Do these data provide evidence, at the 0.02 level of significance, that the crime rate is associated with
the temperature at the time of the crime?solution:I.
H 0 : The crime rate is independent of temperature (or, H 0 : Crime Rate and Temperature are not
associated).H (^) A : The crime rate is not independent of temperature (or, H (^) A : Crime Rate and Temperature are
associated).
II . We will use a chi-square test for independence.
The cities were randomly selected.
A matrix of expected values (using the TI-83/84 as explained in the previous Calculator Tip) is
found to be:
. Since all expected values are greater than 5, the conditionsare present for a chi-square test.
III. , df = (3 – 1)(3 – 1) = 4
0.01 < P -value < 0.02 (from Table C; or P- value = DISTR χ^2 cdf(12.92,1000,410)=0.012
).
(Note that the entire problem could be done by entering the observed values in MATRIX [A] and
using STAT TESTS χ^2 -Test .]
IV . Since P < 0.02, we reject H 0 . We have strong evidence that the number of crimes committed is
related to the temperature at the time of the crime.Chi-Square Test for Homogeneity of Proportions (or Homogeneity of Populations)
In the previous section, we tested for the independence of two categorical variables measured on a single
population. In this section we again use the chi-square statistic but will investigate whether or not the
values of a single categorical variable are proportional among two or more populations. In the previous
section, we considered a situation in which a sample of 36 students was selected and were then
categorized according to gender and political party preference. We then asked if gender and party
preference are independent in the population. Now suppose instead that we had selected a random sample
of 20 males from the population of males in the school and another, independent, random sample of 16
females from the population of females in the school. Within each sample we classify the students as
Democrat, Republican, or Independent. The results are presented in the following table, which you should
notice is exactly the same table we presented earlier when gender was a category.