Question

7. A graduate student in criminal science suspects that there is a link between hair color and crime.

Out of 200 randomly selected humans, she finds that 54 are blond criminals,
26 are criminals, but not blond,
6 are blond, but not criminals, and

114 are neither blond nor criminals.

Construct a contingency table, and conduct a hypothesis test to evaluate the independence of blondness and crime at significance level ? = 0.05.

Answer 1

Solution:

Here, we have to use chi square test for independence of two categorical variables.

Null hypothesis: H₀: The blondness and crime are independent.

Alternative hypothesis: H_a: The blondness and crime are not independent.

We are given level of significance = α = 0.05

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

E = row total * column total / Grand total

We are given

Number of rows = r = 2

Number of columns = c = 2

Degrees of freedom = df = (r – 1)*(c – 1) = 1*1 = 1

α = 0.05

Critical value = 3.841459

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Column variable
Row variable	Blond	No Blond	Total
Criminal	54	26	80
No criminal	6	114	120
Total	60	140	200

Expected Frequencies
	Column variable
Row variable	Blond	No Blond	Total
Criminal	24	56	80
No criminal	36	84	120
Total	60	140	200

Calculations
(O - E)
30	-30
-30	30

(O - E)^2/E
37.5	16.07143
25	10.71429

Chi square = ∑[(O – E)^2/E] = 89.28571

P-value = 0.0000

(By using Chi square table or excel)

P-value < α = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that the blondness and crime are not independent.

There is sufficient evidence to conclude that there is a link between hair color and crime.