Statistical Methods for Psychology

Because we know that the sampling distribution is normally distributed with a mean of

50 and a standard error of 2.58, we can find areas under the distribution by referring to

tables of the standard normal distribution. Thus, for example, because two standard errors

is 2(2.58) 5 5.16, the area to the right of is simply the area under the normal

distribution greater than two standard deviations above the mean.

For our particular situation, we first need to know the probability of a sample mean

greater than or equal to 56, and thus we need to find the area above We can calcu-

late this in the same way we did with individual observations, with only a minor change in

the formula for z:

becomes

which can also be written as

For our data this becomes

Notice that the equation for zused here is in the same form as our earlier formula for z in

Chapter 4. The only differences are that Xhas been replaced by and shas been replaced

by. These differences occur because we are now dealing with a distribution of means,

and thus the data points are now means, and the standard deviation in question is now the

standard error of the mean (the standard deviation of means). The formula for zcontinues to

sX

X

z=

56250

10

115

=

6

2.58

=2.32

z=

X2m

s

1 n

z=

X2m

sX

z=

X2m

s

X=56.

X=55.46

184 Chapter 7 Hypothesis Tests Applied to Means

f(

X

)

CBCL Mean

0.0

40 45 50 55

56

60

0.2

0.1

0.3

0.4

Figure 7.3 Sampling distribution of the mean for n 5 15 drawn from a population

with m 5 50 and s 5 10