Question

Of n1 randomly selected male smokers, X1 smoked filter cigarettes, whereas of n2 randomly selected female smokers, X2 smoked filter cigarettes. Let p1 and p2 denote the probabilities that a randomly selected male and female, respectively, smoke filter cigarettes. (a) Show that (X1/n1) − (X2/n2) is an unbiased estimator for p1 − p2. [Hint: E(Xi)

Answer 1

Answer:

See explanation below

Step-by-step explanation:

An Estimator is unbiased if E(X) = p where E(X) is the expected value of X.

Remember that the Expected value of X is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values.

In this case, we have that:

E(X1) = n1p1  (since we have n1 male smokers with each of them with probability p1 of smoking)

E(X2) = n2p2 (since we have n2 female smokers with each of them with probability p2 of smoking)

So we have to prove that (X1/n1) - (X2/n2) is an unbiased estimator for p1 - p2.

This means that we have to prove (by definition) that

E(x1/n1)- E(x2/n2) = p1 - p2

$E(x)= E(\frac{x1}{n1}) - E(\frac{x2}{n2})\\ =\frac{1}{n1}E(x1) -\frac{1}{n2}E(x2)$

But we know that E(X1) = n1p1 and E(X2)=n2p2

$\frac{1}{n1}E(x1) -\frac{1}{n2}E(x2)\\=\frac{1}{n1}(n1)(p1) -\frac{1}{n2} (n2)(p2)\\=p1 - p2$

Therefore E(x1/n1)- E(x2/n2) = p1 - p2

and ∴ (X1/n1) − (X2/n2) is an unbiased estimator for p1 − p2.