We want to study the distribution of sleep deprivation for people affected by 3 distinct diseases. Random samples of affected subjects of sizes 80, 70 and 60 from disease 1, 2 and 3, respectively were obtained. The results are shown in the following table.
|
|
Very severe
|
Severe
|
Mild
|
None
|
|
|
Disease 1
|
10
|
7
|
30
|
33
|
80
|
|
Disease 2
|
7
|
13
|
20
|
30
|
70
|
|
Disease 3
|
5
|
6
|
15
|
34
|
60
|
For all subsequent parts state the null and alternative hypothesis, write down the test statistics and their degrees of freedom, calculate the test statistics, find the p-values and give interpretations.
a) Test the hypothesis that the proportions of subjects affected by Disease 3 that experience very severe, severe, mild and no sleep deprivation are (0.1, 0.1, 0.3, 0.5) (i.e the distribution of sleep deprivation for subjects affected by Disease 3 is multinomial with probabilities (0.1, 0.1, 0.3, 0.5)). Use R to confirm your result (need chisq.test() command).
b) Test the hypothesis that the proportions of subjects that experience very severe, severe, mild and no sleep deprivation are (0.1, 0.1, 0.3, 0.5) for all three diseases (i.e. the distributions of sleep deprivation are completely specified multinomial with probabilities (0.1, 0.1, 0.3, 0.5)).
c) Test the hypothesis that the proportions of subjects that experience very severe, severe, mild and no sleep deprivation are the same for all three diseases (i.e the distributions of sleep deprivation are multinomial with the same unspecified probabilities).