# Statistics

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Statistics

## 1. 1. Probability

### 1.1. Parts

1.1.1. Outcome

1.1.2. Sample Space

1.1.3. Event

### 1.2. Axioms

1.2.1. 0 <= P(A) <= 1

1.2.2. P(S) = 1

1.2.3. A & B disjoint => P(A or B) = P(A) + P(B)

1.2.4. P(~A) = 1 - P(A)

### 1.3. Properties

1.3.1. Conditionality

1.3.1.1. P(B|A) = P(A & B)/P(A), P(A & B) = P(A)P(B|A)

1.3.2. Inclusion - Exclusion Prnciple

1.3.2.1. P(A or B) = P(A) + P(B) - P(A & B)

1.3.3. Independence

1.3.3.1. P(A & B) = P(A)P(B), P(A|B) = P(A)

### 1.4. Probability Model

1.4.1. Random Variable

1.4.2. Personal Probability

1.4.3. Types

1.4.3.1. Discrete

1.4.3.1.1. Tree Diagram

1.4.3.2. Continuous

1.4.3.2.1. Density Curve

## 2. 2. Getting Data

### 2.1. Types

2.1.1. Sampling

2.1.1.1. Parts

2.1.1.1.1. Population

2.1.1.1.2. Sample

2.1.1.1.3. Population 20 times Larger

2.1.1.2. Bias

2.1.1.2.1. Undercoverage

2.1.1.2.2. Response Bias

2.1.1.2.3. Nonresponse

2.1.1.2.4. Wording Errors

2.1.1.2.5. Small Sample Size

2.1.1.3. Types

2.1.1.3.1. Convenience Sample

2.1.1.3.2. Sample Survey

2.1.1.3.3. Random Sampling

2.1.2. Observational Studies

2.1.2.1. Subject to Confounding

2.1.3. Experiments

2.1.3.1. Parts

2.1.3.1.1. Subjects

2.1.3.1.2. Factors

2.1.3.1.3. Treatment

2.1.3.2. Design

2.1.3.2.1. Control

2.1.3.2.2. Randomization

2.1.3.2.3. Replication

## 3. 3. Making Models

### 3.2. Variables

3.2.1. Distributions

3.2.1.1. Properties

3.2.1.1.1. Shape

3.2.1.1.2. Summary

3.2.1.1.3. Outlier

3.2.1.2. Types

3.2.1.2.1. Single - Variable

3.2.1.2.2. Multi - Variable

## 4. 4. Inference

4.1.1. Parameter

4.1.2. Statistic

### 4.2. Sampling Distribution

4.2.1. Population Distribution

4.2.2. Unbiased Estimator

4.2.3. Law of Large Numbers

4.2.3.1. As number of Observations increases, Mean approaches mu

4.2.4. Central Limit Theorem

4.2.4.1. As Sampling Size Increases, the Sampling Distribution of the Mean is Normal

4.2.4.1.1. Sampling Distribution of Mean mu has Mean mu and Standard Deviation sigma/(mu)^1/2

4.2.4.1.2. Average Less Variable than Individuals

4.2.4.1.3. Means more Normal than Individuals

4.2.4.1.4. Large Samples Less Variable than Small

4.2.4.1.5. To cut S.D. in half, Use four Times as many Observations

4.2.4.1.6. Standard Normal Table

4.2.4.2. Variables not highly correlated

4.2.4.3. Distributions aren't too disparate

### 4.3. Types

4.3.1. Mean:

4.3.1.1. Confidence Interval

4.3.1.1.1. Estimate

4.3.1.1.2. Margin of Error

4.3.1.1.3. t Critical Values Table (Bottom)

4.3.1.2. Test of Significance

4.3.1.2.1. Parts

4.3.1.2.2. Errors

4.3.1.2.3. Properties

4.3.2. Comparing Multiple Means

4.3.2.1. Analysis of Variance F Test

4.3.3. Proportion

4.3.3.1. p Statistic

4.3.4. Two Categorical Variables

4.3.4.1. Chi-Square Test

4.3.4.1.1. Expected Count

4.3.4.1.2. One-Sided

4.3.4.1.3. No More than 20% Expected Counts < 5; All Counts > 1