1. Data Introduction

There are 50 different flavored candies in a large bag of 1300 candy beans. Eight of these flavors contain a green color. According to the company’s website, \(15\%\) of the candies in any given package are green.

You are going to collect a random sample of these candies and evaluate the company’s claim that \(15\%\) are green. Along the way, you are going to practice finding probabilities using a Binomial table of probabilities and a Normal (Z) table of probabilities.

2. Analysis Goal

One goal of this exercise is to explore features of the binomial probability distribution and eventually, to see how a Normal probability distribution can approximate a binomial distribution for a large enough sample size. The other goal is to use the expectation and standard deviation of random variables to characterize whether or not the observed proportions in a sample match up with the claims about the package contents from the candy company.

Click this link to view the worksheet for more experiments with probability.

3. Summary of Results

Random variable \(X \sim Binom(n,p)\) and \(E(X) = np, \quad Var(X) = np(1-p)\). The model parameter is \(p=\) proportion of successes.

From a sample of size \(n\), we can derive the sample estimate \(\hat{p} = \frac{\text{number of green candies}}{n}\).

If \(n\) is large enough, then \(X \approx N(\mu = np, \sigma = \sqrt{np(1-p)})\).