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 36 60 57 75 8 or more 94 (a) Conduct a test of independence to determine whether hours of sleep are independent of age State the null and alternative hypotheses Ho: The proportion of people who get 8 or more hours of sleep per night is not equal across the two age groups Ha: The proportion of people who get 8 or more hours of sleep per night is equal across the two age groups Ho: Hours of sleep per night is not independent of age 0" Ha: Hours of sleep per night is independent of age O Ho: Hours of sleep per night is mutually exclusive from age H: Hours of sleep per night is not mutually exclusive from age al H: Hours of sleep per night is independent of age Ha: Hours of sleep per night is not 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? Do not reject Ho. We cannot conclude that hours of sleep per night and age are not independent. ● Reject Ho. We cannot conclude that hours of sleep per night and age are independent. Reject Ho We conclude that hours of sleep per night and age are not independent Do not reject Ho. We 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, 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

Answer 1

A survey was conducted to determine whether the hours of sleep per night are independent of age.

a.) Hence, the appropriate test of hypothesis would be :

            Option d

           Ho: Hours of sleep per night is independent of age

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

    To test the above hypothesis we make use of the Pearsonian chi-square test.

The value of the test-statistic is given by :

                             $\chi ^{2}=\sum \sum (Oij-Eij)^{2}/Eij$

       Where Oij are the observed cell frequencies and Eij are the expected cell frequencies if they both are to be independent.

If N be the total number of observations,

               Eij=Np_i.* p._j

                                              _where     p_{i.=sum of all observations for the ith row / N}

                                                              p_{.j=sum of all observations for the jth column / N}

The contingency table below provides the following information: the observed cell totals, (the expected cell totals) and [the chi-square statistic for each cell].

       

          

   Hence,   the observed value of the test statistic is 3.8946 and the p-value is 0.2731

Since the p-value is greater than our chosen level of significance which is 0.05, so we cannot reject the null hypothesis and conclude that :

       Option d:

        Do not reject Ho.Hours of sleep per night and age are independent.

b.) To calculate the percentages we use the row totals from the table above.

Hence,

     Fewer than 6 = 70/500*100   = 14%

     6 to 6.9          = 117/500*100 = 23.4%

     7 to 7.9          = 152/500*100 = 30.4%

     8 or more       = 161/500*100 = 32.2%