Question

1. Suppose YPoisson(A) and Y2 ~Poisson(2X) are two independent observations. (a) Derive the MLE of λ based on (Yi,Yo) (b) Show that the estimator λ (Y + Y)/3 is unbiased for λ and compute its variance. (c) With as much rigor as possible, show that if A is large then (A-X)/v is approximately normally distributed. (d) Derive a 95 percent confidence interval for A based on the asymptotic distribution of λ in part (c) (e) Extra Credit Based on part (c), derive the asymptotic distribution of log X. 2. Adult-onset diabetes is known to be highly genetically determined. A study was done comparing frequencies of a particular allele in a sample of such diabetics and a sample of non-diabetics. The data are shown in the following table: Allele Diabetic Normal Bb/bb BB 1239 49 (a) Conduct a binomial test of proportions to determine if the frequency of the Bb/bb allele type differs between diabetics and normals. State the null and alternative hypotheses of your test and interpret your result (b) Conduct a Pearson test of homogeneity to determine if the frequency of the Bb/bb allele type differs between diabetics and normals. State the null and alternative hypotheses f your test and interpret your result. (c) Confirm your calculations above by conducting the chi-square test of homogeneity in R using the following code. Hand in all of your code and output >Bb.bb BB chisq.test cbind (Bb.bb, BB), correct-FALSE) To learn more about the chisq.test function, type help( chisq.test in R. 3. Let Yj ~ Binomial(nj, pj), j = 1,2 independently. For testing the null hypothesis H0 : Pi = P2, a commonly used test statistic (slightly different from the one given in lecture) is P1 P2 where pi = Y5/nj and p = (Yİ + Y)/(m + n2) is the pooled estimate of proportion under Ho. Such data can also be summarized as a 2 x 2 table of counts: Population Successes Failures Y1 ni-1 1 For this table, denote the test statistic for the x2 test of homogeneity by X2, that is, x? = Σǐj(Oij-By)2/By where the sum is over all four cells of the table, Oij is the observed count in cell (i,j), and E is the corresponding expected count under Ho. Show that the test based on W is equivalent to Pearson's chi-squared test of homogeneity by proving that W2X2 algebraically

Answer 1

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