# Sampling distributions and the central limit theorem

03/31/2021

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## Outline

• The following topics will be covered in this lecture:

• Random samples
• Sampling distributions
• Point estimators
• Central limit theorem
• Applications of the central limit theorem
• Approximate sampling distribution of a difference in sample means

## Motivation

• We have now learned about the fundamentals of theoretical probabilistic models.

• Particularly, we have learned about the:

• probability distribution;
• probability mass / density function; and
• cumulative probability distribution function;
• for discrete and continuous random variables.

• We have also learned about several fundamental probability distributions:

• the binomial;
• the uniform; and
• the normal.
• We will begin to combine these models with data to produce statistical inference.

• Our goal in this course is to use statistics from a small, representative sample to say something general about the larger, unobservable population or phenomena.

• The process of saying something general from the smaller representative sample, while qualifying our uncertainty, is what we mean by statistical inference.

### Motivation continued

• The link between the probability models in the earlier chapters and the data is made as follows.

• Suppose we take a sample of $$n = 10$$ observations $$\{x_{1,i}\}_{i=1}^{10}$$ from a population and compute the sample average,

$\overline{x}_1 = \frac{1}{n} \sum_{i=1}^n x_{1,i} = \frac{1}{10}\sum_{i=1}^{10} x_{1,i}$

getting the result $$\overline{x}_1 = 10.2$$.

• Now we repeat this process, taking a second sample of $$n = 10$$ observations from the same population,

$\{x_{2,i}\}_{i=1}^{10}$

and the resulting sample average is $$\overline{x}_2=10.4$$.

• This discrepancy is what we call sampling error, in which the random variation in a sample of a fixed size $$n$$ upon replication produces differences in the computation of a statistic.

• The sample average depends on the observations in the sample, which differ from sample to sample because they are random variables.

• Consequently, the sample average (or any other function of the sample data) is a random variable.

• Because a statistic is a random variable, it has a probability distribution.

### Motivation continued

• Specifically, suppose that we want to obtain an estimate of a population parameter, where the population is modeled with a random variable $$X$$.

• We know that before the data are collected, the observations are considered to be random variables,

• i.e., we treat an independent sequence of measurements of $$X$$,

$X_1, X_2, \cdots , X_n$

• as random variables all drawn from a parent distribution $$X \sim F(x)$$ (where the CDF will define the distribution).
Random sample
The random variables $$X_1 , X_2, \cdots , X_n$$ are a random sample of size $$n$$ if the $$X_i$$’s are independent random variables and every $$X_i$$ has the same probability distribution.
• We then say that the measurements we obtain are possible outcomes of the sample variables $$\{X_i\}_{i=1}^n$$; particularly, if we make a computation of the sample mean,

$\overline{X} = \frac{1}{n} \sum_{i=1}^n X_i$

the above is treated as a random variable (a linear combination of random variables) which has a random outcome, dependent on the realizations of the $$X_i$$.

### Motivation continued

• More generally, any function of the observations, i.e., any statistic, is also modeled as a random variable.

• If $$h$$ is a general function used to compute some statistic, we thus define

$\tilde{X} = h(X_1, \cdots, X_n)$

to be a random variable that will depend on the particular realizations of $$X_1,\cdots, X_n$$.

• We call the probability distribution of a statistic a sampling distribution.

Sampling Distribution
The probability distribution of a statistic is called a sampling distribution.

### Motivation continued

• Given particular realizations of the sample random variables, we obtain a fixed numerical value.

• Each numerical value in a data set is treated as the observed realization of a random variable.

• Given particular realizations $$x_1,\cdots,x_n$$ of the random variables $$X_1, \cdots, X_n$$, the value

$\overline{x} = \frac{1}{n}\sum_{i=1}^n x_i$

is not a random variable, as this is a fixed numerical value.

• Given some particular, observed realizations $$x_1, \cdots,x_n$$,

$\tilde{x} = h(x_1, \cdots, x_n)$

is a fixed numerical value, based on the fixed, observed data values $$x_1, \cdots, x_n$$.

### Motivation continued

• When discussing inference problems, it is convenient to have a general symbol to represent the parameter of interest – we use the Greek symbol $$\theta$$ (theta) to represent the parameter.

• The symbol $$\theta$$ can represent the mean $$\mu$$, the variance $$\sigma^2$$, or any parameter of interest to us.

• The objective of point estimation is to estimate a single number based on sample data that is the most plausible value for $$\theta$$.

• The numerical value of a sample statistic is used as the point estimate.

• Once we describe the process of point estimation, the next step is to describe how we quantify the uncertainty of the estimate.

• If $$X$$ is a random variable with probability distribution $$F(x)$$, characterized by the unknown parameter $$\theta$$,

and if $$X_1 , X_2, \cdots , X_n$$ is a random sample of size $$n$$ from $$X$$,

• the statistic $$\hat{\Theta} = h(X_1 , X_2 , ... , X_n )$$ given as a function of the sample is called a point estimator of $$\theta$$.

• Note that $$\hat{\Theta}$$ is a random variable because it is a function of random variables.

• After the sample has been selected, $$\hat{\Theta}$$ takes on a particular numerical value $$\hat{\Theta}$$ called the point estimate of $$\theta$$.

• The uncertainty of the point estimate $$\hat{\Theta}$$ can be understood as how much will the sampling error cause a discrepancy between $$\hat{\Theta}$$ and the true $$\theta$$.

## Point estimators

• We will now introduce some formal definitions:
• Point estimators
A point estimate of some population parameter $$\theta$$ is a single numerical value $$\hat{\Theta}$$ of a statistic $$\hat{\Theta}$$. This is a particular realization of the random variable $$\hat{\Theta}$$, viewed as a random variable; $$\hat{\Theta}$$ is called the point estimator.
• Estimation problems modeled as above occur frequently in engineering.

• We often need to estimate

• The mean $$\mu$$ of a single population
• The variance $$\sigma^2$$ (or standard deviation $$\sigma$$) of a single population
• The proportion $$p$$ of items in a population that belong to a class of interest
• The difference in means of two populations, $$\mu_1 - \mu_2$$
• The difference in two population proportions, $$p_1 − p_2$$

### Point estimators continued

• Reasonable point estimates of these parameters are as follows:

• For $$\mu$$,
• the estimate is $$\hat{\mu}=\overline{x}$$, the sample mean.
• For $$\sigma^2$$,
• the estimate is $$\hat{\sigma}^2 = s^2$$, the sample variance.
• For $$p$$,
• the estimate is $$\hat{p}=\frac{x}{n}$$, the sample proportion, where $$x$$ is the number of items in a random sample of size $$n$$ that belong to the class of interest.
• For $$\mu_1 -\mu_2$$,
• the estimate $$\hat{\mu}_1 - \hat{\mu}_2 = \overline{x}_1 - \overline{x}_2$$, the difference between the sample means of two independent random samples.
• For $$p_1 − p_2$$ ,
• the estimate is $$\hat{p}_1 - \hat{p}_2$$ , the difference between two sample proportions computed from two independent random samples.
• Generally, however, we may have several different choices for the point estimator of a parameter.

• To decide which point estimator of a particular parameter is the best one to use, we need to examine their statistical properties and develop some criteria for comparing estimators.

## Central limit theorem

• Let's consider a simple argument for the sampling distribution of the sample mean $$X$$.

• Suppose that a random sample of size $$n$$ is taken from a normal population with mean $$\mu$$ and variance $$\sigma^2$$.

• By definition of a random sample each observation in this sample, say, $$X_1, X_2, \cdots, X_n$$, is a normally and independently distributed random variable with mean $$\mu$$ and variance $$\sigma^2$$.

• A special property of the normal distribution is that it can be translated and rescaled while remaining normal;

• similarly, a sum of independent, normally distributed random variables are also normally distributed.
• We conclude that the sample mean

$\overline{X}= \frac{X_1 + X_2 + \cdots + X_n}{n}$

has a normal distribution with mean

$\mu_\overline{X} = \frac{\mu + \mu + \cdots + \mu}{n} = \mu$

• and variance

$\sigma^2_\overline{X} = \frac{\sigma^2 + \sigma^2 + \cdots + \sigma^2}{n^2} = \frac{\sigma^2}{n}$

### Central limit theorem continued

• More generally, if we are sampling from a population that has an unknown probability distribution, the sampling distribution of the sample mean will still be approximately normal with mean $$\mu$$ and variance $$\frac{\sigma^2}{n}$$ if the sample size $$n$$ is large.

• This is one of the most useful theorems in statistics, called the central limit theorem:

The central limit theorem
Let $$X_1 , X_2 , \cdots , X_n$$ be a random sample of size $$n$$ taken from a population with mean $$\mu$$ and finite variance $$\sigma^2$$ and $$\overline{X}$$ be the sample mean. Then the limiting form of the distribution of $Z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}$ as $$n \rightarrow \infty$$ is the standard normal distribution.
• Put another way, for $$n$$ sufficiently large, $$\overline{X}$$ has approximately a $$N\left(\mu, \frac{\sigma^2}{n}\right)$$ distribution – this says the following.

• Suppose we take a sample of size $$n$$ and compute the sample mean $$\overline{X}$$.
• Then suppose we replicate this sample and record the observed realizations for the sample mean $$\overline{x}_1, \overline{x}_2, \cdots$$.
• If the sample size $$n$$ is lage, these data points $$\overline{x}_1, \cdots$$ will be approximately bell shaped with the following properties:
• the bell will be centered approximately at $$\mu$$, the true population mean;
• the spread of the data around the center will be given by approximately by the standard deviation $$\frac{\sigma}{\sqrt{n}}$$.
• Particularly, if $$n$$ is very large, the observed sample means will be very close to the center (the true mean).

### Central limit theorem continued

• As a visualization of the concept, suppose again that we have a random sample indexed by $$j$$ $X_{j,1}, \cdots, X_{j,n}.$
• We will make replications for $$j=1,\cdots,m$$ and get a random variable for sample mean indexed by $$j$$, $\overline{X}_j = \frac{1}{n}\sum_{i=1}^n X_{j,i}.$
• When we observe a realization of $$\overline{X}_j=\overline{x}_j$$ or respectively the sample $X_{j,1}=x_{j,1}, \cdots, X_{j,n}=x_{j,n},$ we record these fixed numerical values.