Abstract
In chapters 8 and 9, tests of statistical significance were presented that did not make assumptions about the population distribution of the characteristics studied. We now turn to a different type of test of statistical significance in which the researcher must make certain assumptions about the population distribution. These tests, called parametric tests, are widely used in criminal justice and criminology because they allow the researcher to test hypotheses in reference to interval-level scales.
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Notes
- 1.
F. H. Allport, “The J-Curve Hypothesis of Conforming Behavior,” Journal of Social Psychology 5 (1934): 141–183.
- 2.
Our hypothesized results mirror those found in prior studies; see R. J. Hernstein, Some Criminogenic Traits of Offenders,” in J. Q. Wilson (ed.), Crime and Public Policy (San Francisco: Institute for Contemporary Studies, 1983). Whether these differences mean that offenders are, on average, less intelligent than nonoffenders is an issue of some controversy in criminology, in part because of the relationship of IQ to other factors, such as education and social status.
- 3.
By implication, we are asking whether it is reasonable to believe that our sample of prisoners was drawn from the general population. For this reason, the z-test can also be used to test for random sampling. If you have reason to doubt the sampling methods of a study, you can conduct this test, comparing the observed characteristics of your sample with the known parameters of the population from which your sample was drawn.
- 4.
In principle, any distribution may be arranged in such a way that it conforms to a normal shape. This can be done simply by ranking scores and then placing the appropriate number within standard deviation units appropriate for constructing a standard normal distribution.
- 5.
It would not make sense, however, to use a normal distribution test for nominal-scale measures with more than two categories. The normal distribution assumes scores above and below a mean. The sampling distribution of a proportion follows this pattern because it includes only two potential outcomes, which then are associated with each tail of the distribution. In a multicategory nominal-scale measure, we have more than two outcomes and thus cannot fit each outcome to a tail of the normal curve. Because the order of these outcomes is not defined, we also cannot place them on a continuum within the normal distribution. This latter possibility would suggest that the normal distribution could be applied to ordinal-level measures. However, because we do not assume a constant unit of measurement between ordinal categories, the normal distribution is often considered inappropriate for hypothesis testing with ordinal scales. In the case of a proportion, there is a constant unit of measurement between scores simply because there are only two possible outcomes (e.g., success and failure).
- 6.
As noted on page 105 (footnote 1), computerized statistical analysis packages, such as SPSS, use this corrected estimate in calculating the variance and standard deviation for sample estimates.
- 7.
Our statistical problem is that we assume that _ and _ are independent in developing the t distribution. When a distribution is normal, this is indeed the case. However, for other types of distributions, we cannot make this assumption, and when N is small, a violation of this assumption is likely to lead to misleading approximations of the observed significance level of a test.
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Weisburd, D., Britt, C. (2014). The Normal Distribution and Its Application to Tests of Statistical Significance. In: Statistics in Criminal Justice. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-9170-5_10
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DOI: https://doi.org/10.1007/978-1-4614-9170-5_10
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