Basic Statistics

CONFIDENCE INTERVALS 81

population under the new diet will be quite close to that under the old diet. We

assume, therefore, that the population’s standard deviation, under the new diet, is

120g. The data for this example are given in Table 7.1 in the columns labeled

“Supplemented Diet.”

Since the assumed population standard deviation is a = 120 g, the standard

deviation for the population of means of samples of size 16 is ax = a/& =

120/m = 120/4 = 30 g (see Section 5.3, where sampling properties of the mean

and variance are discussed). In order to compute a 95% confidence interval for p,

we find the distance on each side of p within which 95% of all the sample means

lie. This is illustrated in Figure 7.1. From Table A.2 we find that 95% of all the z’s

lie between -1.96 and t-1.96. The value of the sample mean f? corresponding to

z = 1.96 is obtained by solving 1.96 = (x - ,LL)/~F for x. This gives

• X = p + 1.96(30)

or -
X = ,LL + 58.8

Similarly, corresponding to z = -1.96, we have = p - 1.96(30). Thus 95% of
all the f?’s lie within a distance of 1.96(30) = 58.8 g from the mean p. The 95%
confidence interval for p is
X f 58.8

where & signifies plus or minus. Substituting in the numerical value of the sample
mean, we have
311.9 f 58.8

or
253.1 to 370.7 g

The measured sample mean f?, 31 1.9 g, may or may not be close to the population
mean p. If it is one of the x’s falling under the shaded area in Figure 7.1 the interval
311.9 f 58.88 includes the population p. On the other hand, if 311.9 is one of the
sample means lying farther away from p in the unshaded area, the confidence interval
does not include p. We do not know whether or not this particular is close to p,
but we do know that in repeated sampling 95% of all the x’s obtained will be close
enough to p so that the interval i 58.8 contains p.
If we take a second sample of 16 infants, a different x will be obtained and the
95% confidence interval will then be a different interval. The intervals obtained vary
from one sample to the next. They are formed in such a way, however, that in the
long run about 95% of the intervals contain 1-1.

7.1.2 Definition of Confidence Interval

A 95% confidence interval for a population parameter is an interval obtained from

a sample by some specified method such that, in repeated sampling, 95% of the

intervals thus obtained include the value of the parameter. For further discussion of

the meaning of confidence limits, see Rothman [ 19861 or van Belle et al. [2004].