Distribution of the Sample Proportion: Binomial and Bernoulli Distributions

Distribution of the Sample Proportion

Introduction

In statistics, researchers are often interested in studying a process or population to make inferences about a characteristic of interest. When the population is too large or infinite, a sample is taken to estimate the population parameters. The sample proportion is a key statistic used to estimate the proportion of individuals in a population with a given characteristic. This chapter explores the distribution of the sample proportion, its relationship to the binomial and Bernoulli distributions, and how sample size affects the precision of estimates.

Random Sample

Definition and Importance

A random sample is a subset of individuals chosen from a population such that every member has an equal chance of being selected. This ensures that the sample is representative of the population, allowing valid statistical inference.

• Random sampling reduces bias and increases the reliability of estimates.

• Measurements on individuals in a random sample are assumed to be independent and identically distributed.

• Random samples are essential for applying probability models to sample statistics.

Example: A researcher selects 100 individuals at random from a city to estimate the proportion who support a new policy.

Bernoulli Distribution

Definition and Properties

The Bernoulli distribution models a random experiment with only two possible outcomes: "success" (usually coded as 1) and "failure" (coded as 0). If the probability of success is $p$, then the probability mass function is:

• $P(X = 1) = p$

• $P(X = 0) = 1 - p$

The mean and variance of a Bernoulli random variable $X$ are:

• Mean: $\mu_X = p$

• Variance: $\sigma_X^2 = p(1-p)$

Example: The probability that a randomly selected patient responds to a drug is 0.8. $X$ is 1 if the patient responds, 0 otherwise.

The Family of Binomial Distributions

Definition and Properties

The binomial distribution generalizes the Bernoulli distribution to $n$ independent trials, each with probability $p$ of success. The binomial random variable $Y$ counts the number of successes in $n$ trials.

• Probability mass function: $P(Y = y) = \binom{n}{y} p^y (1-p)^{n-y}$ for $y = 0, 1, ..., n$

• Mean: $\mu_Y = np$

• Variance: $\sigma_Y^2 = np(1-p)$

• Standard deviation: $\sigma_Y = \sqrt{np(1-p)}$

Example: In a sample of 10 children, each has a 0.4 probability of inheriting a trait. The probability that exactly 3 children inherit the trait is:

• $P(Y = 3) = \binom{10}{3} (0.4)^3 (0.6)^7$

Binomial Distribution Table

Parameter	Symbol	Formula
Number of trials	n	-
Probability of success	p	-
Mean	$\mu_Y$	$np$
Variance	$\sigma_Y^2$	$np(1-p)$
Standard deviation	$\sigma_Y$	$\sqrt{np(1-p)}$

Distribution of the Sample Proportion

Definition and Calculation

The sample proportion $\hat{p}$ is the proportion of individuals in the sample with the characteristic of interest. If $Y$ is the number of successes in a sample of size $n$, then:

• $\hat{p} = \frac{Y}{n}$

The sample proportion is a random variable with its own distribution, mean, and variance:

• Mean: $\mu_{\hat{p}} = p$

• Variance: $\sigma_{\hat{p}}^2 = \frac{p(1-p)}{n}$

• Standard deviation: $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

Example: In a sample of 10 individuals, if $p = 0.4$, the probability that the sample proportion is 0.2 is:

• $P(\hat{p} = 0.2) = P(Y = 2) = \binom{10}{2} (0.4)^2 (0.6)^8$

Effect of Sample Size

As the sample size $n$ increases, the variance of the sample proportion decreases, resulting in more precise estimates of $p$.

• For $n = 10$, $\sigma_{\hat{p}}^2 = \frac{0.25(1-0.25)}{10} = 0.01875$

• For $n = 20$, $\sigma_{\hat{p}}^2 = \frac{0.25(1-0.25)}{20} = 0.009375$

Graph: The variance of $\hat{p}$ decreases as $n$ increases, as shown in the provided figure.

Conclusion

In summary, the distribution of the sample proportion $\hat{p}$ is a key tool for making inferences about the proportion $p$ of individuals in a population with a given characteristic. Under the assumption of independence and a common Bernoulli distribution, $\hat{p}$ is an unbiased estimator of $p$. Increasing the sample size reduces the variance of $\hat{p}$, leading to more precise estimates.

Key Formulas

• Bernoulli mean: $\mu_X = p$

• Bernoulli variance: $\sigma_X^2 = p(1-p)$

• Binomial mean: $\mu_Y = np$

• Binomial variance: $\sigma_Y^2 = np(1-p)$

• Sample proportion: $\hat{p} = \frac{Y}{n}$

• Sample proportion mean: $\mu_{\hat{p}} = p$

• Sample proportion variance: $\sigma_{\hat{p}}^2 = \frac{p(1-p)}{n}$

Summary Table: Distributions and Properties

Distribution	Random Variable	Mean	Variance
Bernoulli	$X$	$p$	$p(1-p)$
Binomial	$Y$	$np$	$np(1-p)$
Sample Proportion	$\hat{p}$	$p$	$\frac{p(1-p)}{n}$

Additional info: The notes include exercises and calculator instructions for computing binomial probabilities, which are useful for practical applications in statistics.