Question

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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	60	57
7 to 7.9	77	75
8 or more	67	94

(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 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.

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

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a)

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

Using excel<data<megastat<chi square<contigency table

Here is the output:

		39 or younger	40 or older	Total
Fewer than 6	Observed	36	34	70
	Expected	33.60	36.40	70.00
	O - E	2.40	-2.40	0.00
	(O - E)² / E	0.17	0.16	0.33
6 to 6.9	Observed	60	57	117
	Expected	56.16	60.84	117.00
	O - E	3.84	-3.84	0.00
	(O - E)² / E	0.26	0.24	0.50
7 to 7.9	Observed	77	75	152
	Expected	72.96	79.04	152.00
	O - E	4.04	-4.04	0.00
	(O - E)² / E	0.22	0.21	0.43
8 or more	Observed	67	94	161
	Expected	77.28	83.72	161.00
	O - E	-10.28	10.28	0.00
	(O - E)² / E	1.37	1.26	2.63
Total	Observed	240	260	500
	Expected	240.00	260.00	500.00
	O - E	0.00	0.00	0.00
	(O - E)² / E	2.03	1.87	3.89
		3.89	chi-square
		3	df
		.2731	p-value

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

Test Statistic= 3.895

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

p-value = 0.2731

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

Since p value>alpha(0.05). So we do not reject h0

Do not reject H₀. We cannot conclude that hours of sleep per night and age are not 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 hours	14
6 to 6.9 hours	23.4
7 to 7.9 hours	30.4
8 or more hours	32.2

Fewer than 6 hours=(70/500)*100=14

6 to 6.9 hours= (117/500)*100=23.4

7 to 7.9 hours=152/500*100=30.4

8 or more=161/500*100=32.2