Statistical Methods for Psychology

in this case the results also indicate that Kaufman and Rock’s experimental apparatus

performed as it should.) For those who like technology, a probability calculator at http://www

.danielsoper.com/statcalc/calc40.aspx gives the two-tailed probability as .001483.

Confidence Interval on m

Confidence intervals are a useful way to convey the meaning of an experimental result that

goes beyond the simple hypothesis test. The data on the moon illusion offer an excellent

example of a case in which we are particularly interested in estimating the true value

of m—in this case, the true ratio of the perceived size of the horizon moon to the per-

ceived size of the zenith moon. The sample mean ( ), as you already know, is an unbi-

ased estimate of m. When we have one specific estimate of a parameter, we call this a

point estimate.There are also interval estimates, which are attempts to set limits that have

a high probability of encompassing the true (population) value of the mean [the mean (m)

of a whole population of observations]. What we want here are confidence limitson m.

These limits enclose what is called a confidence interval.^5 In Chapter 3, we saw how to

set “probable limits” on an observation. A similar line of reasoning will apply here, where

we attempt to set confidence limits on a parameter.

If we want to set limits that are likely to include m given the data at hand, what we re-

ally want is to ask how large, or small, the true value of m could be without causing us to

reject H 0 if we ran at test on the obtained sample mean. For example, when we tested the

null hypothesis that m 51.00 we rejected that hypothesis. What if we tested the null hy-

pothesis that m 5 1.15? We would again reject that null. We can keep increasing the value

of m to the point where we just barely do not reject H 0 , and that is the smallest value of m

for which we would be likely to obtain our data at p .025. Then we could start with large

values of m (e.g., 2.2) and keep lowering m until we again just barely fail to reject H 0. That

is the largest value of mfor which we would expect to obtain the data at p .025. Now

any estimate of m between those upper and lower limits would lead us to retain the null hy-

pothesis. Although we could do things this way, there is a shortcut that makes life easier.

But it will come to the same answer.

An easy way to see what we are doing is to start with the formula for tfor the one-

sample case:

From the moon illusion data we know 5 1.463, s 5 0.341, n 5 10. We also know

that the critical two-tailed value fort at a 5 .05 is t.025(9)5 62.262. We will substitute

these values in the formula fort, but this time we will solve for the m associated with this

value of t.

Rearranging to solve for m, we have

m 5 62.262(0.108) 1 1.4635 60.244 1 1.463

t=

X2m

s

1 n

6 2.262=

1.4632m

0.341

110

=

1.4632m

0.108

X

t=

X2m

sX

=

X2m

s

1 n

...

Ú

X

192 Chapter 7 Hypothesis Tests Applied to Means

(^5) We often speak of “confidence limits” and “confidence interval” as if they were synonymous. The pretty much
are, except that the limits are the end points of the interval. Don’t be confused when you see them used
interchangeably.
point estimate
confidence limits
confidence
interval