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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 38 36
6 to 6.9 60 57
7 to 7.9 75 73
8 or more 67 94
(a)
Conduct a test of independence to determine whether hours of sleep are 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?
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 %

Sagot :

For given study the hours of sleep per night is independent of the age.

Also, estimate of the percentages of individuals

                           Hours of sleep

Age       <6      6-6.9       7-7.9         ≥8         Total

≤ 39       38        60           75            67          240

≥ 40       36        57           73            94          260

Total      74        117         148          161        500

%       14.8%   23.4%     29.6%      32.2%

Let the null hypothesis H0 : Hours of sleep per night is independent of age

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

Expected frequencies :

E1: ((38+60+75+67)*(38+36))/500 = 35.52

E2: ((38+60+75+67)*(60+57))/500 = 56.16

E3: ((38+60+75+67)*(77+75))/500 = 72.96

E4: ((38+60+75+67)*(65+92))/500 = 75.36

E5: ((36+57+73+94)*(38+36))/500 = 38.48

E6: ((36+57+73+94)*(60+57))/500 = 60.84

E7: ((36+57+73+94)*(77+75))/500 = 79.04

E8: ((36+57+73+94)*(65+92))/500 = 81.64

Then performing the Chi-square independence test X²

X² = Σ[(f1-e1)²/e1 + (f2-e2)²/e2... (fn - en)²/en]

F1 to f8 = 38,60,75,67,36,57,73,94

X² = ((38-49.92)^2)/35.52 + ((60-56.16)^2)/56.16 + ((77-72.96)^2)/72.96 + ((65-75.36)^2)/75.36 + ((36-38.48)^2)/38.48 + ((57-60.84)^2)/60.84 + ((75-79.04)^2)/79.04 + ((92-81.64)^2)/81.64

X² = 4.00

α = 0.05

Reject H0, if p < α

df = (row - 1)*( column - 1)

df = (2 - 1)*(4-1)

   = 3

To find the p value from the X² square score at α = 0.05 and 3 degree of freedom

p = 0.261

Since, p > α ( 0.261 > 0.05) ; The result is not significant at α = 0.05 ; Hence we fail to reject the H0 (null hypothesis).

Now we need to 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.

                          Hours of sleep

Age       <6      6-6.9       7-7.9         ≥8         Total

≤ 39       38        60           75            67          240

≥ 40       36        57           73            94          260

Total      74        117         148          161        500

%       14.8%   23.4%     29.6%      32.2%

Therefore, we can conclude that the hours of sleep per night is independent of the age.

Learn more about the test statistic here:

https://brainly.com/question/14128303

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