Question

Are gender and age-associated with each other for emergency room visits? Do a chi-squared test for the following.   Please report the expected counts as a matrix. Also, is this a test of independence or not? Make the 4 steps cleared as outlined on page 325 of your textbook.

The following data represent the gender of 489 randomly selected patients who visited the emergency room with an injury-related emergency.

Gender	(Age) Under 15	15-24	25-44	45-64	65-74	75 and older
Male	66	56	92	37	8	11
Female	44	39	66	34	12	24

Answer 1

Solution:

This is a test of independence.

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

Null hypothesis: H₀: Gender and age are not associated with each other for emergency room visits.

Alternative hypothesis: H_a: Gender and age are associated with each other for emergency room visits.

We assume 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 = 6

Degrees of freedom = df = (r – 1)*(c – 1) = 1*5 = 5

α = 0.05

Critical value = 11.0705

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Age Group
Gender	<15	15 to 24	25 to 44	45 to 64	65 to 74	> 75	Total
Male	66	56	92	37	8	11	270
Female	44	39	66	34	12	24	219
Total	110	95	158	71	20	35	489

Expected Frequencies
	Age Group
Gender	<15	15 to 24	25 to 44	45 to 64	65 to 74	> 75	Total
Male	60.7362	52.45399	87.23926	39.20245	11.04294	19.32515	270
Female	49.2638	42.54601	70.76074	31.79755	8.957055	15.67485	219
Total	110	95	158	71	20	35	489

Calculations
(O - E)
5.263804	3.546012	4.760736	-2.20245	-3.04294	-8.32515
-5.2638	-3.54601	-4.76074	2.202454	3.042945	8.325153

(O - E)^2/E
0.456196	0.239719	0.259798	0.123737	0.8385	3.586423
0.562434	0.295544	0.320299	0.152553	1.033768	4.421618

Chi square = ∑[(O – E)^2/E] = 12.29059

P-value = 0.031016

(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 Gender and age are associated with each other for emergency room visits.