Lab6: Goodness of Fit

1 7 Exercise

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

treatment<- read.csv("treatment.csv")
table(treatment$treatment, treatment$improvement)

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

# Chi-square test
chisq.test(treatment$treatment, treatment$improvement, correct=FALSE)

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

The chi-squared value is 5.55. Since we get a p-Value less than the significance level of 0.05, we could reject the null hypothesis and conclude that the two variables are in fact dependent

7.2 Exercice 2 Find out if the cyl and carb variables in mtcars dataset are dependent or not.

mtcars_table<-table(mtcars$carb, mtcars$cyl)
print (mtcars_table)

##    
##     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 test
chisq.test(mtcars$carb, mtcars$cyl)

## Warning in chisq.test(mtcars$carb, mtcars$cyl): 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

The chi-squared value is 24.389 and we got a p-value is less than 0.05 significance level. So we reject the null hypothesis and we know that carb and cyl have a significant relationship.

7.3 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)

Zodiac<-readxl::read_xlsx("Zodiac.xlsx")
print(Zodiac)

## # A tibble: 12 x 3
##    Category    Observed Expected
##    <chr>          <dbl>    <dbl>
##  1 Aries             29     21.3
##  2 Taurus            24     21.3
##  3 Gemini            22     21.3
##  4 Cancer            19     21.3
##  5 Leo               21     21.3
##  6 Virgo             18     21.3
##  7 Libra             19     21.3
##  8 Scorpio           20     21.3
##  9 Sagittarius       23     21.3
## 10 Capricorn         18     21.3
## 11 Aquarius          20     21.3
## 12 Pisces            23     21.3

#chisq-test
Residual<-(Zodiac$Observed - Zodiac$Expected)
x<-(Residual)^2
Component<-(x/Zodiac$Expected)
print(Component)

##  [1] 2.755490976 0.333421882 0.020854498 0.255139408 0.005198003 0.520737308
##  [7] 0.255139408 0.083292973 0.130262457 0.520737308 0.083292973 0.130262457

chisq_test<-sum(Component)
chisq_test

## [1] 5.09383