Question

Knowing your blood type is important not only because it determines who you can donate blood to but also who you can receive blood from. The second most common blood type in America is A positive and 36 percent of Americans share this blood type. The least common blood type is AB negative and only 0.5 percent of Americans have this blood type. Suppose a random sample of 196 American donors has been chosen at random. In this sample, let X be the number of donors with A positive blood, and let Y be the number of donors with AB negative blood.

(a)	Find the mean of X.

(b)	Find the variance of X.

(c)	Use either the normal or the Poisson approximation, whichever is appropriate, to find the simplest estimate for the probability P(X  ≥  81).

(d)	Find the mean of Y.

(e)	Find the variance of Y.

(f)	Use either the normal or the Poisson approximation, whichever is appropriate, to find the simplest estimate for the probability P(Y  ≥  2).

Answer 1

It is given that 36% of Americans have A positive blood type.

So, the chance that a randomly selected American has A positive blood type, is 0.36.

A random sample of 196 American donors have been chosen.

X is the number of donors with A positive blood.

Question a

Mean of X

Question b

Variance of X

Question c

We approximate the distribution by normal distribution because the sample size is too large, and the mean is not small positive value either.

So, after normal approximation, X follows normal with mean 70.56 and variance 45.1584, ie. standard deviation of 6.72.

Now, we have to find

So, the answer is 0.9606.

It is given that 0.5% of Americans have AB negative blood type.

So, the chance that a randomly selected American has AB negative blood type, is 0.005.

A random sample of 196 American donors have been chosen.

Y is the number of donors with AB negartive blood.

Question d

Mean of Y

Question e

Variance of Y

Question f

In this case, the sample size is again large, but the mean is a very small positive value, and the variance are very close.

So, we approximate this binomial distribution by a poisson distribution.

We have to find

Where, Y is poisson 0.98.

So, by calculating, this becomes

So, the answer is 0.2569.