11.3 Test of independence  (Page 2/20)

The observed table and the question at the end of the problem, "Is the number of hours volunteered independent of the type of volunteer?" tell you this is a test of independence. The two factors are number of hours volunteered and type of volunteer . This test is always right-tailed.

H ₀ : The number of hours volunteered is independent of the type of volunteer.

H _a : The number of hours volunteered is dependent on the type of volunteer.

The expected result are in [link] .

For example, the calculation for the expected frequency for the top left cell is

$$E=\frac{(\text{row total})(\text{column total})}{\text{total number surveyed}}=\frac{\left(255\right)\left(298\right)}{839}=90.57$$

Calculate the test statistic: χ ² = 12.99 (calculator or computer)

Distribution for the test: ${\chi }_4^2$

df = (3 columns – 1)(3 rows – 1) = (2)(2) = 4

Graph:

Probability statement: p -value= P ( χ ² >12.99) = 0.0113

Compare α and the p -value: Since no α is given, assume α = 0.05. p -value = 0.0113. α > p -value.

Make a decision: Since α > p -value, reject H ₀ . This means that the factors are not independent.

Conclusion: At a 5% level of significance, from the data, there is sufficient evidence to conclude that the number of hours volunteered and the type of volunteer are dependent on one another.

For the example in [link] , if there had been another type of volunteer, teenagers, what would the degrees of freedom be?