Section 7.5 Summary
Subsection 7.5.1 Glossary
- Random Sample
-
The probabilistic model for the values of a measurements in the sample, before the measurement is taken.
- Sampling Distribution
-
The distribution of a random sample.
- Sampling Distribution of a Statistic
-
A statistic is a function of the data; i.e. a formula applied to the data. The statistic becomes a random variable when the formula is applied to a random sample. The distribution of this random variable, which is inherited from the distribution of the sample, is its sampling distribution.
- Sampling Distribution of the Sample Average
-
The distribution of the sample average, considered as a random variable.
- The Law of Large Numbers
-
A mathematical result regarding the sampling distribution of the sample average. States that the distribution of the average of measurements is highly concentrated in the vicinity of the expectation of a measurement when the sample size is large.
- The Central Limit Theorem
-
A mathematical result regarding the sampling distribution of the sample average. States that the distribution of the average is approximately Normal when the sample size is large.
Subsection 7.5.2 Discussion in the Forum
Limit theorems in mathematics deal with the convergence of some property to a limit as some indexing parameter goes to infinity. The Law of Large Numbers and the Central Limit Theorem are examples of limit theorems. The property they consider is the sampling distribution of the sample average. The indexing parameter that goes to infinity is the sample size \(n\text{.}\)
Some people say that the Law of Large Numbers and the Central Limit Theorem are useless for practical purposes. These theorems deal with a sample size that goes to infinity. However, all sample sizes one finds in reality are necessarily finite. What is your opinion?
When forming your answer to this question you may give an example of a situation from your own field of interest in which conclusions of an abstract mathematical theory are used in order to solve a practical problem. Identify the merits and weaknesses of the application of the mathematical theory.
For example, in making statistical inference one frequently needs to make statements regarding the sampling distribution of the sample average. For instant, one may want to identify the central region that contains 95% of the distribution. The Normal distribution is used in the computation. The justification is the Central Limit Theorem.
Subsection 7.5.3 Summary of Formulas
- Expectation of the sample average
-
\(\displaystyle \Expec(\bar X) = \Expec(X)\)
- Variance of the sample average
-
\(\displaystyle \Var(\bar X) = \Var(X)/n\)
