4.1.3 Introduction to Confidence Estimation
Statistical inferenceis a procedure whereby inferences about a population are
made on the basis of the results obtained from a sample drawn from that pop-
ulation. Professionals in health science are often interested in a parameter of a
certain population. For example, a health professional may be interested in
knowing what proportion of a certain type of person, treated with a particular
drug, su¤ers undesirable side e¤ects. The process of estimation entails calcu-
lating, from the data of a sample, some statistic that is o¤ered as an estimate of
the corresponding parameter of the population from which the sample was
drawn.
A point estimate is a single numerical value used to estimate the corre-
sponding population parameter. For example, the sample mean is a point esti-
mate for the population mean, and the sample proportion is a point estimate
for the population proportion. However, having access to the data of a sample
and a knowledge of statistical theory, we can do more than just providing a
point estimate. The sampling distribution of a statistic—if available—would
provide information on biasedness/unbiasedness (several statistics, such asx,p,
ands^2 , are unbiased) and variance.
Variance is important; a small variance for a sampling distribution indicates
that most possible values for the statistic are close to each other, so that a par-
ticular value is more likely to be reproduced. In other words, the variance of a
sampling distribution of a statistic can be used as a measure of precision or
reproducibility of that statistic; the smaller this quantity, the better the statistic
as an estimate of the corresponding parameter. The square root of this variance
is called thestandard errorof the statistic; for example, we will have the stan-
dard error of the sample mean, or SEðxÞ; the standard error of the sample
proportion, SEðpÞ; and so on. It is the same quantity, but we use the term
standard deviationfor measurements and the termstandard errorwhen we refer
to the standard deviation of a statistic. In the next few sections we introduce a
process whereby the point estimate and its standard error are combined to form
an interval estimate orconfidence interval. A confidence interval consists of two
numerical values, defining an interval which, with a specified degree of confi-
dence, we believe includes the parameter being estimated.
4.2 ESTIMATION OF MEANS
The results of Example 4.1 are not coincidences but are examples of the char-
acteristics of sampling distributions in general. The key tool here is thecentral
limit theorem, introduced in Section 3.2.1, which may be summarized as fol-
lows: Given any population with meanmand variances^2 , the sampling distri-
bution ofxwill be approximately normal with meanmand variances^2 =nwhen
the sample sizenis large (of course, the larger the sample size, the better the
152 ESTIMATION OF PARAMETERS