Question

Mark noticed that the probability that a certain player hits a home run in a single game is 0.175. Mark is interested in the variability of the number of home runs if this player plays 200 games. If Mark uses the normal approximation of the binomial distribution to model the number of home runs, what is the standard deviation for a total of 200 games? Answer choices are rounded to the hundredths place.

Answer 1

Solution

Let X = number of home runs of this player in 200 games played by him. Then, X ~ B(200, p), where p = probability that a this player hits a home run in a single game and this is given to be 0.175.

Thus, X ~ B(200, 0.175) …………………………………………………………………………… (1)

Back-up Theory

Normal approximation

If X ~ B(n, p) np ≥ 5 and np(1 - p) ≥ 5, (X – np)/√{np(1 - p)} ~ N(0, 1) [approximately] ……… (2)

Where np = Mean of X and √{np(1 - p)} is the standard deviation of X.

Now to work out the solution

Here n = 200 and p = 0.175. So, standard deviation for a total of 200 games is the standard deviation for a total of X

= √{200 x 0.175 x 0.825) [vide (2)]

= 5.3735 Answer

DONE