04/05/2021
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The following topics will be covered in this lecture:
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,
\[ X_1, X_2, \cdots , X_n \]
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.