Question

 

A survey was conducted to determine whether hours of sleep per night are independent of age. A sample of individuals was asked to indicate the number of hours of sleep per night with categorical options: fewer than 6 hours, 6 to 6.9 hours, 7 to 7.9 hours, and 8 hours or more. Later in the survey, the individuals were asked to indicate their age with categorical options: age 39 or younger and age 40 or older. Sample data follow.

Hours of Sleep	Age Group
	39 or younger	40 or older
Fewer than 6	37	35
6 to 6.9	61	58
7 to 7.9	77	75
8 or more	65	92

(a)

Conduct a test of independence to determine whether hours of sleep are independent of age.

(b)

What is your estimate of the percentages of individuals who sleep fewer than 6 hours, 6 to 6.9 hours, 7 to 7.9 hours, and 8 hours or more per night?

Answer 1

a)

Ho: hours of sleep are independent of age

Ha: hours of sleep are dependent of age

Applying chi square test of independence:

Expected	Ei=row total*column total/grand total	<39	>40	Total
	<6	34.5600	37.4400	72
	6-6.9	57.1200	61.8800	119
	7-7.9	72.9600	79.0400	152
	>8	75.3600	81.6400	157
	total	240	260	500
chi square    χ2	=(Oi-Ei)2/Ei	<39	>40	Total
	<6	0.172	0.159	0.3313
	6-6.9	0.264	0.243	0.5068
	7-7.9	0.224	0.206	0.4302
	>8	1.424	1.315	2.7389
	total	2.0838	1.9235	4.0072
test statistic X2 =		4.007
p value =		0.2607

since p value >0.05 , we fail to reject the null hypothesis.

b)

less than 6 hours=(72/500)*100=		14.4%
6 to 6.9 hours=(119/500)*100=		23.8%
7 to 7.9 hours=(152/500)*100=		30.4%
8 or more hours=(157/500)*100=		31.4%