Question

3. Suppose a luxury ship colluded and sank. The ship had 2224 passengers and crew and 1502 people died. The ship had three classes of passengers, 323 passengers in the first class, 277 in the second class, and 709 in the third class. The number of passengers survived were 200 in the first class, 119 in the second, and 181 in the third, respectively.

   a. Find the chi-square statistic and the p-value.
   b. What are the hypotheses?
   c. Construct the table for expected frequency and explain it. d. What is the conclusion of this test?

Answer 1

Solution:

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

b. What are the hypotheses?

Null hypothesis: H0: Two variables class of passenger and whether he/she survives are independent.

Alternative hypothesis: Ha: Two variables class of passenger and whether he/she survives are dependent.

We assume/given level of significance = α = 0.05

a. Find the chi-square statistic and the p-value.

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 = 3

Degrees of freedom = df = (r – 1)*(c – 1) = 1*2 = 2

α = 0.05

Critical value = 5.991465

(by using Chi square table or excel)

c. Construct the table for expected frequency and explain it.

Calculation tables for test statistic are given as below:

Observed Frequencies
	Class of Passenger
Survive or died?	First	Second	Third	Total
Survive	200	119	181	500
Died	123	158	528	809
Total	323	277	709	1309

Expected Frequencies
	Class of Passenger
Survive or died?	First	Second	Third	Total
Survive	123.3766	105.806	270.8174	500
Died	199.6234	171.194	438.1826	809
Total	323	277	709	1309

	(O - E)^2/E
	First	Second	Third
Survive	47.58715	1.645302	29.78822
Died	29.41109	1.016874	18.41052

Test Statistic = Chi square = ∑[(O – E)^2/E] = 127.8592

χ2 test statistic = 127.8592

P-value = 0.0000

(By using Chi square table or excel)

d. What is the conclusion of this test?

P-value < α = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that two variables class of passenger and whether he/she survives are dependent.