Question

1. A random sample of American was asked if they believed that everyone has an equal opportunity to obtain a quality education in the U.S.A. Result shows below in the table.

Response	less High School	High School	Associate Degree	Bachelor Degree	Graduate Degree
Yes	312	542	31	102	44
No	88	209	22	70	33

At the 0.05 level of significance, test the claim that the response is independent of a person’s level of education.

I did the work and got p-value 0.0002

chi-square=32.52

-I'm not sure I did the work correct I put it in stat crunch with my observations and expectations

Please correct me if I am wrong

Answer 1

Solution:

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

Null hypothesis: H0: Two categorical variables are independent.

Alternative hypothesis: Ha: Two categorical variables are dependent.

We assume/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 = 2

Number of columns = c = 5

Degrees of freedom = df = (r – 1)*(c – 1) = 1*4 = 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
	Level of education
Response	Less High School	High School	Associate Degree	Bachelor Degree	Graduate Degree	Total
Yes	312	542	31	102	44	1031
No	88	209	22	70	33	422
Total	400	751	53	172	77	1453

Expected Frequencies
	Level of education
Response	Less High School	High School	Associate Degree	Bachelor Degree	Graduate Degree	Total
Yes	283.8266	532.8844	37.60702	122.0454	54.63661	1031
No	116.1734	218.1156	15.39298	49.95458	22.36339	422
Total	400	751	53	172	77	1453

Calculations
(O - E)
28.17343	9.115623	-6.60702	-20.0454	-10.6366
-28.1734	-9.11562	6.60702	20.04542	10.63661

(O - E)^2/E
2.796575	0.155934	1.16076	3.292372	2.070728
6.832392	0.380966	2.835884	8.043687	5.059053

Test Statistic = Chi square = ∑[(O – E)^2/E] = 32.62835

χ2 statistic = 32.62835

P-value = 0.00000142

(By using Chi square table or excel)

P-value < α = 0.05

So, we reject the null hypothesis

There is not sufficient evidence to conclude that the response is independent of a person’s level of education.