A survey was conducted to determine whether hours of sleep per night are independent of age. A sample ofindividuals was asked to indicate the number of hours of sleep per night with categorical options: fewer than 6hours, 6 to 6.9 hours, 7 to 7.9 hours, and 8 hours or more. Later in the survey, the individuals were asked toindicate their age with categorical options: age 39 or younger and age 40 or older. Sample data follow.Age GroupHours of Sleep39 or younger40 or olderFewer than 637356 to 6.960577 to 7.978768 or more6592(a) Conduct a test of independence to determine whether 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, 7to 7.9 hours, and 8 hours or more per night? Now that we have the observed and expected frequencies, the other components for the test statistic can befound. The following table gives the given observed data and the calculated expected frequencies. The sum ofall the values in the final column will be the x test statistic.Complete the table, rounding values in the final column to four decimal places and x" to three decimal places.SquaredDifferenceDivided byExpectedFrequencySquaredDifferenceObservedExpectedFrequencyHoursDifferenceAgeGroupFrequencyof-e,Sleepz("a - ")5.953639 or2.442 =5.9536Fewer37 - 34.56 =3734.5634.56than 6younger2.440.1723Fewerthan 640 or0.1593537.44-2.445.9536older6 to39 or3.846056.1614.74560.26266.9younger6 to6.940 or5760.84-3.840.2424older7 to39 or7873.924.0816.64647.9younger7 to40 orolder7680.08-4.0816.64647.98 or39 or|-10.386575.36107.32961.4242mоreyounger40 orolder8 or9281.6410.36107.3296more(f, - e,4.008

The computation is done as follow: Hours of sleep Age group Observed frequencies (Oij) Expected…

Answered: A survey was conducted to determine…

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section: Chapter Questions

Problem 13PT

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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
37
35
6 to 6.9
60
57
7 to 7.9
78
76
8 or more
65
92
(a) Conduct a test of independence to determine whether 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?

Now that we have the observed and expected frequencies, the other components for the test statistic can be
found. The following table gives the given observed data and the calculated expected frequencies. The sum of
all the values in the final column will be the x test statistic.
Complete the table, rounding values in the final column to four decimal places and x" to three decimal places.
Squared
Difference
Divided by
Expected
Frequency
Squared
Difference
Observed
Expected
Frequency
Hours
Difference
Age
Group
Frequency
of
-e,
Sleep
z("a - ")
5.9536
39 or
2.442 =
5.9536
Fewer
37 - 34.56 =
37
34.56
34.56
than 6
younger
2.44
0.1723
Fewer
than 6
40 or
0.159
35
37.44
-2.44
5.9536
older
6 to
39 or
3.84
60
56.16
14.7456
0.2626
6.9
younger
6 to
6.9
40 or
57
60.84
-3.84
0.2424
older
7 to
39 or
78
73.92
4.08
16.6464
7.9
younger
7 to
40 or
older
76
80.08
-4.08
16.6464
7.9
8 or
39 or
|-10.38
65
75.36
107.3296
1.4242
mоre
younger
40 or
older
8 or
92
81.64
10.36
107.3296
more
(f, - e,
4.008

Expert Solution

Step 1 Part a

The computation is done as follow:

Hours of sleep	Age group	Observed frequencies (O_ij)	Expected frequencies (E_ij)	E_ij-O_ij	${(E i j - O i j)}^{2}$	$\frac{{(E i j - O i j)}^{2}}{E_{i j}}$
Fewer than 6	39 or younger	37	34.56	2.44	5.9536	0.172269
6 to 6.9	39 or younger	60	56.16	3.84	14.7456	0.262564
7 to 7.9	39 or younger	78	73.92	4.08	16.6464	0.225195
8 or more	39 or younger	65	75.36	-10.36	107.3296	1.424225
Fewer than 6	40 or older	35	37.44	-2.44	5.9536	0.159017
6 to 6.9	40 or older	57	60.84	-3.84	14.7456	0.242367
7 to 7.9	40 or older	76	80.08	-4.08	16.6464	0.207872
8 or more	40 or older	92	81.64	10.36	107.3296	1.314669

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