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Example 1
observation on an approximately normally distributed random variable centered
around the population covariance with a standard deviation of about 1.16, that is, if the
assumptions in the section “Distribution Assumptions for Amos Models” on p. 35 are
met. For example, you can use these figures to construct a 95% confidence interval on
the population covariance by computing . Later, you
will see that you can use Amos to estimate many kinds of population parameters
besides covariances and can follow the same procedure to set a confidence interval on
any one of them.
Next to the standard error, in the
C.R. column, is the critical ratio obtained by
dividing the covariance estimate by its standard error . This ratio
is relevant to the null hypothesis that, in the population from which Attig’s 40 subjects
came, the covariance between recall1 and recall2 is 0. If this hypothesis is true, and
still under the assumptions in the section “Distribution Assumptions for Amos
Models” on p. 35, the critical ratio is an observation on a random variable that has an
approximate standard normal distribution. Thus, using a significance level of 0.05, any
critical ratio that exceeds 1.96 in magnitude would be called significant. In this
example, since 2.20 is greater than 1.96, you would say that the covariance between
recall1 and recall2 is significantly different from 0 at the 0.05 level.
The P column, to the right of C.R., gives an approximate two-tailed p value for
testing the null hypothesis that the parameter value is 0 in the population. The table
shows that the covariance between recall1 and recall2 is significantly different from 0
with . The calculation of P assumes that parameter estimates are normally
distributed, and it is correct only in large samples. See Appendix A for more
information.
The assertion that the parameter estimates are normally distributed is only an
approximation. Moreover, the standard errors reported in the S.E. column are only
approximations and may not be the best available. Consequently, the confidence
interval and the hypothesis test just discussed are also only approximate. This is
because the theory on which these results are based is asymptotic. Asymptotic means
that it can be made to apply with any desired degree of accuracy, but only by using a
sufficiently large sample. We will not discuss whether the approximation is
satisfactory with the present sample size because there would be no way to generalize
the conclusions to the many other kinds of analyses that you can do with Amos.
However, you may want to re-examine the null hypothesis that recall1 and recall2 are
uncorrelated, just to see what is meant by an approximate test. We previously
concluded that the covariance is significantly different from 0 because 2.20 exceeds
1.96. The p value associated with a standard normal deviate of 2.20 is 0.028 (two-
tailed), which, of course, is less than 0.05. By contrast, the conventional t statistic (for
2.56 1.96 1.160 2.56 2.27±=×±
2.20 2.56 1.16⁄=()
p 0.03=