formula_sheet_Q3.Rmd

---
title: "Formula Sheet for Quiz 3"
subtitle: "STAT 011"
output: pdf_document
---


```{r setup_pres, include=FALSE, echo=FALSE}
rm(list=ls())
library('tidyverse')

#setwd("~/Google Drive Swat/Swat docs/Stat 21/Data")
options(htmltools.dir.version = FALSE)
```

# Sample Statistics 

## For a sample of data

If $\{x_1,x_2,\dots,x_n\}$ is a data set of $n$ observational units, we have the following:

Sample mean 
$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i$$ 

Sample variance 
$$Var(x_1, \dots, x_n) = s^2 =  \frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2$$ 

Sample standard deviation 
$$sd(x_1, \dots, x_n) = s = \sqrt{s^2}$$ 

If we want to standardize the data set $X$, to create a new standardized data set $Z = \{ z_1, z_2, \dots, z_n \}$ we preform
$$z_i = \frac{x_i-\bar{x}}{sd(x_1, \dots, x_n)}, \text{ for $i=1,\dots,n$}.$$  


## Simple linear regression notation 

The fitted/estimated  regression model is $\hat{y}_i = b_0 + b_1 x_i$ where $b_0 = \bar{y} - b_1 \bar{x}$ and $b_1 = \frac{s_{xy}}{\sqrt{s_x s_y}} \cdot \frac{s_y}{s_x}$. 


$$\text{Residual} = e = y - \hat{y} = \text{observed value} - \text{predicted value}$$

Standard error of the residuals: $s_e = \sqrt{\frac{\sum_{i=1}^{n} e_i^2 }{n-2}}$

### Sum of squares terms

$$s_x = \sum_{i=1}^{n}(x_i - \bar{x})^2, \quad s_y = \sum_{i=1}^{n}(y_i - \bar{y})^2, \quad s_{xy} = \sum_{i=1}^{n}(x_i - \bar{x})(y_i-\bar{y})$$

### Correlation coefficient 

$$r = \frac{s_{xy}}{\sqrt{s_x s_y}}$$



# Probability 

## Five Laws of Probability 

### 1) A probability is a number between 0 and 1. 

$$0 \leq Pr(A) \leq 1, \quad\text{for $A \in S$}$$ 

### 2) The probability of the set of all possible outcomes of a trial is 1. 

$$Pr(S)=1$$

### 3) The probability of an event not occuring is equal to 1 minus the probability the event does occur. 

$$Pr(A^{C}) = 1 - Pr(A)$$

### 4) For any events in the sample space of a random variable, say, $A$ and $B$, we compute the probability of event A or event B or both events A and B occurring with the  formula:

$$Pr(A \text{ or } B) = Pr(A) + Pr(B) - Pr(A \text{ and } B)$$

### 5) If an event $A$ is independent of another event $B$, then the probability that both events occur is the product of the probabilities of the two individual events:

$$Pr(A\text{ and } B) = Pr(A)\times Pr(B).$$ 


## Definition of conditional probability

$$Pr(B \mid A) = \frac{Pr(A \text{ and } B)}{Pr(A)}$$

## General multiplication rule 

For any random events $A$ and $B$ (that need not be independent),
$$Pr(A \text{ and }B) = Pr(A) \times Pr(B \mid A).$$ 

## Law of total probability 

$$Pr(B) = Pr(B\text{ and }A) + Pr(B \text{ and } A^C)$$ 

# Random Variables 

For a random variable $X$, 

$$E(X) = \sum_{x\in S} \left[ x \times Pr(x) \right],\quad   Var(X) = \sum_{x \in S} \left[(x-E(X))^2\times Pr(x)\right], \quad st.dev(X) = \sqrt{Var(X)}.$$

For two random variables, $X$ and $Y$:

$$Cov(X,Y) = E\left[(X-E(X))\cdot(Y-E(Y)) \right], \quad Cor(X,Y) = \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}.$$



## Linear transformations of a random Variable 

Suppose $a$ is some number between $-\infty$ and $+\infty$. The following are properties of expectation and variance for linear transformations of a random variable $X$. 

* $E(aX) = aE(X), \quad E(a \pm X) = a \pm E(X)$ 

* $Var(aX) = a^2 Var(X), \quad Var(a \pm X) = Var(X)$
  
  
## Linear transformations of two random variables 

Suppose both $X$ and $Y$ are random variables that may or may not be related to one another. The following are properties of expectation and variance for linear transformations involving both random variables.   
  
* $E(X \pm Y) = E(X) \pm E(Y)$

* $Var(X \pm Y) = Var(X) + Var(Y) \pm 2Cov(X,Y)$

* If $X$ and $Y$ are independent random variables, then $Cov(X,Y)=0$. 

## Normal Random Variable

If $X \sim N(\mu, \sigma^2)$ then $Z = \frac{X - \mu}{\sigma} \sim N(0, 1).$ 

## Binomial Random Variable 

If $X \sim Bin(n,p)$ then $Pr(X = x) = nCx \cdot p^x \cdot (1-p)^{n-x}$, where $nCx = \frac{n!}{x!(n-x)!}$.


# Sampling Distributions

Under appropriate conditions, the sampling distribution for the sample proportion is 
$$\hat{p} \sim N\left(p, \sqrt{\frac{p(1-p)}{n}} \right).$$ 
The standard error for the sample proportion is $SE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.$


Under appropriate conditions, the sampling distribution for the sample mean is
$$\bar{X} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}} \right).$$
The standard error for the sample mean is $SE(\bar{x}) = \frac{s}{\sqrt{n}}$.

# Confidence Intervals 

## For a single proportion 

$$\hat{p} \pm [z^*_{a} \times SE(\hat{p})]$$ 
where $z^*_{a}$ is the lower (or upper) $\left(\frac{1-a}{2}\right)^{th}$ quantile of a $N(0,1)$ distribution for confidence level $a$. 

## For a single mean 

$$\bar{x} \pm [t^*_{a,(n-1)} \times  SE(\bar{x})]$$ 
where $t^*_{a, (n-1)}$ is the lower (or upper) $\left(\frac{1-a}{2}\right)^{th}$ quantile of a t-distribution with $n-1$ degrees of freedom, for confidence level $a$. 

## For a difference in proportions 

$$(\hat{p}_1 - \hat{p}_2) \pm [z^*_{a} \times SE(\hat{p}_1 - \hat{p}_2)]$$ 
where $SE(\hat{p}_1 - \hat{p}_2) =\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$ and  $z^*_{a}$ is the lower (or upper) $\left(\frac{1-a}{2}\right)^{th}$ quantile of a $N(0,1)$ distribution for confidence level $a$. 

## For a difference in means 

### Independent samples 

$$(\bar{x}_1 - \bar{x}_2) \pm [t^*_{a, (\nu)} \times SE(\bar{x}_1 - \bar{x}_2)]$$ 
where $SE(\bar{x}_1 - \bar{x}_2) =\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$ and  $t^*_{a, (\nu)}$ is the lower (or upper) $\left(\frac{1-a}{2}\right)^{th}$ quantile of a t-distribution with $\nu$ degrees of freedom, for confidence level $a$. (These degrees of freedom will always be provided to you as they are complicated to derive.)

### Paired samples 

$$\bar{d} \pm [t^*_{a, (n-1)} \times SE(\bar{d})]$$ 
where $SE(\bar{d}) =\frac{s_{d}}{\sqrt{n}}$ and  $t^*_{a, (n-1)}$ is the lower (or upper) $\left(\frac{1-a}{2}\right)^{th}$ quantile of a t-distribution with $n-1$ degrees of freedom, for confidence level $a$. 

# Hypothesis Tests 

## For a single proportion 

We can test $H_0: p = p_0$ with the test statistic $T.S. = \frac{\hat{p} - p_0}{st.dev(\hat{p})}$, where $st.dev(\hat{p}) = \sqrt{\frac{p_0 (1-p_0)}{n}}$.

## For a single mean 

We can test $H_0: \mu = \mu_0$ with the test statistic $T.S. = \frac{\bar{x} - \mu_0}{SE(\bar{x})}$.

## For a difference in proportions 

We can test $H_0: p_1 - p_2 = 0$ with the test statistic $T.S. = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{SE(\hat{p}_1 - \hat{p}_2)}$.

## For a difference in means 

### Independent samples 

We can test $H_0: \mu_1 - \mu_2 = \Delta_0$ with the test statistic $T.S. = \frac{(\bar{x}_1 - \bar{x}_2) - \Delta_0}{SE(\bar{x}_1 - \bar{x}_2)}$.

### Paired samples 

We can test $H_0: \mu_d = \Delta_0$ with the test statistic $T.S. = \frac{\bar{d} - \Delta_0}{SE(\bar{d})}$.