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	38	36
6 to 6.9	58	55
7 to 7.9	79	77
8 or more	65	92

(a)

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

State the null and alternative hypotheses.

Find the value of the test statistic. (Round your answer to three decimal places.)

What is the p-value? (Round your answer to four decimal places.)

Using a 0.05 level of significance, what is your conclusion?

(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?

Fewer than 6 ___%

6 to 6.9 ___%

7 to 7.9 ___%

8 or more ___%

Answer 1

Null hypothesis: Ho: hours of sleep are independent of age

Alternate hypothesis: Ho: hours of sleep are not independent of age

Applying chi square test of independence:

Expected	Ei=row total*column total/grand total	39 or younger	40 or older	Total
	<6	35.5	38.5	74
	6-6.9	54.2	58.8	113
	7-7.9	74.9	81.1	156
	8 or more	75.4	81.6	157
	total	240	260	500
chi square    χ2	=(Oi-Ei)2/Ei	39 or younger	40 or older	Total
	<6	0.173	0.160	0.3330
	6-6.9	0.261	0.241	0.5012
	7-7.9	0.227	0.209	0.4359
	8 or more	1.424	1.315	2.7389
	total	2.0847	1.9244	4.0091
test statistic X2=		4.009
p value =	0.2605	from excel: chidist(4.0091,3)

since p value >0.05 ; we can not reject null hypothesis

b)

less than 6 hours		14.8%
6 to 6.9 hours		22.6%
7 to 7.9 hours		31.2%
8 or more hours		31.4%