3.15 Let Ya and Yb denote Bernoulli random variables from two different populations, denoted a and b.

Question:

3.15 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 pn

a, and a random sample of size nb is chosen from population b, with sample average denoted pn

b. Suppose the sample from population a is independent of the sample from population b.

a. Show that E( pn

a) = pa and var(pn

a) = pa(1 - pa)>na. Show that E(pn

b) = pb and var(pn

b) = pb(1 - pb)>nb.

b. Show that var(pn a - pn

b) = 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 (pn a - pn

b) { 1.964pn a(1 - pn a)
na + pn b(1 - pn b)
nb .
How would you construct a 90% confidence interval for pa - pb?

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