Question

Problem 3. (15 points) 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. Mild 30 20 None Very severe Seve Discase1 Disease 2 Disease 3 80 70 60 30 34 For all subsequent parts state the null and altemative hypothesis, write down the test statistics and their degrees of freedom, calculate the test statistics, find the p-values and give interpretations. a) 5 points) Test the hypothesis that the proportions of subjects affected by Disease 3 that severe, severe, mild and no s leep deprivation are (0 .1, 0.1, 0.3, 0.5) (ie the experience very 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.tcst0 command). b) (5 points) 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.c. the distributions of sleep deprivation are completely specified multinomial with probabilities (0.1, 0.1, 0.3, 0.5)) c) 5 points) 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 (is the distributions of sleep deprivation are multinomial with the same unspecified probabilities).

Answer 1

SOLUTON.

Ho:Proportion of Disease 3is 0.1,0.1,0.3,0.5 Ha : Proporitons are not as give above z1.2 pvalue = 0.753 We fail to reject Ho and conclude that proporiton of disea se 3are 0.1,0.1,0.3,0.5 Observed P expected 0.1 6 0.166667 15 34 60 0.3 0.5 0.5 30 0.533333 1.2 18 60 value 0.753004 R-code >observed c5,6,15,34) > expected cC0.1,0.1,0.3,0.5) > chisq.test(x observed, p-expected) Chi-squared test for given probabilities data: observed X-squared 1.2, df 3, p-value0.753 b. Ho:Proporti on of verysevere, severe, mild and none of all three dise ases are 0.1,0.1,0.3,0.5 Ha Proporitonsare not 0.1,0.1,0.3,0.5 pvalue 0.59 We fail toreject Ho since pvalue is veyr high We conclude that Proportion of verysevere ,severe, mild and none of all three diseases are 0.1.0.1.0.3.0.5 0l- Observed P expectedEi)A2/Ei 0.1 21 0.0476 21 1.1905 63 0.0635 105 0.6095 210 1.9111 26 65 97 210 0.3 0.5 pvalue 0.591059

Ho:Proporti on of verysevere, severe, mild and none of all three diseases are same 0.25 Ha Proporitonsare not same 71.795 pvalue 0.00000 We reject Ho since pvalue is veyr low We conclude that Proportion of verysevere, severe, mild and none of all three diseases are not the same 0.25 0.25 0.25 0.25 525 17.7190 525 13.3762 525 525 37.7190 210 71.7905 26 65 97 210 2.9762 value 1.77E-15