Question

(9) A study looking at the education level and diabetes status of patients was completed and the following contingency table was obtained. At the a = 0.05 significance level, run a hypothesis test to determine if diabetes status is independent of education level. Diabetes | No Diabetes 218 389 393 178 Less than HS education HS grad Some college College grad 33 25 17

Answer 1

Given table data is as below

MATRIX	col1	col2	TOTALS
row 1	33	218	251
row 2	25	389	414
row 3	20	393	413
row 4	17	178	195
TOTALS	95	1178	1273

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calculation formula for E table matrix

E-TABLE	col1	col2
row 1	row1*col1/N	row1*col2/N
row 2	row2*col1/N	row2*col2/N
row 3	row3*col1/N	row3*col2/N
row 4	row4*col1/N	row4*col2/N

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expected frequencies calculated by applying E - table matrix formula

E-TABLE	col1	col2
row 1	18.731	232.269
row 2	30.896	383.104
row 3	30.821	382.179
row 4	14.552	180.448

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calculate chi-square test statistic using given observed frequencies, calculated expected frequencies from above

Oi	Ei	Oi-Ei	(Oi-Ei)^2	(Oi-Ei)^2/Ei
33	18.731	14.269	203.604	10.87
218	232.269	-14.269	203.604	0.877
25	30.896	-5.896	34.763	1.125
389	383.104	5.896	34.763	0.091
20	30.821	-10.821	117.094	3.799
393	382.179	10.821	117.094	0.306
17	14.552	2.448	5.993	0.412
178	180.448	-2.448	5.993	0.033
				ᴪ^2 o = 17.513

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set up null vs alternative as
null, Ho: diabetes status is independent of education level
alternative, H1: diabetes status is dependent of education level
level of significance, alpha = 0.05
from standard normal table, chi square value at right tailed, ᴪ^2 alpha/2 =7.815
since our test is right tailed,reject Ho when ᴪ^2 o > 7.815
we use test statistic ᴪ^2 o = Σ(Oi-Ei)^2/Ei
from the table , ᴪ^2 o = 17.513
critical value
the value of |ᴪ^2 alpha| at los 0.05 with d.f (r-1)(c-1)= ( 4 -1 ) * ( 2 - 1 ) = 3 * 1 = 3 is 7.815
we got | ᴪ^2| =17.513 & | ᴪ^2 alpha | =7.815
make decision
hence value of | ᴪ^2 o | > | ᴪ^2 alpha| and here we reject Ho
ᴪ^2 p_value =0.001
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[ANSWERS]
null, Ho: diabetes status is independent of education level
alternative, H1: diabetes status is dependent of education level
test statistic: 17.513
critical value: 7.815
p-value:0.001
decision: reject Ho                                                  
evidence to support that diabetes status is dependent of education level