04/05/2021

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The following topics will be covered in this lecture:

- A review of the central limit theorem
- Applications of the central limit theorem
- Approximate sampling distribution of a difference in sample means
- General concepts in point estimation
- Bias of estimators
- Variance of estimators
- Standard Error

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 \( X_i \).

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 \[ Z = \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}} \] 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.