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top entry; the value of 9.33, which appears below it, gives the .01 breaking
point). Since our observed (computed) F of 18.75 exceeds the value of
4.75, we know that it lies to the right of the 5 percent benchmark in the
5 percent area under the curve, and we can conclude that our obtained F
value is statistically significant at the .05 alpha level. We would therefore
reject the null hypothesis that the two groups of scores were drawn from
the same population, and instead interpret the mean difference as a valid
difference between the Blue Room and Red Room groups.
Technological advances now enable users of major statistical software
packages to bypass the Critical Values table. SPSS and SAS compute the
exact probability of obtaining the F ratio by chance alone with the given
degrees of freedom on the presumption that the null hypothesis is true.
The column in the summary table (Table 3.2) labeled “Probability” repre-
sents this exact probability truncated to a value of .001 because that is the
number of decimal places SPSS is designed to report (SAS provides prob-
ability values to four decimal places). Thus, the exact probability is not
actually zero but some very small value that cannot be displayed with only
the three decimal places that are permitted in the output. Our obtained F
value would therefore ordinarily occur less than once in every thousand
occasions if the null hypothesis is true. Students should not report the
probablity as .000 but should instead evaluate and report it with respect
to the alpha level they are using (e.g., p
<.
05).
4.4.3 DIFFERENCES OF GROUP MEANS
In our room color study, we evaluated the obtained probability value
against our alpha level, .05, and determined that we have a statistically
significant effect of the independent variable. Because there are only two
group means compared, we know automatically that they are significantly
different. In looking at the actual values of the means of the two groups,
we can conclude that the students completing the mood survey in the blue
room reported significantly more relaxation than those completing the
survey in the red room, thus confirming the experimental hypothesis.
Had there been more than two group means, that is, if there were three
or more groups in the design, we would no longer know which means were
significantly different from which others given a statistically significant F
ratio. It is possible that only some of the means would differ significantly
from others; in fact, although it is not common, it is possible that no two
means, when taken individually, differ significantly from each other. As a
result of this ambiguity, when there are three or more means to compare,
it is necessary to perform a follow-up, post-ANOVA analysis to examine
the difference between means. We will elaborate on this in Chapter 7.
4.4.4 ALPHA LEVEL CONSIDERATIONS
Most of the statistical analyses that we perform will be run under an alpha
level of .05, a standard across the scientific disciplines. But there are a few
times when circumstances will either suggest or demand that you modify
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