Lab6: Goodness of Fit

library(ggplot2)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(tidyr)
library(ggthemes)

1 Exercise

1.1 Exercise 1

Please work out in R by doing a chi-squared test on the treatment (X) and improvement (Y) columns in treatment.csv.

data <- read.csv("treatment.csv")

table(data$treatment, data$improvement)

##              
##               improved not-improved
##   not-treated       26           29
##   treated           35           15

#chi-sq test
chisq.test(data$treatment, data$improvement, correct=FALSE)

## 
##  Pearson's Chi-squared test
## 
## data:  data$treatment and data$improvement
## X-squared = 5.5569, df = 1, p-value = 0.01841

We now have a chi-squared value of 5.5569 and we get a p-Value less than the significance level of 0.05 which is 0.01841.

1.2 Exercise 2

Find out if the cyl and carb variables in mtcars dataset are dependent or not.

data(mtcars)
table(mtcars$carb, mtcars$cyl)

##    
##     4 6 8
##   1 5 2 0
##   2 6 0 4
##   3 0 0 3
##   4 0 4 6
##   6 0 1 0
##   8 0 0 1

#chi-sq tes
chisq.test(mtcars$carb, mtcars$cyl, correct = FALSE)

## Warning in chisq.test(mtcars$carb, mtcars$cyl, correct = FALSE): Chi-squared
## approximation may be incorrect

## 
##  Pearson's Chi-squared test
## 
## data:  mtcars$carb and mtcars$cyl
## X-squared = 24.389, df = 10, p-value = 0.006632

We now have a high chi-squared value and a p-value of less that 0.05 significance level. So we reject the null hypothesis and conclude that carb and cyl have a significant relationship which means they are independent.

1.3 Exercise 3

256 visual artists were surveyed to find out their zodiac sign. The results were: Aries (29), Taurus (24), Gemini (22), Cancer (19), Leo (21), Virgo (18), Libra (19), Scorpio (20), Sagittarius (23), Capricorn (18), Aquarius (20), Pisces (23). Test the hypothesis that zodiac signs are evenly distributed across visual artists. (Reference)

# Here we input/make the data as a data frame
Births <-c(29,24,22,19,21,18,19,20,23,18,20,23)

# The Simple Method 
chisq.test(Births)

## 
##  Chi-squared test for given probabilities
## 
## data:  Births
## X-squared = 5.0938, df = 11, p-value = 0.9265

# Manual Method
n<-256
expected <- c(1/12) * n
alpha <- .05
r <- c(1 , 2 , 3, 4 , 5 , 6 , 7, 8 , 9 , 10 , 11 , 12)
df <- 12 - 1
(chisq <- sum((Births - expected)^2 / expected))

## [1] 5.09375

(p_value <- pchisq(q = chisq, df = df, lower.tail = F))

## [1] 0.9265414

Both methods shows that now we have low chi-square value and a p-value more than 0.05 significance level, so we accept the null hypothesis which means all the zodiac signs distributed evenly across visual artists.