Researches want to determine whether criminals are more likely to commit crimes

Question

Researches want to determine whether criminals are more likely to commit crimes on certain days of the week. They randomly selected the records of 154 crimes and determined the day each crime was committed. The results are summarized in the table below.

If criminals do not tend to commit crimes on certain day, then crimes should be uniformly distributed in the week. In the case, the distribution of crimes should be as follows in the table below.

A). State the appropriate hypotheses (in English) needed to test whether the random sample of crimes observed came from a population that is uniformly distributed.

B). Calculate the observed value of the chi-square statistic.

C). Fine the appropriate critical value for testing the hypotheses above at the 5% significance level.

D). State a conclusion of the 5% significance level.

E). Calculate the p-value for this test.

Day Mon Tue Wed Thu Fri Sat Sun Total Count 17 22 23 21 27 28 16 154

Explanation / Answer

Expected frequency = 154/7
= 22

a.
H0: random sample of crimes observed came from a population that is uniformly distributed
H1: they are not from uniformly distributed population

b.
test statistic,X^2 = 5.636

c.
n = 7
alpha,a = 0.05

critical value = X^2(a,n-1) = X^2(0.05,6)
= 12.59

d.
Since X^2 < 12.59 hence H0 can not be rejected.
So, we conclude that random sample of crimes observed came from a population that is uniformly distributed.

e.
P_value = 0.465

f0 fe (f0-fe)^2 (f0-fe)^2 / fe 17 22 25 1.136 22 22 0 0.000 23 22 1 0.045 21 22 1 0.045 27 22 25 1.136 28 22 36 1.636 16 22 36 1.636 sum = 5.636

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