formula_sheet_Q1.Rmd

---
title: "Formula Sheet for Quiz 1"
subtitle: "STAT 011"
output: pdf_document
---


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

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


## 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 

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

### Coefficient of determination/R-squared

$$R = \left(\frac{s_{xy}}{\sqrt{s_{x}s_{y}}} \right)^2$$