Statistical Methods for Psychology

in the sample. As. (The symbol is read

“approaches.”) Since the skewness of the sampling distribution of disappears as the

number of degrees of freedom increases, the tendency for sto underestimate swill also

disappear. Thus, for an infinitely large number of degrees of freedom, t will be normally

distributed and equivalent to z.

The test of one sample mean against a known population mean, which we have just per-

formed, is based on the assumption that the sample was drawn from a normally distributed

population. This assumption is required primarily because Gosset derived the tdistribution

assuming that the mean and variance are independent, which they are with a normal distri-

bution. In practice, however, our t statistic can reasonably be compared to the tdistribution

whenever the sample size is sufficiently large to produce a normal sampling distribution of

the mean. Most people would suggest that annof 25 or 30 is “sufficiently large” for most

situations, and for many situations it can be considerably smaller than that.

On the other hand, Wuensch (1993, personal communication) has argued convincingly

that, at least with veryskewed distributions, the fact that nis large enough to lead to a sam-

pling distribution of the mean that appears to be normal does not guarantee that the result-

ing sampling distribution of t follows Student’s t distribution. The derivation of t makes

assumptions both about the distribution of means (which is under the control of the Central

Limit Theorem), and the variance, which is not controlled by that theorem.

Degrees of Freedom

I have mentioned that the t distribution is a function of the degrees of freedom (df). For the

one-sample case, df 5 n 2 1; the one degree of freedom has been lost because we used the

sample mean in calculating. To be more precise, we obtained the variance ( ) by calcu-

lating the deviations of the observations from their own mean (X 2 ), rather than from the

population mean (X2 m). Because the sum of the deviations about the mean

is always zero, only n 2 1 of the deviations are free to vary (the nth deviation is determined

if the sum of the deviations is to be zero).

Psychomotor Abilities of Low-Birthweight Infants

An example drawn from an actual study of low-birthweight (LBW) infants will be useful at

this point because that same general study can serve to illustrate both this particular t test

and other t tests to be discussed later in the chapter. Nurcombe et al. (1984) reported on an

intervention program for the mothers of LBW infants. These infants present special prob-

lems for their parents because they are (superficially) unresponsive and unpredictable, in

Cg(X^2 X)D

X

s^2 s^2

s^2

nQ q, p(s^2 ,s^2 )Qp(s^2 .s^2 ) Q

Section 7.3 Testing a Sample Mean When sIs Unknown—The One-Sample tTest 187

Figure 7.5 tdistribution for 1, 30, and `degrees of freedom

–3

f(

t)

–2 3

t 30

t 1

t = z

2

t

–1 0 1