Sampling Distributions, Normal Approximation, and Probability in Statistics

Sampling Distributions and the Central Limit Theorem

Definition and Properties

The sampling distribution of a statistic (such as the mean or proportion) describes the probability distribution of that statistic over all possible samples of a fixed size from a population. The Central Limit Theorem (CLT) states that, for sufficiently large sample sizes, the sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution, provided the population has a finite mean and variance.

• Sample Mean ($\bar{X}$): The average value from a sample of size $n$.

• Mean of Sampling Distribution: $\mu_{\bar{X}} = \mu$ (population mean)

• Standard Deviation of Sampling Distribution (Standard Error): $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$

• Normal Approximation: If the population is normal or $n \geq 30$, the sampling distribution of $\bar{X}$ is approximately normal.

Example: If the height of 20-year-old American women is normally distributed with mean 64 inches and standard deviation 3 inches, the sampling distribution for the average height of 36 randomly chosen women will be normal with mean 64 inches and standard deviation $\frac{3}{\sqrt{36}} = 0.5$ inches.

Normal Distribution and Probability Calculations

Standard Normal Distribution

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Probabilities for normal distributions are often calculated by converting values to z-scores:

• Z-score formula: $z = \frac{X - \mu}{\sigma}$

• Use standard normal tables or software to find probabilities associated with z-scores.

Example: To find $P(X > 120)$ for $X \sim N(110, 8^2)$, calculate $z = \frac{120 - 110}{8} = 1.25$ and use the normal table to find $P(Z > 1.25)$.

Binomial Distribution and Normal Approximation

Conditions for Normal Approximation

The binomial distribution models the number of successes in $n$ independent Bernoulli trials with probability $p$ of success. For large $n$, the binomial can be approximated by a normal distribution if:

• $np \geq 5$ and $n(1-p) \geq 5$

• Mean: $\mu = np$

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

Example: For a sample of 49 customers with $p = 0.16$, $np = 7.84$ and $n(1-p) = 41.16$, so normal approximation is valid.

Probability Functions and R Commands

Using R for Probability Calculations

Statistical software such as R provides functions to compute probabilities for normal and binomial distributions:

• pnorm(x, mean, sd): Computes $P(X \leq x)$ for normal distribution.

• qnorm(p, mean, sd): Computes the quantile (inverse cumulative probability).

• 1 - pnorm(x, mean, sd): Computes $P(X > x)$.

Example: pnorm(60, mean=54, sd=4) computes $P(X \leq 60)$ for $X \sim N(54, 4^2)$.

Sampling Distribution of Proportion

Sample Proportion and Its Distribution

The sample proportion $\hat{p}$ is the fraction of successes in a sample. Its sampling distribution is approximately normal if $np \geq 5$ and $n(1-p) \geq 5$:

• Mean: $\mu_{\hat{p}} = p$

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

Example: For a sample of 620 customers with $p = 0.16$, $\mu_{\hat{p}} = 0.16$, $\sigma_{\hat{p}} = \sqrt{\frac{0.16 \times 0.84}{620}}$.

Application: Real-World Scenarios

Interpreting Statistical Results

Statistical methods are used to answer questions about populations based on sample data, such as estimating average blood pressure, average time spent on the internet, or the proportion of customers outside a city.

• Always check conditions for normal approximation before applying it.

• Use sample statistics to estimate population parameters and compute probabilities.

Tables: Summary of Key Formulas

Statistic	Mean	Standard Deviation	Distribution (if normal approx. valid)
Sample Mean ($\bar{X}$)	$\mu$	$\frac{\sigma}{\sqrt{n}}$	Normal
Sample Proportion ($\hat{p}$)	$p$	$\sqrt{\frac{p(1-p)}{n}}$	Normal
Binomial ($X$)	$np$	$\sqrt{np(1-p)}$	Normal (if $np, n(1-p) \geq 5$)

Additional info:

• Some questions reference R code for probability calculations, which is common in statistics courses.

• All examples and scenarios are typical applications of sampling distributions, normal approximation, and probability in introductory statistics.