SECTION 5.1 Summary

• A count $X$ of successes has the binomial distribution $B\left(n,p\right)$ in the binomial setting: there are $n$ trials, all independent, each resulting in a success or a failure, and each having the same probability $p$ of a success.

• If $X$ has the binomial distribution with parameters $n$ and $p$, the possible values of $X$ are the whole numbers $\mathrm{0,}\mathrm{1,}\mathrm{2,\; .\; .\; .\; ,}n$. The binomial probability that $X$ takes any value is

  $P\left(X=k\right)=\left(\begin{array}{c}n\\ k\end{array}\right)p^k{\left(1-p\right)}^{n-k}$

  Binomial probabilities are most easily found by software. This formula is practical for calculations when $n$ is small. Table C contains binomial probabilities for some values of $n$ and $p$.

• The binomial coefficient

  $\left(\begin{array}{c}n\\ k\end{array}\right)={\displaystyle \frac{n!}{k!\left(n-k\right)!}}$

  counts the number of ways $k$ successes can be arranged among $n$ observations. Here, the factorial $n!$ is

  $n!=n\times \left(n-1\right)\times \left(n-2\right)\times \cdots \times 3\times 2\times 1$

  for positive whole numbers $n$, and $0!=1$.

• The mean and standard deviation of a binomial count $X$ and a sample proportion $\hat{p}=X/n$ are
  $\begin{array}{ll}{\mu }_X=np& {\mu }_{\hat{p}}=p\\ {\sigma }_X=\sqrt{np\left(1-p\right)}& {\sigma }_{\hat{p}}=\sqrt{{\displaystyle \frac{p\left(1-p\right)}{n}}}\end{array}$
  Page 264

• The Normal approximation to the binomial distribution says that if $X$ is a count having the $B\left(n,p\right)$ distribution, then when $n$ is large,

  $\begin{array}{l}X\text{\hspace{1em}is\hspace{1em}approximately\hspace{1em}}N\left(np,\sqrt{np\left(1-p\right)}\right)\\ \hat{p}\text{\hspace{1em}is\hspace{1em}approximately\hspace{1em}}N\left(p,\sqrt{{\displaystyle \frac{p\left(1-p\right)}{n}}}\right)\end{array}$

  We will use this approximation when $np\ge 10$ and $n\left(1-p\right)\ge 10$. It allows us to approximate probability calculations about $X$ and $\hat{p}$ using the Normal distribution. The continuity correction improves the accuracy of the Normal approximations.

• A simple check for the adequacy of the binomial model is to compare the binomial-based standard deviation (or variance) with the observed count standard deviation (or variance). In addition, some software packages provide fitting of the binomial model on the observed histogram to assess compatibility.