03/31/2021

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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

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**.

- the
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**.

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.

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 \( X_i \).

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**.

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 \( x_1, \cdots, x_n \).

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 \( \theta \).

- 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, \( p_1 − p_2 \)

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.

- For \( \mu \),
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.

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.

- 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

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

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).

- 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.