Sampling Distributions: Mean and Proportion (Fundamentals of Statistics, Chapter 8)

Sampling Distributions

Introduction to Sampling Distributions

Sampling distributions are a foundational concept in statistics, describing the probability distribution of a statistic (such as the mean or proportion) computed from a random sample of a given size. Understanding the shape, center, and spread of these distributions is essential for making inferences about populations from sample data.

• Statistic: A numerical measure computed from sample data (e.g., sample mean, sample proportion).

• Random Variable: Statistics are random variables because their values vary from sample to sample.

• Sampling Distribution: The probability distribution of a statistic for all possible samples of a given size from a population.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean describes the distribution of means from all possible samples of a fixed size drawn from a population. This distribution is crucial for understanding how sample means relate to the population mean.

• Definition: The sampling distribution of the sample mean is the probability distribution of all possible values of the sample mean computed from samples of size $n$ from a population with mean $bc$ and standard deviation $c3$.

• Procedure:

  1. Obtain a simple random sample of size $n$.

  2. Compute the sample mean.

  3. Repeat for all possible distinct samples of size $n$.

Distribution of the Sample Mean: Normal Population

When the population is normally distributed, the sampling distribution of the sample mean is also normal, regardless of sample size. The mean and standard deviation of the sampling distribution are related to those of the population.

• Shape: The sampling distribution of the sample mean is normal.

• Center: The mean of the sampling distribution equals the population mean ($bc$).

• Spread: The standard deviation of the sampling distribution (standard error) is smaller than the population standard deviation and is given by: $c3_{ar{x}} = rac{c3}{sqrt{n}}$

• Example: For IQ scores (mean = 100, SD = 15), the sampling distribution of the mean for $n = 10$ is normal with mean 100 and standard deviation $rac{15}{sqrt{10}} approx 4.74$.

Effect of Sample Size on the Sampling Distribution

Increasing the sample size reduces the standard error, making the sampling distribution more concentrated around the population mean.

• Key Point: As $n$ increases, $c3_{ar{x}}$ decreases.

• Example: For $n = 25$, $c3_{ar{x}} = rac{15}{sqrt{25}} = 3$.

Distribution of the Sample Mean: Nonnormal Population

When the population is not normal, the sampling distribution of the sample mean becomes approximately normal as the sample size increases, due to the Central Limit Theorem.

• Central Limit Theorem (CLT): Regardless of the population's shape, the sampling distribution of the sample mean approaches normality as $n$ increases (typically $n geq 30$ is sufficient).

• Mean: The mean of the sampling distribution equals the population mean.

• Standard Error: $c3_{ar{x}} = rac{c3}{sqrt{n}}$

• Example: For a skewed population (e.g., household size), sample means for $n = 30$ are approximately normal.

Central Limit Theorem

The Central Limit Theorem is a fundamental result in statistics, stating that the sampling distribution of the sample mean becomes approximately normal as the sample size increases, regardless of the population's distribution.

• Application: Enables the use of normal probability calculations for sample means when $n$ is large.

• Implication: Highly skewed populations require larger $n$ for normality.

Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion describes the distribution of proportions from all possible samples of a fixed size. This is used to estimate population proportions and compute probabilities.

• Definition: The sample proportion $hat{p}$ is $hat{p} = rac{x}{n}$, where $x$ is the number of individuals with a specified characteristic in a sample of size $n$.

• Mean: $mu_{hat{p}} = p$

• Standard Deviation (Standard Error): $sigma_{hat{p}} = sqrt{ rac{p(1-p)}{n}}$

• Shape: The sampling distribution of $hat{p}$ is approximately normal if $np geq 10$ and $n(1-p) geq 10$.

• Example: In a survey of 1200 adults, 552 plan to work while on vacation. $hat{p} = rac{552}{1200} = 0.46$.

Computing Probabilities for Sample Means and Proportions

Probabilities involving sample means or proportions can be computed using the normal distribution, provided the sampling distribution is approximately normal.

• For Sample Mean:

  • Convert the sample mean to a z-score: $z = rac{ar{x} - mu}{sigma_{ar{x}}}$

  • Use the standard normal table or technology to find probabilities.

• For Sample Proportion:

  • Convert the sample proportion to a z-score: $z = rac{hat{p} - p}{sigma_{hat{p}}}$

  • Use the standard normal table or technology to find probabilities.

• Example: If $p = 0.15$, $n = 120$, probability that at most 12% have hearing trouble: $z = rac{0.12 - 0.15}{sqrt{ rac{0.15 times 0.85}{120}}} approx -0.92$

Summary Table: Properties of Sampling Distributions

Statistic	Mean	Standard Error	Shape
Sample Mean ($bar{x}$)	$mu$	$frac{sigma}{sqrt{n}}$	Normal if population is normal; approximately normal if $n geq 30$
Sample Proportion ($hat{p}$)	$p$	$sqrt{frac{p(1-p)}{n}}$	Approximately normal if $np geq 10$ and $n(1-p) geq 10$

Key Takeaways

• Sampling distributions allow us to make probabilistic statements about sample statistics.

• The Central Limit Theorem ensures normality of the sampling distribution of the mean for large samples.

• Standard error quantifies the variability of a statistic across samples.

• Probabilities for sample means and proportions can be computed using z-scores and the normal distribution.