Chapter 5: Sampling Distributions

SECTION 5.3 SUMMARY

• A count X of successes has the binomial distribution B(n, p) 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.
• The binomial distribution B(n, p) is a good approximation to the sampling distribution of the count of successes in an SRS of size n from a large population containing proportion p of successes. We will use this approximation when the population is at least 20 times larger than the sample.
• The sample proportion of successes $\hat{p} = X / n$ is an estimator of the population proportion p. It does not have a binomial distribution, but we can do probability calculations about $\hat{p}$ by restating them in terms of X.
• Binomial probabilities are most easily found by software. There is an exact formula that is practical for calculations when n is small. Table C contains binomial probabilities for some values of n and p. For large n, you can use the Normal approximation.
• The mean and standard deviation of a binomial count X and a sample proportion $\hat{p} = X / n$ are

μ_X = np $μ_{\hat{p}} = p$

$σ_{X} = \sqrt{n p (1 - p)} σ_{\hat{p}} = \sqrt{\frac{p (1 - p)}{n}}$

The sample proportion $\hat{p}$ is, therefore, an unbiased estimator of the population proportion p.

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

$X is approximately N (n p, \sqrt{n p (1 - p)})$

$\hat{p} is approximately N (p, \sqrt{\frac{p (1 - p)}{n}})$

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We will use this approximation when $n p \geq 10$ and $n (1 - p) \geq 10$ . 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.
• The exact binomial probability formula is

$P (X = k) = (\begin{matrix} n \\ k \end{matrix}) p^{k} {(1 - p)}^{n - k}$

where the possible values of X are k = 0,1, . . . , n. The binomial probability formula uses the binomial coefficient

$(\begin{matrix} n \\ k \end{matrix}) = \frac{n!}{k! (n - k)!}$

• Here the factorial n! is

$n! = n \times (n - 1) \times (n - 2) \times . . . \times 3 \times 2 \times 1$

for positive whole numbers n and 0! = 1. The binomial coefficient counts the number of ways of distributing k successes among n trials.

• A count X of successes has a Poisson distribution in the Poisson setting: the number of successes that occur in two nonoverlapping units of measure are independent; the probability that a success will occur in a unit of measure is the same for all units of equal size and is proportional to the size of the unit; the probability that more than one event occurs in a unit of measure is negligible for very small-sized units. In other words, the events occur one at a time.
• If X has the Poisson distribution with mean μ, then the standard deviation of X is $\sqrt{μ}$ , and the possible values of X are the whole numbers 0, 1, 2, 3, and so on.
• The Poisson probability that X takes any of these values is

$P (X = k) = \frac{e^{- μ} μ^{k}}{k!} k = 0, 1, 2, 3, ...$

Sums of independent Poisson random variables also have the Poisson distribution. For example, in a Poisson model with mean μ per unit of measure, the count of successes in a units is a Poisson random variable with mean aμ.