Question

The Sleep Heart Health Study enrolled a simple random sample of 688 adults not treated for sleep-disordered breathing. The men and women in the study were classified into four groups depending on the extent of their sleep-disordered breathing (none, mild, moderate, or severe). We will use a chi-square test to test the competing hypotheses:

H0: There is no association between the severity of sleep-disordered breathing and sex
versus
H1: There is some association between the severity of sleep-disordered breathing and sex

The observed results of the study are found in the following table:

Sleep disordered breathing	women	men	total
None	231	174	405
Mild	79	62	141
Moderate	34	55	89
Severe	20	33	53
Total	364	324	688

(a) Under the null hypothesis of no relationship between sex and sleep-disordered breathing, what is the expected count of women with severe sleep-disordered breathing?

(b) Under the null hypothesis of no relationship between sex and sleep-disordered breathing, what is the contribution to the chi-square statistic (i.e., chi-square component) that comes from the women with severe sleep-disordered breathing?

(c) What is the degrees of freedom for this chi-square test?

(d) What is the test statistic for this test?

(e) Based on your calculations, what should you conclude?

A. There is not enough evidence (P-value > 0.05) to conclude that there is an association between the severity of sleep-disordered breathing and sex.

B. There is no association between the severity of sleep-disordered breathing and sex (P-value < 0.05).

C. There is a significant (P-value < 0.05) association between the severity of sleep-disordered breathing and sex.

D. There is a significant (P-value > 0.05) association between the severity of sleep-disordered breathing and sex.

(f) What can we state about the chi-square test in this situation?

A. The test is not valid because the sample sizes are small.

B. The test is not valid because some observed counts are too small.

C. The test is valid because the expected cell counts are large enough, and the participants are a simple random sample.

D. The test may be valid because the observed cell counts are large enough, and the participants are a simple random sample.

Answer 1

(a)

Expected count of Women:

None	Mild	Moderate	Severe
405*364/688=214.2733	141*364/688=74.5988	89*364/688=47.0872	53*364/688=28.0407

(b) The contribution to the chi-square statistic that comes from the women with severe sleep-disordered breathing

(c)

Degrees of freedom=(2-1)*(4-1)=3

(d)

Expected count of men:

None	Mild	Moderate	Severe
405*324/688=190.7267	141*324/688=66.4012	89*324/688=41.9128	53*324/688=24.9593

The contribution to the chi-square statistic that comes from the men with severe sleep-disordered breathing

Option: C. There is a significant (P-value < 0.05) association between the severity of sleep-disordered breathing and sex.

f. Option: C. The test is valid because the expected cell counts are large enough, and the participants are a simple random sample.