Question: 3.15 Ya and Yb are Bernoulli random variables from two different populations, denoted a and b. Suppose E(Ya) = pa and E(Yb) = pb. A

3.15 Ya and Yb are Bernoulli random variables from two different populations, denoted a and

b. Suppose E(Ya) = pa and E(Yb) = pb. A random sample of size na is chosen from population

a, with a sample average denoted pn

a, and a random sample of size nb is chosen from population

b, with a sample average denoted pn

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

a. Show that E1pn a2 = pa and var1pn a2 = pa11 - pa2 > na. Show that E1pn b2 = pb and var1pn b2 = pb11 - pb2 > nb.

b. Show that var1pn a - pn b2 =

pa11 - pa2 na

+

pb 11 - pb2 nb

.

(Hint: Remember that the samples are independent.)

c. Suppose na and nb are large. Show that a 95% confidence interval for pa - pb is given by 1pn a - pn b2 { 1.96 A

pn a11 - pn a2 na

+

pn b11 - pn b2 nb

.

How would you construct a 90% confidence interval for pa - pb?

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