CHAPTER 10 Having Confidence in Your Results 133
In some situations, you may want all the failures to be on one side. In other words,
you want a one-sided confidence limit. Cars that run on gasoline may have a dec-
laration by their manufacturer that they go an average distance of at least 40 miles
per gallon (mpg). If you were to test this by keeping track of distance traveled and
gas usage on a sample of car trips, you may only be concerned if the average was
below the lower confidence limit, but not care if it was above the upper confidence
limit. This makes the boundary on one side infinite (which would really save you
money on gas!). For example, from the results of your study, you could have an
observed value of 45 mpg, with a one-sided confidence interval that goes from
42 mpg to plus infinity mpg!
In biostatistics, it is traditional to always use two-way CIs rather than one-way
CIs, as these are seen as most conservative.
Calculating Confidence Intervals
Although an SE and a CI are different calculations intended to express different
information, they are related in that the SE is used in the CI calculation. SEs and
CIs are calculated using different formulas (depending on the type of sample sta-
tistic for which you are calculating the SE and CI). In the following sections, we
describe methods of calculating SEs and CIs for commonly used sample
statistics.
Before you begin: Formulas for confidence
limits in large samples
Most of the methods we describe in the following sections are based on the
assumption that your sample statistic has a sampling distribution that’s approxi-
mately normal (Chapter 3 covers sampling distributions). There are strong theo-
retical reasons to assume a normal or nearly normal sampling distribution if you
draw a large enough samples.
For any normally distributed sample statistic, the lower and upper confidence
limits can be calculated from the observed value of the statistic (V) and standard
error (SE) of the statistic:
CL
V
k
SE
L
CL
V
k
SE
U