Question

A national organization that promotes good local government management (ICMA) is interested if a city’s region is related to privatized waste collection.

Privatized                     Government
EAST 20                                40

WEST 10                               30

SOUTH 30                             40

1. Provide null and alternative hypotheses in formal terms and layperson's terms for the chi-square two sample test.

2. Do the math and reject/accept at a=.05

3. Explain the results in layperson's terms

Answer 1

Solution:

1. Provide null and alternative hypotheses in formal terms and layperson's terms for the chi-square two sample test.

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

Null hypothesis: H₀: Region of city and privatized waste collection are independent.

Alternative hypothesis: H_a: Region of city and privatized waste collection are not independent.

In simple words, we can also write these hypotheses as below:

Null hypothesis: H₀: A city’s region is not related to privatize waste collection.

Alternative hypothesis: H_a: A city’s region is related to privatize waste collection.

2. Do the math and reject/accept at a=.05

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 = 2

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

α = 0.05

Critical value = 5.991464547

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Column variable
Row variable	Privatized	Government	Total
East	20	40	60
West	10	30	40
South	30	40	70
Total	60	110	170

Expected Frequencies
	Column variable
Row variable	Privatized	Government	Total
East	21.17647059	38.82352941	60
West	14.11764706	25.88235294	40
South	24.70588235	45.29411765	70
Total	60	110	170

Calculations
(O - E)
-1.176470588	1.176470588
-4.117647059	4.117647059
5.294117647	-5.294117647

(O - E)^2/E
0.065359477	0.035650624
1.200980392	0.655080214
1.134453782	0.618792972

Chi square = ∑[(O – E)^2/E] = 3.71031746

P-value = 0.156428111

(By using Chi square table or excel)

P-value > α = 0.05

So, we do not reject the null hypothesis

3. Explain the results in layperson's terms.

There is insufficient evidence to conclude that city’s region is related to privatize waste collection.

Conclusion: A city’s region is not related to privatize waste collection.