INCHAPTERS 5 AND6 we considered tests dealing with frequency (categorical) data.
In those situations, the results of any experiment can usually be represented by a few
subtotals—the frequency of occurrence of each category of response. In this and subse-
quent chapters, we will deal with a different type of data, that which I have previously
termed measurement or quantitative data.
In analyzing measurement data, our interest can focus either on differences between
groups of subjects or on the relationship between two or more variables. The question of
relationships between variables will be postponed until Chapters 9, 10, 15, and 16. This
chapter will be concerned with the question of differences, and the statistic we will be most
interested in will be the sample mean.
Low-birthweight (LBW) infants (who are often premature) are considered to be at risk
for a variety of developmental difficulties. As part of an example we will return to later,
Nurcombe et al. (1984) took 25 LBW infants in an experimental group and 31 LBW in-
fants in a control group, provided training to the parents of those in the experimental
group on how to recognize the needs of LBW infants, and, when these children were
2 years old, obtained a measure of cognitive ability. Suppose that we found that the LBW
infants in the experimental group had a mean score of 117.2, whereas those in the control
group had a mean score of 106.7. Is the observed mean difference sufficient evidence for
us to conclude that 2-year-old LBW children in the experimental group score higher, on
average, than do 2-year-old LBW control children? We will answer this particular ques-
tion later; I mention the problem here to illustrate the kind of question we will discuss in
this chapter.7.1 Sampling Distribution of the Mean
As you should recall from Chapter 4, the sampling distribution of any statistic is the distri-
bution of values we would expect to obtain for that statistic if we drew an infinite number
of samples from the population in question and calculated the statistic on each sample. Be-
cause we are concerned in this chapter with sample means, we need to know something
about the sampling distribution of the mean.Fortunately, all the important information
about the sampling distribution of the mean can be summed up in one very important theo-
rem: the central limit theorem. The central limit theoremis a factual statement about the
distribution of means. In an extended form it states:
Given a population with mean mand variance s^2 , the sampling distribution of the mean
(the distribution of sample means) will have a mean equal to m(i.e., ), a vari-
ance ( ) equal to , and a standard deviation ( ) equal to. The distribution
will approach the normal distribution as n, the sample size, increases.^1
This is one of the most important theorems in statistics. It not only tells us what the
mean and variance of the sampling distribution of the mean must be for any given sample
size, but also states that as nincreases, the shape of this sampling distribution approaches
normal, whateverthe shape of the parent population. The importance of these facts will
become clear shortly.sX^2 s^2 >n sX s> 1 nmX=m180 Chapter 7 Hypothesis Tests Applied to Means
(^1) The central limit theorem can be found stated in a variety of forms. The simplest form merely says that the sam-
pling distribution of the mean approaches normal as nincreases. The more extended form given here includes all
the important information about the sampling distribution of the mean.
sampling
distribution of
the mean
central limit
theorem