Question

The National Sleep Foundation used a survey to determine whether hours of sleeping per night are independent of age (Newsweek, January 19, 2004). The following show the hours of sleep on weeknights for a sample of individuals age 49 and younger and for a sample of individuals age 50 and older Age 49 or younger 50 or older a. Conduct a test of independence to determine whether the hours of sleep on weeknights are independent of age. Use α = .05. Use Table 12.4 Fewer than 6 36 38 Hours of Sleep 6 to 6.9 57 56 7 to 7.9 76 72 8 or more Total 240 260 71 94 Compute the value of the χ 2 test statistic (to 2 decimals) The p value is greater than .10 What is your conclusion? Cannot reject the assumption that age and hours of sleep are independent ▼ b. Using the total sample of 500, estimate the percentage of people who sleep less than 6, 6 to 6.9, 7 to 7.9, and 8 or more hours on weeknights (to 1 decimal) Less than 6 hours 6 to 6.9 hours 7 to 7.9 hours 8 or more hours

Answer 1

(a)

(i)

Observed Frequencies:

	Fewer than 6	6 to 6.9	7 to 7.9	8 or more	Total
49 or younger	36	57	76	71	240
50 or older	38	56	72	94	260
Total	74	113	148	165	500

Expected Frequencies:

	Fewer than 6	6 to 6.9	7 to 7.9	8 or more	Total
49 or younger	74X240/500=35.52	54.24	71.04	79.20	240
50 or older	38.48	58.76	76.96	85.80	260
Total	74	113	148	165	500

The value of $\chi ^{2}$ statistic is obtained as follows:

O	E	(O -E)2/E
36	35.52	0.01
57	54.24	0.14
76	71.04	0.35
71	79.20	0.85
38	38.48	0.01
56	58.76	0.13
72	76.96	0.32
94	85.80	0.78
Total = $\chi ^{2}$ =		2.58

The value of the $\chi ^{2}$ statistic is 2.58

(ii)

ndf = (r - 1) X (c - 1) = (2 - 1) X (3 - 1) = 2

By Technology, p - value = 0.4608

So,

Correct option:

The p - value is greater than .10

(iii)

Since p - value is greater than $\alpha$ , the difference is not significant. Fail to reject null hypothesis.

So,

Correct option:

cannot reject the assumption that age and hours of sleep are independent.

(b)

Less than 6 hours = (74/500) X 100 = 14.8 %

6 to 6.9 hours =(113/500) X 100 = 22.6 %

7 to 7.9 hours = (148/500) X 500 = 29.6 %

8 or more hours = (165/500) X 100 = 33.0 %