Sampling Distribution Models

Binomial Model and Its Properties

The Binomial model is a fundamental probability model used to describe the number of successes in a fixed number of independent trials, each with the same probability of success. Understanding its properties is essential for applying normal approximations and exploring sampling distributions.

• Definition: A Binomial model satisfies four key conditions:

  1. A sequence of n trials is performed, with n fixed in advance.

  2. Trials are identical—each results in either a success (S) or a failure (F).

  3. Trials are independent.

  4. The probability of success, denoted p, is the same for each trial.

• Random Variable: If X is the number of successes out of n trials, then X follows a Binomial distribution: X ~ Binomial(n, p).

Example: In a speed dating event with 10 people, if the probability of a good impression is 10% per person, questions may include:

• What is the probability that you like exactly 5 of them?

• What is the probability that you like at most 5 of them?

• What is the expected number of candidates you will like?

Binomial Probability Formula

The probability of observing exactly k successes in n independent trials is given by:

This formula is used to compute probabilities for binomial outcomes.

Normal Approximation to the Binomial

When and Why to Use Normal Approximation

For large sample sizes, the Binomial distribution can be approximated by a Normal distribution, which simplifies probability calculations.

• Conditions: The normal approximation is appropriate when both and .

• Approximation: If , then for large :

  • , where

• Mean:

• Variance:

Why use normal approximation? It greatly simplifies calculations, especially for large , where direct computation using the binomial formula is impractical.

Example: If you browse 1,000 profiles with , the probability of liking at most 100 can be approximated using the normal model.

Shape of Binomial Distribution

The shape of the binomial distribution depends on the sample size and probability:

• For large and moderate , the distribution is approximately normal (bell-shaped).

• For small or extreme , the distribution is skewed and far from normal.

Example: Simulations in R show that with , , the histogram of successes is bell-shaped. With , , the histogram is skewed and not normal.

Sampling Distributions

Definition and Importance

A sampling distribution describes the distribution of a statistic (such as the sample mean or proportion) computed from random samples of a fixed size from a population.

• The value of a statistic varies from sample to sample.

• The sampling distribution allows us to make probabilistic statements about statistics.

Sample Proportion

When interested in the proportion of successes, the sample proportion is used:

• Sample proportion: , where is the number of successes in trials.

• For large (with and ), the sampling distribution of is approximately normal:

Example: An insurance company claims 75% of clients have smoke detectors. For 80 applications:

• Mean of :

• Standard deviation:

• Assumptions: Independence, fixed , constant

• Probability that over 60% get discounts can be found using the normal approximation.

Central Limit Theorem (CLT)

Statement and Implications

The Central Limit Theorem (CLT) is a foundational result in statistics, describing the behavior of the sampling distribution of the sample mean.

• For an i.i.d. sample from any distribution with mean and variance :

• As gets large, the sampling distribution of is approximately normal with mean and variance , regardless of the original distribution.

Remarks on CLT

• The CLT does not require the original data to be normal.

• If the original data is normal, the sample mean is always normal, regardless of .

• The more the original distribution deviates from normal, the larger must be for the normal approximation to be accurate.

Example: Human gestation times have mean 266 days, SD 16 days. For samples of 100 women:

• The histogram of sample means will be approximately normal.

• Center: 266 days

• Standard deviation: days

Summary Table: Binomial vs. Normal Approximation

Key Takeaways

• Binomial models describe the number of successes in fixed, independent trials.

• For large samples, binomial distributions can be approximated by normal distributions, simplifying calculations.

• Sampling distributions allow us to understand the variability of statistics from sample to sample.

• The Central Limit Theorem ensures that sample means are approximately normal for large samples, regardless of the original data distribution.

Additional info: R code examples and histograms in the original notes illustrate the transition from binomial to normal shape as sample size increases, and the application of simulation to visualize sampling distributions.