Statistical Methods for Psychology

Using the 1 0.244 and 2 0.244 separately to obtain the upper and lower limits for m, we have

mupper5 10.244 1 1.463 5 1.707

mlower5 20.244 1 1.463 5 1.219

and thus we can write the 95% confidence limits as 1.219 and 1.707 and the confidence
interval as

CI.95 5 1.219 m 1.707
Testing a null hypothesis about any value of m outside these limits would lead to rejec-
tion of H 0 , while testing a null hypothesis about any value of minside those limits would
not lead to rejection. The general expression is

We have a 95% confidence interval because we used the two-tailed critical value oft at
a 5 .05. For the 99% limits we would take Then the 99% confi-
dence interval is

We can now say that the probability is 0.95 that intervals calculated as we have calcu-
lated the 95% interval above include the true mean ratio for the moon illusion. It is very
tempting to say that the probability is .95 that the interval 1.219 to 1.707 includes the true
mean ratio for the moon illusion, and the probability is .99 that the interval 1.112 to 1.814
includes m. However, most statisticians would object to the statement of a confidence limit
expressed in this way. They would argue that before the experiment is runand the calcula-
tions are made, an interval of the form

has a probability of .95 of encompassing m. However, m is a fixed (though unknown) quan-
tity, and once the data are in, the specific interval 1.219 to 1.707 either includes the value
of m (p 5 1.00) or it does not (p 5 .00). Put in slightly different form,

is a random variable (it will vary from one experiment to the next), but the specific interval
1.219 to 1.707 is not a random variable and therefore does not have a probability associ-
ated with it. Good (1999) has made the point that we place our confidence in the method,
and not in the interval. Many would maintain that it is perfectly reasonable to say that my
confidence is .95 that if you were to tell me the true value of m, it would be found to lie be-
tween 1.219 and 1.707. But there are many people just lying in wait for you to say that the
probabilityis .95 that mlies between 1.219 and 1.707. When you do, they will pounce!
Note that neither the 95% nor the 99% confidence intervals that I computed include the
value of 1.00, which represents no illusion. We already knew this for the 95% confidence
interval because we had rejected that null hypothesis when we ran ourt test at that signifi-
cance level.
I should add another way of looking at the interpretation of confidence limits. State-
ments of the form p(1.219 ,m ,1.707) 5 .95 are not interpreted in the usual way. (In
fact, I probably shouldn’t use pin that equation.) The parameter m is not a variable—it does
not jump around from experiment to experiment. Rather, m is a constant, and the interval is
what varies from experiment to experiment. Thus, we can think of the parameter as a stake
and the experimenter, in computing confidence limits, as tossing rings at it. Ninety-five

X 6 t.025 (sX)

X 6 t.025 (sX)

CI.99=X 6 t.01> 2 (sX)=1.463 6 3.250(0.108)=1.112...m...1.814

t.01/2=t.005= 6 3.250.

CI 1 2a=X 6 ta> 2 (sX)=X 6 ta> 2

s

1 n

... ...

Section 7.3 Testing a Sample Mean When sIs Unknown—The One-Sample tTest 193