Question

1. There are 10,000 pieces of candy. Suppose we want to know the probability that any sample of 500 pieces from the bowl will contain at least 50 pieces Reese’s cups on average. (Do not calculate this.) A discrete distribution could calculate this problem, but it would be really hard. What theorem could help approximate the probability? Describe the process for how the theorem could be applied to this problem, but do not calculate the probability.

Answer 1

We can use Binomial Distributio to calcuate this probability, but that's a daunting task.

For large values of n, the distributions of the count X and the sample proportion are approximately normal. This result follows from the Central Limit Theorem. The mean and variance for the approximately normal distribution of X are np and np(1-p), identical to the mean and variance of the binomial(n,p) distribution. Similarly, the mean and variance for the approximately normal distribution of the sample proportion are p and (p(1-p)/n).

Keep in mind that np > 10.

Central Limit Theorem: