplest method to calculate a Poisson CI is based on approximating the

Poisson distribution by a normal distribution (see Chapter 24). It should be used only when N is large

(at least 50). You first calculate the SE of the event rate using this formula:

. Next, you

use the normal-based formulas in the earlier section “Before you begin: Formulas for confidence limits

in large samples” to calculate the lower and upper confidence limits.

Using the numbers from hospital falls example,

and

, so the SE for the event rate is

, which is the same as the square root of 2, which is 1.41. According to Table 10-1, k is 1.96 for 95

percent CLs. So CLL = 12.0 – 1.96 × 1.41 and CLU = 12.0 + 1.96 × 1.41, which works out to 95

percent confidence limits of 9.24 and 14.76. You report your result this way: “The serious fall rate