Statistical Methods for Psychology

percent of the time, a ring of specified width will encircle the parameter; 5% of the time, it

will miss. A confidence statement is a statement of the probability that the ring has been on

target; it is not a statement of the probability that the target (parameter) landed in the ring.

A graphic demonstration of confidence limits is shown in Figure 7.6. To generate this

figure, I drew 25 samples of n 5 4 from a population with a mean (m) of 5. For every

sample, a 95% confidence limit on m was calculated and plotted. For example, the limits

produced from the first sample (the top horizontal line) were approximately 4.46 and 5.72,

whereas those for the second sample were 4.83 and 5.80. Since in this case we know that

the value of m equals 5, I have drawn a vertical line at that point. Notice that the limits for

samples 12 and 14 do not include m 55. We would expect that 95% confidence limits

would encompass m 95 times out of 100. Therefore, two misses out of 25 seems reason-

able. Notice also that the confidence intervals vary in width. This variability is due to the

fact that the width of an interval is a function of the standard deviation of the sample, and

some samples have larger standard deviations than others.

Using SPSS to Run One-Sample tTests

With a large data set, it is often convenient to use a program such as SPSS to compute t values.

Exhibit 7.1 shows how SPSS can be used to obtain a one-sample t test and confidence lim-

its for the moon-illusion data. To compute tfor the moon illusion example you simply

choose Analyze/Compare Means/One SampletTestfrom the pull down menus, and

then specify the dependent variable in the resulting dialog box. Notice that SPSS’s result

for the ttest agrees, within rounding error, with the value we obtained by hand. Notice also

that SPSS computes the exact probability of a Type I error (the plevel), rather than com-

paring t to a tabled value. Thus, whereas we concluded that the probability of a Type I er-

ror was less than.05, SPSS reveals that the actual probability is .0020. Most computer

programs operate in this way.

But there is a difference between the confidence limits we calculated by hand and those

produced by SPSS, though both are correct. When I calculated the confidence limits by

hand I calculated limits based on the mean moon illusion estimate, which was 1.463. But

SPSS is testing the difference between 1.463 and an illusion mean of 1.00 (no illusion),

and its confidence limits are on this difference. In other words I calculated limits around

1.463, whereas SPSS calculated limits around (1.463 2 1.00 5 0.463). Therefore the SPSS

limits are 1.00 less than my limits. Once you realize that the two procedures are calculat-

ing something slightly different, the difference in the result is explained.^6

7.4 Hypothesis Tests Applied to Means—Two Matched Samples

In Section 7.3 we considered the situation in which we had one sample mean ( ) and

wished to test to see whether it was reasonable to believe that such a sample mean would

have occurred if we had been sampling from a population with some specified mean (often

denoted ). Another way of phrasing this is to say that we were testing to determine

whether the mean of the population from which we sampled (call it ) was equal to some

particular value given by the null hypothesis ( ). In this section we will consider the case

in which we have two matched samples(often called repeated measures,when the same

subjects respond on two occasions, or related samples, correlated samples, paired

m 0

m 1

m 0

X

194 Chapter 7 Hypothesis Tests Applied to Means

(^6) SPSS will give you the confidence limits that I calculated if you use Analyze, Descriptive statistics/Explorer.
matched
samples
repeated
measures
related samples
plevel