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	36	34
6 to 6.9	62	59
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 independent of age.
H_a: Hours of sleep per night is not independent of age. CORRECT!

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?

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

(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

he National Sleep Foundation used a survey to determine whether hours of sleep per night are independent of age (Newsweek, January 19, 2004). 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	39 or Younger	40 or Older
Fewer than 6	36	34
6 to 6.9	62	59
7 to 7.9	77	75
8 or More	65	92

Conduct a test of independence to determine whether hours of sleep are independent of age. Using a .05 level of significance.

	#observed frequency
	Hours Of Sleep	39 or Younger	40 or Older	Total
	Fewer than 6	36	34	70
	6 to 6.9	62	59	121
	7 to 7.9	77	75	152
	8 or More	65	92	157
	total	240	260	500
Expected	Ei=Σrow*Σcolumn/Σtotal	39 or Younger	40 or Older	Total
	Hours Of Sleep	33.6	36.4	70
	Fewer than 6	58.08	62.92	121
	6 to 6.9	72.96	79.04	152
	7 to 7.9	75.36	81.64	157
	8 or More	240	260	500
	total
chi square    χ2	=(Oi-Ei)2/Ei	39 or Younger	40 or Older	Total
	Fewer than 6	0.171428571	0.158241758	0.32967
	6 to 6.9	0.264573003	0.244221233	0.508794
	7 to 7.9	0.22370614	0.206497976	0.430204
	8 or More	1.424225053	1.31466928	2.738894
	total	2.083932768	1.923630247	4.007563

Data
Level of Significance	0.05
Number of Rows	4
Number of Columns	2
Degrees of Freedom	3
Results
Critical Value	7.8147
Chi-Square Test Statistic	4.007
p-Value	0.2607

ompute the value of the ² test statistic (to 3decimals): 4.007

The p-value is:   0.2607

.

What is your conclusion?

Calculated P=0.2607 > 0.05 level of significance.

The null hypothesis is not rejected.

We conclude that 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? Round your answer to one decimal place.

Fewer than 6	14.0%
6 to 6.9	24.2%
7 to 7.9	30.4%
8 or more	31.4%