Killed by Police: Another Approach to Age/Race

Intersectionality:  Age and Race/Ethnicity

In another activity, we asked whether there might be a relationship between the age and the race/ethnicity of persons killed by  police.  Now we can use ANOVA to study that question.

Using a database of persons killed by police from 2016-2020 and random.org Links to an external site., independent simple random samples were generated of Asian, Black, Hispanic, Native American and White males, and their ages were recorded.  

For ANOVA, it is not necessary for the independent samples to have the same size.  A sample size n=10 was used for Black, Hispanic and white males.  Due to the smaller numbers of Asian and Native American males in the population (database), smaller samples were generated to keep n < .05N.  Persons whose gender and/or race/ethnicity were not recorded were omitted from analysis.  So were Pacific Islanders (44 individuals).

The sample data (age in years) are recorded in the following table:

Conditions for an ANOVA Test

1. Each of the k samples is independent and randomly selected.  The sampling process conformed to this. 
2. Each of the k samples came from a population with a normal distribution of values of the variable.
3. The variables from each group come from populations with approximately the same standard deviation.

QUESTION 1:  How can we verify condition B?

We can use technology to create a normal probability plot Links to an external site.If our app or device shows us "bounds," the answer will be clear; otherwise, we "eyeball it."    If you have not studied this, here is a video about confidence intervals for linear regression Links to an external site. that explains it in more detail than you want -- but if you start at 2:30 and skip the technical parts, you will get the basic idea.   Your statistical calculator or app probably has a 'normal probability plot" function.

Or we can do a chi-square goodness-of-fit test for each sample and hope that we do NOT reject the null hypothesis (that the data fits the normal distribution).  

QUESTION 2:  Use either of these methods to estimate whether the five samples in our data set could have come from normally distributed populations.  What is your conclusion?

QUESTION 3:  How can we determine whether the data fulfill Condition C?

Note:  In a way, this is a trick question.  You almost certainly have not studied any method for comparing three or more standard deviations or variances.  You might have learned already how to test two standard deviations or variances using a 2-Sample F test.    If not, this 17-minute video Links to an external site. spells it out.  It's helpful because it introduces the F distribution, which we also use in ANOVA.  Don't be scared by the hand calculations.  Your statistical calculator or app should have a "2-sample F test function.   

QUESTION 3a:  Which two variances should we test?  We might as well use the samples with the largest and the smallest variances.  If they are "approximately equal," then the other ones will be too.  Enter the data in the table into L1 - L5, and use 1-variable statistics on each list to get the SAMPLE standard deviations. 

QUESTION 3b:  Use a 2-sample F test where the null hypothesis is that "the variances are equal" and the alternate hypothesis is that "the variances are not equal."  Write down all steps of the hypothesis test and use  a 5% level of significance.  Since the calculator or app will give you a p-value, it's easier to use the p-value method.    What is your conclusion?

We Have a Problem

Comparing the Asian men and the white men, you will reject the null hypothesis.  At a 5% level of significance, the variances are different.  Condition C is NOT satisfied.  And if you look at the table, you will probably see that the Asian men are closer to each other in age.  We should leave the Asian sample out of this analysis, although their mean age does seem to be quite different from any of the other groups.

QUESTION 3c:  The sample with the second-lowest standard deviation is the Native American group.  Repeat the 2-sample F test with the Native American men and the white men.  What do you conclude?

QUESTION 4:  Conduct an ANOVA hypothesis test on the remaining four racial/ethnic groups:  Black, Hispanic, Native American, white.  Use a 5% level of significance and make sure to include all steps of the hypothesis test.  What is the conclusion?  Based on these data, is there a relationship between the age and the race/ethnicity of men killed by the police?

Post final comments, answers to questions, etc. in the Killed by Police: Another Approach to Age/Race Discussion.