Question

In: Statistics and Probability

The National Sleep Foundation used a survey to determine whether hours of sleeping per night are...

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.

Hours of Sleep
Age Fewer than 6 6 to 6.9 7 to 7.9 8 or more Total
49 or younger 35 65 71 69 240
50 or older 37 63 80 80 260
  1. Conduct a test of independence to determine whether the hours of sleep on weeknights are independent of age. Use = .05. Use Table 12.4.

    Compute the value of the 2 test statistic (to 2 decimals).


    The p value is Selectless than .005between .005 and .01between .01 and .025between .025 and .05between .05 and .10greater than .10

    What is your conclusion?
    SelectConclude age and hours of sleep are not independentCannot reject the assumption that age and hours of sleep are independent
  2. 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 %

Solutions

Expert Solution

a;

Hypotheses are:

H0;  The hours of sleep on weeknights are independent of age.

Ha;  The hours of sleep on weeknights are not independent of age.

Following table shows the row total and column total:

Hours of Sleep
Age Fewer than 6 6 to 6.9 7 to 7.9 8 or more Total
49 or younger 35 65 71 69 240
50 or older 37 63 80 80 260
Total 72 128 151 149 500


Expected frequencies will be calculated as follows:

Following table shows the expected frequency:

Hours of Sleep
Age Fewer than 6 6 to 6.9 7 to 7.9 8 or more Total
49 or younger 34.56 61.44 72.48 71.52 240
50 or older 37.44 66.56 78.52 77.48 260
Total 72 128 151 149 500

Following table shows the calculations for chi square test statistics:

O E (O-E)^2/E
35 34.56 0.005601852
37 37.44 0.00517094
65 61.44 0.206276042
63 66.56 0.190408654
71 72.48 0.030220751
80 78.52 0.027896077
69 71.52 0.088791946
80 77.48 0.081961797
Total 0.636328058

The test statistics is:

Degree of freedom: df =( number of rows -1)*(number of columns-1) = (2-1)*(4-1)=4

The p-value is: 0.8881

Since p-value is greater than 0.05 so we fail to reject the null hypothesis.

b)

Following table shows the required percentage:

Frequency, f (f*100)/500%
Fewer than 6   72 14.4
6 to 6.9   128 25.6
7 to 7.9   151 30.2
8 or more   149 29.8

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