Question

The California Transit system would like to see if the type of transportation used depended on the time of day: Please use the following table to answer questions:

Transportation Total		Bus		Carpool		Own Car		Total
Time of Day		Observed	Expected	Observed	Expected	Observed	Expected
	Morning	60		30		30		120
	Night	20		20		40		80
	Total	80		50		70		200

1. What is the value of χ², for independent samples?
2. What is the critical value of χ²?
3. What is the conclusion in APA format?

Answer 1

Solution:

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

Null hypothesis: H₀: The type of transportation used not depended on the time of day.

Alternative hypothesis: H_a: The type of transportation used depended on the time of day.

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

What is the critical value of χ²?

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)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Type of transportation
Time of Day	Bus	Carpool	Own Car	Total
Morning	60	30	30	120
Night	20	20	40	80
Total	80	50	70	200

Expected Frequencies
	Type of transportation
Time of Day	Bus	Carpool	Own Car	Total
Morning	48	30	42	120
Night	32	20	28	80
Total	80	50	70	200

Transportation Total		Bus		Carpool		Own Car		Total
Time of Day		Observed	Expected	Observed	Expected	Observed	Expected
	Morning	60	80*120/200 = 48	30	50*120/200 = 30	30	70*120/200 = 42	120
	Night	20	80*80/200 = 32	20	50*80/200 = 20	40	70*80/200 = 28	80
	Total	80	80	50	50	70	70	200

The variable time of the day represents rows.

The variable type of transportation represents columns.

Calculations
(O - E)
12	0	-12
-12	0	12

(O - E)^2/E
3	0	3.428571
4.5	0	5.142857

What is the value of χ², for independent samples?

Chi square = ∑[(O – E)^2/E] = 16.07143

P-value = 0.000324

(By using Chi square table or excel)

P-value < α = 0.05

So, we reject the null hypothesis

Conclusion:

There is sufficient evidence to conclude that the type of transportation used depended on the time of day.