Question

Here are the figures for the loyalty of a sample of imperial soldiers on the Death Star:

	traitors	loyal
stormtroopers	250	2000
officers	190	1200

Please fill in the appropriate totals:

	traitors	loyal	total
stormtroopers	250	2000
officers	190	1200
total

Assuming these totals, what counts would be expected in each cell (to two places after the decimal) if soldier type and loyalty status were independent?

	traitors	loyal
stormtroopers
officers

Assuming soldier type and loyalty status are independent, fill in (two two places after the decimal) each chi-square contribution:

	traitors	loyal
stormtroopers
officers

What is the value of the chi-square test-statistic (to one place after the decimal)?

Using an online chi-square calculator, what is the p-value for this rounded chi-square value?

Do we have evidence at the 10% level that soldier type and loyalty status (for all Death Star soldiers) are dependent (0 = no, 1 = yes)?
Do we have evidence at the 5% level that soldier type and loyalty status (for all Death Star soldiers) are dependent (0 = no, 1 = yes)?
Do we have evidence at the 1% level that soldier type and loyalty status (for all Death Star soldiers) are dependent (0 = no, 1 = yes)?

Answer 1

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

Null hypothesis: H0: Two variables soldier type and loyalty status are independent.

Alternative hypothesis: Ha: Two variables soldier type and loyalty status are dependent.

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

Calculation tables for test statistic are given as below:

Observed Frequencies
	Column variable
Row variable	Traitors	Loyal	Total
Storm troopers	250	2000	2250
Officers	190	1200	1390
Total	440	3200	3640

Expected Frequencies
	Column variable
Row variable	Traitors	Loyal	Total
Storm troopers	271.978	1978.022	2250
Officers	168.022	1221.978	1390
Total	440	3200	3640

Calculations
	(O - E)
Row variable	Traitors	Loyal	Total
Storm troopers	-21.978022	21.97802198
Officers	21.97802198	-21.97802198
Total

Chi-square contribution:
	(O - E)^2/E
Row variable	Traitors	Loyal	Total
Storm troopers	1.776001776	0.244200244	2.02020202
Officers	2.874823019	0.395288165	3.270111184
Total			5.290313204

Test Statistic = Chi square = ∑[(O – E)^2/E] = 5.290313204

χ2 test statistic = 5.290313204

P-value = 0.021444

(By using Chi square table or excel)

For α = 0.10

P-value < α = 0.10

So, we reject the null hypothesis

Yes, we have sufficient evidence at 10% that two variables soldier type and loyalty status are dependent.

For α = 0.05

P-value < α = 0.05

So, we reject the null hypothesis

Yes, we have sufficient evidence at 10% that two variables soldier type and loyalty status are dependent.

For α = 0.01

P-value > α = 0.01

So, we do not reject the null hypothesis

No, we don’t have sufficient evidence at 10% that two variables soldier type and loyalty status are dependent.