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. Age Group Hours of Sleep 39 or younger 40 or older Fewer than 6 6 to 6.9 7 to 7.9 8 or more 38 62 75 65 36 59 73 92 (a) Conduct a test of independence to determine whether hours of sleep are independent of age. State the null and altemative hypotheses Ho: Hours of sleep per night is mutually exdusive from age. Ha: Hours of sleep per night is nat mutually exclusive from age. H: Hours of sleep per night is independent of age. Ha: Hours of sleep per night is not independent of age. H.: The proport on of people who get 8 or more hours of sleep per night is not equal across the two age groups. : The proportion of people who get 8 or more hours of sleep per night is equal across the two age groups. H: Hours of sleep per night is not independent of age. Ma: Hours of sleep per night is 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.) D-value Using a 0.05 level af significance, what is your condlusion? Reject H0. We conclude that hours of sleep per night and age are not independent. Do nat reject . We cennot conclude thet hours of sleep per night and age are not independent. O Do not reject H. We conclude thet hours of sleep per night and age are independent. o Reject Ho. We cannot 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, 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) right choice is second

H0:Hours of sleep per night is independent of age

Ha:Hours of sleep per night is not independent of age

test statistic=4.01

chi-square=sum((O-E)²/E)=4.01 with (r-1)(c-1)=(4-1)(2-1)=3 df

p-value=0.260

Donot reject H0: We conclude that hours of sleep per night and age are independent

(since the p-value=0.260 is more than level of significance =0.05)

(b) please look

	%
<=6	14.8
6-6.9	24.2
7-7.9	29.6
>=8	31.4
total	100

following data is given

39 or younger 40 or older total % <=6 38 36 74 14.8 6-6.9 62 59 121 24.2 7-7.9 75 73 148 29.6 >=8 65 92 157 31.4 total 240 260 500 100

here we use chi-square test and

chi-square=sum((O-E)²/E)=4.01 with (r-1)(c-1)=(4-1)(2-1)=3 df

	Observed(O)	Expected(E)	E	(O-E)	(O-E)2/E
	38	74*240/500	35.52	2.48	0.17
	62	121*240/500	58.08	3.92	0.26
	75	148*240/500	71.04	3.96	0.22
	65	157*240/500	75.36	-10.36	1.42
	36	74*260/500	38.48	-2.48	0.16
	59	121*260/500	62.92	-3.92	0.24
	73	148*260/500	76.96	-3.96	0.20
	92	157*260/500	81.64	10.36	1.31
sum=	500	0	500	0	4.01
	chi-square=	4.01 with 3 df
	p-value=	0.2604