Basic Statistics

82 ESTIMATION OF POPULATION MEANS: CONFIDENCE INTERVALS

Figure 7.1 Distribution of means from sample of size 16.

The confidence level is 95%, and we say that we have 95% confidence that 1-1.

lies within the interval. For the interval 253.1-370.7 g, 253.1 g is the lower confi-

dence limit, and 370.7 g is the upper confidence limit. The rule for obtaining a 95%

confidence interval in this situation is

or

Certain assumptions are made in obtaining this confidence interval. We assume that

the 16 infants are a simple random sample from a population with a standard deviation

equal to c. Each infant measured is assumed to be independent of the other infants

measured (e.g., twins are excluded from the study). We also assume that (x- p)/q

is normally distributed, as is the case if X is normally distributed. With large samples

or with a distribution close to the normal, we know that x is normally distributed

and most investigators assume normality unless the sample is quite nonnormal in

appearance.

7.1.3 Choice of Confidence Level

There was no necessary reason for choosing a 95% confidence interval; we might

compute a 90% confidence interval or a 99% confidence interval.

For a 90% interval, we find from Table A.2 that 90% of the z’s lie between il.645,

so that the 90% confidence interval is

311.9 * 1.645(30) = 311.9 & 49.4

or 262.5 to 361.3 g. In repeated sampling, about 90% of all such intervals obtained

cover p. A 99% confidence interval is

311.9 2.575(30) = 311.9 & 77.2

or 234.7-389.1 g.