Let Ya and Yb denote Bernoulli random variables from two dif

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Let Ya and Yb denote Bernoulli random variables from two different populations, denoted a and b. Suppose that E(Ya) = pa

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Let Ya and Yb denote Bernoulli random variables from two different populations, denoted a and b. Suppose that E(Ya) = pa and E(Yb) = pb. A random sample of size na is chosen from population a, with sample average denoted pna, and a random sample of size nb is chosen from population b, with sample average denoted pnb. Suppose the sample from population a is independent of the sample from population b. a. Show that E( pna) = pa and var(pna) = pa(1 - pa)>na. Show that E(pnb) = pb and var(pnb) = pb(1 - pb)>nb. b. Show that var(pna - pnb) = pa(1 - pa) na + pb(1 - pb) nb . (Hint: Remember that the samples are independent.) c. Suppose that na and nb are large. Show that a 95% confidence interval for pa - pb is given by (pna - pnb) { 1.964pna(1 - pna) na + pnb(1 - pnb) nb . How would you construct a 90% confidence interval for pa - pb?

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