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	60	57
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.

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: Hours of sleep per night is mutually exclusive from age.
H_a: Hours of sleep per night is not mutually exclusive from age.    

H₀: Hours of sleep per night is independent of age.
H_a: Hours of sleep per night is not 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.

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?

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 cannot 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

Observed Frequencies

	B1	B2	Total
A1	38	36	74
A2	60	57	117
A3	77	75	152
A4	65	92	157
Total	240	260	500

Expected Frequencies

	B1	B2	Total
A1	35.52	38.48	74
A2	56.16	60.84	117
A3	72.96	79.04	152
A4	75.36	81.64	157
Total	240	260	500

Compute Chi-square

Compute the degrees of freedom (df).
df=(4-1)⋅(2-1)=3