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.

H₀: Hours of sleep per night is not independent of age.
H_a: Hours of sleep per night is independent of age.H₀: The proportion of people who get 8 or more hours of sleep per night is not equal across the two age groups.
H_a: The proportion of people who get 8 or more hours of sleep per night is equal across the two age groups.    H₀: Hours of sleep per night is independent of age.
H_a: Hours of sleep per night is not independent of age.H₀: Hours of sleep per night is mutually exclusive from age.
H_a: Hours of sleep per night is not mutually exclusive from age.

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.)

p-value =

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

Do not reject H₀. We cannot conclude that hours of sleep per night and age are not independent.Reject H₀. We cannot conclude that hours of sleep per night and age are independent.    Reject H₀. We conclude that hours of sleep per night and age are not independent.Do not reject H₀. We conclude that hours of sleep per night and age are independent.

(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

a)

H₀: Hours of sleep per night is independent of age.
H_a: Hours of sleep per night is 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	35.5200	38.4800	74
	6-6.9	54.2400	58.7600	113
	7-7.9	74.8800	81.1200	156
	>8	75.3600	81.6400	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	1.424	1.315	2.7389
	total	2.0847	1.9244	4.0091
test statistic X2 =		4.009

p value =

0.2605

Do not reject H₀. We cannot conclude that hours of sleep per night and age are not independent.

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%