Question

2. A pediatrician speculated that frequency of visits to his office may be influenced by type of medical insurance coverage. As an exploratory study, she randomly chose 15 patients: 5 whose parents belonged to a health maintenance organization (HMO) 5 whose parents had traditional medical insurance, and 5 whose parents were uninsured. The data are presented below: Traditional None n2-5 32.8 HMO n-5 =7.8 Si 2.49 2-4.8 S2 2.28 S 1.48 Overall mean ()=5.13 a) Test the nul hypothesis that type of insurance coverage has no effect on frequency of visits (set a=.05 level) (12 points) b) Conduct a multiple comparison of means by Scheffe method to determine exactly where the significant differences occur. (10 points) c) Calculate the correlation ratio and interpret the result. (3 points

Answer 1

$(a)~Let~\mu_1=true~mean~effect~due~to~HMO\\ \mu_2=true~mean~effect~due~to~Traditional\\ \mu_3=true~mean~effect~due~to~None.\\ Null~hypothesis,~H_0:\mu_1=\mu_2=\mu_3\\ Alternative~hypothesis,~H_1:~at~least~one~\mu_i~is~different~from~others.\\ SSG=sum~of~square~due~to~group=\sum_{i=1}^kn_i(\bar{x}_i-\bar{x})^2\\ =5*((7.8-5.13)^2&plus;(4.8-5.13)^2&plus;(2.8-5.13)^2)=63.3335~with~df=3-1=2\\ where,~k=3,~n_1=n_2=n_3=5,~\bar{x}_1=7.8,~\bar{x}_2=4.8,~\bar{x}_3=2.8,~\bar{x}=5.13\\ SSE=sum~of~square~due~to~error=\sum_{i=1}^k(n_i-1)S_i^2\\ =4*(2.49^2&plus;2.28^2&plus;1.48^2)=54.3556~with~df=15-3=12\\ MSG=SSG/2=63.3335/2=31.6668\\ MSE=SSE/12=54.3556/12=4.5296\\ F-ratio=\frac{MSG}{MSE}=31.6668/4.5296=6.9911\\ p-value=P(F>6.9911|F\sim F_{2,12})=0.0097<0.05.\\ Hence~we~reject~H_0~at~5\%~level~of~significance~and~conclude~that~type~of~insurance~coverage~has~\\ significant~effect~on~frequency~of~visits.\\$$(b)\\ Significant~difference=\sqrt{MSE\times F_{0.05,2,12}\times \frac{2}{5}}=2.6532\\ where,~F_{0.05,2,12}=3.8853\\ |\bar{x}_1-\bar{x}_2|=|7.8-4.8|=3>2.6532\\ |\bar{x}_1-\bar{x}_3|=|7.8-2.8|=5>2.6532\\ |\bar{x}_2-\bar{x}_3|=|4.8-2.8|=2<2.6532\\ Hence~HMO~is~significantly~different~from~others.\\ (c)~Correlation~ratio=\frac{SSG}{SST}=63.3335/117.6891=0.5381\\ SST=sum~of~square~due~to~total=SSG&plus;SSE=63.3335&plus;54.3556\\ =117.6891\\ Type~of~insurance~coverage~is~positively~related~with~frequency~of~visits.$