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Statistical Tests

All specific statements of comparisons have been tested for statistical significance
at the .05 level using Student's *t*-statistics to ensure that the differences
are larger than those that might be expected due to sampling variation. Adjustments
for multiple comparisons were not included. Readers are cautioned not to draw causal
inferences based on the univariate and bivariate results presented. Many of the
variables examined in this report may be related to one another, and complex interactions
and relationships among the variables have not been explored. The variables examined
here are also just a few of those that can be examined in these data.

The tests of significance used in this report are based on Student's t statistics
for the comparisons of means and of percentages. To test for a difference between
two subgroups in the population percentage having a particular characteristic, say
*p*_{1} versus *p*_{2}, the test statistic is computed
as:

where *p*_{i} is the estimated percentage of subgroup *i*(*i*
= 1, 2) having the particular characteristic and *s.e.(p*_{i}) is
the standard error of that estimate. Thus, if *p*_{1} is the 74 percent
of students attending assigned public schools in 2003, with a standard error of
0.6, and *p*_{2} is the 73 percent of students attending assigned
public schools in 2007, with a standard error of 0.7, then the *t*-value
is equal to 0.74. The decision rule is to reject the null hypothesis (i.e., there
is no measurable difference between the two groups in the population in terms of
the percentage having the characteristic) if |*t*| > *t*_{α/2}_{;df},
where *t*_{α/2}_{;df} is the value such that the probability
a Student's *t* random variable with *df* degrees of freedom exceeds
that value is α/2 . All tests in this report are based on a significance level
of 0.05, i.e., α = 0.05. When the degrees of freedom are large, greater than
120, *t*_{0.025;df} ≈ 1.96. Regarding the example given above,
the *t*-value of 0.74, which is less than 1.96, indicates that the null hypothesis
cannot be rejected. Simply put, there is no statistically measurable difference
between the percentage of students attending assigned public schools in 2003 compared
with 2007.

Tests of significant differences in estimates for students in assigned public schools
across more than two years are based on regressions. Regression tests were used
to verify that there was a positive or negative trend over time. We made one exception
to discussing trends for only assigned public schools. We tested the overall trend
in percentage enrollment for both types of private schools and chosen public schools
with a regression for summary statements. However, in the detailed findings by subpopulations,
we discuss only point-to-point comparisons.

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