Question

1. Consider the following data for the abundance of a certain species of birds.

Sex	Spring	Summer	Fall	Winter
Males	163	135	71	43
Females	86	77	40	48

Using chi-square, test the null hypothesis that the ratio of numbers of males to females was the same in all four seasons. (Use alpha = 0.05).

If you rejected the null in 1a, identify the season/seasons that has resulted in the rejection. What conclusion can you draw from your analysis? You may use R to conduct the sub-chi square tests.

Answer 1

Solution:

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

Null hypothesis: H0: the ratio of numbers of males to females was the same in all four seasons.

Alternative hypothesis: Ha: the ratio of numbers of males to females was not same in all four seasons.

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 = 4

Degrees of freedom = df = (r – 1)*(c – 1) = 1*3 = 3

α = 0.05

Critical value = 7.814728

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Column variable
Row variable	Spring	Summer	Fall	Winter	Total
Males	163	135	71	43	412
Females	86	77	40	48	251
Total	249	212	111	91	663

Expected Frequencies
	Column variable
Row variable	Spring	Summer	Fall	Winter	Total
Males	154.733	131.7406	68.97738	56.54902	412
Females	94.26697	80.25943	42.02262	34.45098	251
Total	249	212	111	91	663

Calculations
(O - E)
8.266968	3.259427	2.022624	-13.549
-8.26697	-3.25943	-2.02262	13.54902

(O - E)^2/E
0.441682	0.080642	0.059309	3.246315
0.724992	0.132369	0.097353	5.328613

Test Statistic = Chi square = ∑[(O – E)^2/E] = 10.11127

χ2 statistic = 10.11127

P-value = 0.017644

(By using Chi square table or excel)

P-value < α = 0.05

So, we reject the null hypothesis

There is not sufficient evidence to conclude that the ratio of numbers of males to females was the same in all four seasons.

The winter season that has resulted in the rejection because corresponding chi square contribution is more than other seasons.