Stat 13 Lecture 22 comparingproportions• Estimation of population proportion• Confidence interval ; hypothesis testing• Two independent samples• One sample, competitive categories(negative covariance)• One sample, non-competitive categories (usually, positive covariance)An Example1035%52%9%11%29%51%1000Connecticut8%31%59%9%7%25%59%1000New York9%36%53%8%8%33%51%1000New JerseyPeDoClotherPerotDoleClintonnresulttionElecpollselectionPre-1996 USAssume the sample is simple random.Does the poll result significantly show the majority favor Clinton in NewJersey? For Dole, is there is a significant difference between NY andConn ? Find a 95% confidence interval for the difference of supportbetween Clinton and Dole in New Jersey?Do you play• Tennis ? Yes, No• Golf? Yes, No• Basketball? Yes, No60%B 40%70%75%T 30%G 25%No Yesn=100 persons are involved in the surveyGene Ontology200 genes randomly selected100203560unknownothersnucleuscytoplasmCentral limit theorem implies thatbinomial is approximatelynormal when n is large• Sample proportion is approximately normal• The variance of sample proportion is equal to p(1-p)/n• If two random variables,X, Y are independent, thenvariance of (X-Y) = var (X) + var(Y)• If two random variables, X,Y are dependent, thenvariance of (X-Y)=var (X) + var(Y)-2cov(X,Y)• May apply the z-score formula to obtain confidenceinterval as done before.One sample, Competitivecategories• X=votes for Clinton, Y=votes for Dole• Suppose sample size is n=1, then only threepossibilities P(X=1, Y=0)=p1; P(X=0,Y=1)=p2;P(X=0,Y=0)=1-p1-p2• E(X)=p1; E(Y)=p2• Cov(X,Y) = E(X-p1)(Y-p2) = (1-p1)(0-p2)p1 + (0-p1)(1-p2)p2 + (0-p1) (0-p2) (1-p1-p2)• = -p1p2, which is negative• In general, cov(X,Y)= -n p1 p2 ; therefore• Var (X/n - Y/n)=n-2 (Var X + Var Y + 2np1p2)• =n-2(np1(1-p1) + np2(1-p2) + 2np1p2)=• (p1 + p2- p12- p22 + 2p1p2)/n = (p1+p2- (p1-p2)2)/nFormula for confidence interval• Let p1 = X/n, p2=Y/n• Then the interval runs from• p1-p2 - z sd(p1-p2), to• p1-p2 + z sd(p1-p2)• Where sd is the square root of variance,plug in the variance
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