Abstract
In chapter 10, we used parametric significance tests to compare the mean or proportion of a single sample with a population goal or parameter. In this chapter, we turn to a more commonly used application of parametric tests of statistical significance: comparisons between samples. Let's say, for example, that you are interested in whether there is a difference in the mean salaries of male and female police officers or in the proportions of African Americans and others arrested last year. Your question in either of these cases is not whether the population parameters have particular values, but whether the parameters for the groups examined in each case are different. This involves comparing means and proportions for two populations. If you take samples from these populations, you can make inferences regarding the differences between them by building on the normal distribution tests covered in Chapter 10.
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Notes
- 1.
Michael Pendleton, Ezra Stotland, Philip Spiers, and Edward Kirsch, "Stress and Strain among Police, Firefighters, and Government Workers: A Comparative Analysis," Criminal Justice and Behavior 16 (1989): 196–210.
- 2.
The logic here follows simple common sense. If you select each case independently and randomly from a population, on each selection you have an equal probability of choosing any individual, whether male or female, college-educated or not, and so on. From the perspective of a particular group—for example, males—each time you choose a man, the method can be seen as independent and random. That is, the likelihood of choosing any male from the sample is the same each time you draw a case. Of course, sometimes you will draw a female. However, within the population of males, each male has an equal chance of selection on each draw. And if the sampling method is independent, then each male has an equal chance of being selected every time a case is selected.
- 3.
See H. M. Blalock, Social Statistics (New York: McGraw-Hill, 1979), p. 231.
- 4.
A test of statistical significance may be performed to assess differences in variances. It is based on the F distribution, which is discussed in detail in Chapter 12. The test takes a ratio of the two variances being examined:
$$F = \frac{{\hat \sigma _{{\rm{larger variance}}}^2 }}{{\hat \sigma _{{\rm{smaller variance}}}^2 }}$$ - 5.
These data are available through the National Archive of Criminal Justice Data and can be accessed at http://www.icpsr.umich.edu/NACJD.
- 6.
In practice, many statistics texts use the z-test for examples involving proportions. Generally this is done because a difference of proportions test is appropriate only for larger samples, and with larger samples, there is substantively little difference between the outcomes of these two normal distribution tests. We illustrate a difference of proportions problem using a t-test because it follows the logic outlined in Chapter 10. That is, in the case where σ is unknown, a t-test should be used. Moreover, most packaged statistical programs provide outcomes only in terms of t-tests.
- 7.
See Chester Britt III, Michael Gottfredson, and John S. Goldkamp, "Drug Testing and Pretrial Misconduct: An Experiment on the Specific Deterrent Effects of Drug Monitoring Defendants on Pretrial Release," Journal of Research in Crime and Delinquency 29 (1992): 62–78.
- 8.
In fact, although we do not examine their findings here, Britt and colleagues conducted their study in two Arizona counties.
- 9.
Here we examine the t-test for dependent samples only in reference to mean differences for interval-level data. However, this test may also be used for dichotomous nominal-level data. Suppose you were assessing the absence or presence of some characteristic or behavior at two points in time. If each observation were coded as 0 or 1, then you would calculate the mean difference (X d ) and the standard deviation of the difference (s d ) using the same equations as in this section. The only difference from the example discussed in the text is that you would work only with zeros and ones.
- 10.
See Chapter 12 for an example of a rank-order test (the Kruskal-Wallis one-way analysis of variance).
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Weisburd, D., Britt, C. (2014). Comparing Means and Proportions in Two Samples. In: Statistics in Criminal Justice. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-9170-5_11
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