Question

Suppose the incidence of ketosis in Holstein dairy cows is 11.8% in the state of Michigan (Dyk and Emery, 1996; Proc Tri-State Nutrition Conference). Assume the state of ketosis (1 = diseased, 0 = not-diseased) is independent between all animals; i.e., the occurrence or non-occurence of ketosis on any one animal is in no way related to whether ketosis occurs in any other animal. Furthermore, suppose that there are a large number of herds in the state for which the probability of ketosis (11.8%) was the same for each herd.

A)Within any one herd of size 60, what is the exact probability of having 4 or less cows being afflicted with ketosis.

B)Redo (a) but use a normal approximation to determine this probability. Do this with and without the continuity adjustment. Are either of these results consistent with the exact probability you provided in your answer to a)?

Answer 1

(a)

Let X is a random variable shows the number of cows being afflicted with ketosis. Here X has binomial distribution with following parameters

n = 60 and p=0.118

The  exact probability of having 4 or less cows being afflicted with ketosis is

B)

Since np = 7.08 and n(1-p) = 52.92 both are greater than 5 so we can use normal approximation here.

Using normal approximation, X has approximately normal distribution with mean and SD as follows:

Using continuity correction factor:

z-score for X = 4+ 0.5 =4.5 is

Using z table, the required probability is

Without continuity correction factor:

z-score for X = 4 is

Using z table, the required probability is

Using continuity correction factor, results are approximately equal to result of part a.