Question

Students in grade school were asked whether they believed Good Grades, Popularity, or Athletic Ability was the most important to them. The students responses are divided into three school locations: Rural, Suburban, or Urban. The resulting contingency table was produced from the frequencies of observations. Run a hypothesis test to determine if the goals of student are independent of the school location using α = 0.05.

Describe the dependencies if they exist.

	School Area
Goals	Rural	Suburban	Urban
Grades	57	87	24
Popular	50	42	6
Sports	42	22	5

Answer 1

Chi-square test of independence

Solution:

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

Null hypothesis: H₀: The goals of student are independent of the school location.

Alternative hypothesis: H_a: The goals of student are not independent of the school location.

We are given 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 = 3

Number of columns = c = 3

Degrees of freedom = df = (r – 1)*(c – 1) = 2*2 = 4

α = 0.05

Critical value = 9.487729

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Column variable
Row variable	Rural	Suburban	Urban	Total
Grades	57	87	24	168
Popular	50	42	6	98
Sports	42	22	5	69
Total	149	151	35	335

Expected Frequencies
	Column variable
Row variable	Rural	Suburban	Urban	Total
Grades	74.72239	75.72537313	17.55224	168
Popular	43.58806	44.17313433	10.23881	98
Sports	30.68955	31.10149254	7.208955	69
Total	149	151	35	335

Calculations
(O - E)
-17.7224	11.27463	6.447761
6.41194	-2.17313	-4.23881
11.31045	-9.10149	-2.20896

(O - E)^2/E
4.203332	1.678661	2.368565
0.943217	0.106909	1.754841
4.168397	2.663447	0.676864

Chi square = ∑[(O – E)^2/E] = 18.56423

P-value = 0.000957

(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 the goals of student are not independent of the school location.

There is insufficient evidence to conclude that the goals of student are independent of the school location.