Question

12. A study reported by D. Goldberg in “The Detection of Psychiatric Illness by Questionnaire”(Oxford University, London, 1972) examined the relationship between mental-health classification and thoughts of suicide. The mental health of each person was classified and each person was asked, “Have you recently found that the idea of taking your own life kept coming into your mind?” Following are the results. Transfer the table to your blue book.

                            Mental Health

Response	Normal	Mild Illness	Severe Illness
Definitely not	90	43	34
Don’t think so	5	18	8
Crossed my mind	3	21	21
Definitely yes	1	15	36

Is there evidence that an association exists between response to the suicide question and mental-health classification?
a) What kind of chi-square test is appropriate; goodness-of-fit, homogeneity or independence?  
b) State the Hypotheses.  
      c) Compute the expected cell values.   
      d) Calculate the test statistic .   
      e) Make a sketch, give the degrees of freedom.
f) Find the p-value and state your conclusion followed by a summary sentence.

Answer 1

(a) The chi-square test of independence is appropriate here.

(b) The hypothesis being tested is:

H₀: There is no association between response to the suicide question and mental-health classification

H_a: There is an association between response to the suicide question and mental-health classification

(c)

		Normal	Mild Illness	Severe Illness	Total
Definitely not	Observed	90	43	34	167
	Expected	56.04	54.91	56.04	167.00
	O - E	33.96	-11.91	-22.04	0.00
	(O - E)² / E	20.57	2.58	8.67	31.83
Don’t think so	Observed	5	18	8	31
	Expected	10.40	10.19	10.40	31.00
	O - E	-5.40	7.81	-2.40	0.00
	(O - E)² / E	2.81	5.98	0.56	9.34
Crossed my mind	Observed	3	21	21	45
	Expected	15.10	14.80	15.10	45.00
	O - E	-12.10	6.20	5.90	0.00
	(O - E)² / E	9.70	2.60	2.30	14.60
Definitely yes	Observed	1	15	36	52
	Expected	17.45	17.10	17.45	52.00
	O - E	-16.45	-2.10	18.55	0.00
	(O - E)² / E	15.51	0.26	19.72	35.48
Total	Observed	99	97	99	295
	Expected	99.00	97.00	99.00	295.00
	O - E	0.00	0.00	0.00	0.00
	(O - E)² / E	48.59	11.42	31.25	91.25
		91.25	chi-square
		6	df
		1.66E-17	p-value

(d) The test statistic is 91.25.

(e) df = 6

(f) The p-value is 0.0000.

Since the p-value (0.0000) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that there is an association between response to the suicide question and mental-health classification.

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