Question

A candy company claims that 17% of the jelly beans in its spring mix are pink. Suppose that the candies are packaged at random in bags containing about 400 jelly beans. A class of students opens several bags, counts the various colours of jelly beans, and calculates the proportion that are pink. In one bag, the students found 14% of the jelly beans were pink. Is this an unusually small proportion of pink jelly beans? Explain your response.

Answer 1

Let X be the random variable denoting the number of pink jelly beans.

E(X) = 400 * 0.17 = 68, Var(X) = 400 * 0.17 * (1-0.17) = 56.44 i.e. s.d.(X) = = 7.5127

WLG we can assume, X ~ N(68, 7.5127) i.e. (X - 68)/7.5127 ~ N(0,1)

If 14% of the jelly beans are pink out of 400 jelly beans, then (400 * 0.14) = 56 jelly beans are pink.

The probability that less than 14% of the jelly beans are pink = P(X < 56) = P[(X - 68)/7.5127 < (56 - 68)/7.5127] = P[(X - 68)/7.5127 < - 1.5973] = (-1.5973) = 0.0551

[(.) is the cdf of N(0,1)]

Since, the probability that 14% or less than 14% jells beans are pink is 0.0551, which is pretty low, it is an unusual small proportion of pink jelly beans.