Question

## Code in R ##

S17: Work on the data set survey.csv. Test Ho: the two attributes “Smoke” and “Sex” are independent vs H : the two attributes “Smoke" and "Sex” are dependent.

survey.csv

"Sex","Wr.Hnd","NW.Hnd","W.Hnd","Fold","Pulse","Clap","Exer","Smoke","Height","M.I","Age"
"Female",18.5,18,"Right","R on L",92,"Left","Some","Never",173,"Metric",18.25
"Male",19.5,20.5,"Left","R on L",104,"Left","None","Regul",177.8,"Imperial",17.583
"Male",18,13.3,"Right","L on R",87,"Neither","None","Occas",NA,NA,16.917
"Male",18.8,18.9,"Right","R on L",NA,"Neither","None","Never",160,"Metric",20.333
"Male",20,20,"Right","Neither",35,"Right","Some","Never",165,"Metric",23.667
"Female",18,17.7,"Right","L on R",64,"Right","Some","Never",172.72,"Imperial",21
"Male",17.7,17.7,"Right","L on R",83,"Right","Freq","Never",182.88,"Imperial",18.833
"Female",17,17.3,"Right","R on L",74,"Right","Freq","Never",157,"Metric",35.833
"Male",20,19.5,"Right","R on L",72,"Right","Some","Never",175,"Metric",19
"Male",18.5,18.5,"Right","R on L",90,"Right","Some","Never",167,"Metric",22.333
"Female",17,17.2,"Right","L on R",80,"Right","Freq","Never",156.2,"Imperial",28.5
"Male",21,21,"Right","R on L",68,"Left","Freq","Never",NA,NA,18.25
"Female",16,16,"Right","L on R",NA,"Right","Some","Never",155,"Metric",18.75
"Female",19.5,20.2,"Right","L on R",66,"Neither","Some","Never",155,"Metric",17.5
"Male",16,15.5,"Right","R on L",60,"Right","Some","Never",NA,NA,17.167
"Female",17.5,17,"Right","R on L",NA,"Right","Freq","Never",156,"Metric",17.167
"Female",18,18,"Right","L on R",89,"Neither","Freq","Never",157,"Metric",19.333
"Male",19.4,19.2,"Left","R on L",74,"Right","Some","Never",182.88,"Imperial",18.333
"Male",20.5,20.5,"Right","L on R",NA,"Left","Some","Never",190.5,"Imperial",19.75
"Male",21,20.9,"Right","R on L",78,"Right","Freq","Never",177,"Metric",17.917
"Male",21.5,22,"Right","R on L",72,"Left","Freq","Never",190.5,"Imperial",17.917
"Male",20.1,20.7,"Right","L on R",72,"Right","Freq","Never",180.34,"Imperial",18.167
"Male",18.5,18,"Right","L on R",64,"Right","Freq","Never",180.34,"Imperial",17.833
"Male",21.5,21.2,"Right","R on L",62,"Right","Some","Never",184,"Metric",18.25
"Female",17,17.5,"Right","R on L",64,"Left","Some","Never",NA,NA,19.167
"Male",18.5,18.5,"Right","Neither",90,"Neither","Some","Never",NA,NA,17.583
"Male",21,20.7,"Right","R on L",90,"Right","Some","Never",172.72,"Imperial",17.5
"Male",20.8,21.4,"Right","R on L",62,"Neither","Freq","Never",175.26,"Imperial",18.083
"Male",17.8,17.8,"Right","L on R",76,"Neither","Freq","Never",NA,NA,21.917
"Male",19.5,19.5,"Right","L on R",79,"Right","Some","Never",167,"Metric",19.25
"Female",18.5,18,"Right","R on L",76,"Right","None","Occas",NA,NA,41.583
"Male",18.8,18.2,"Right","L on R",78,"Right","Freq","Never",180,"Metric",17.5
"Female",17.1,17.5,"Right","R on L",72,"Right","Freq","Heavy",166.4,"Imperial",39.75
"Male",20.1,20,"Right","R on L",70,"Right","Some","Never",180,"Metric",17.167
"Male",18,19,"Right","L on R",54,"Neither","Some","Regul",NA,NA,17.75
"Male",22.2,21,"Right","L on R",66,"Right","Freq","Occas",190,"Metric",18
"Female",16,16.5,"Right","L on R",NA,"Right","Some","Never",168,"Metric",19
"Male",19.4,18.5,"Right","R on L",72,"Neither","Freq","Never",182.5,"Metric",17.917
"Male",22,22,"Right","R on L",80,"Right","Some","Never",185,"Metric",35.5
"Male",19,19,"Right","R on L",NA,"Neither","Freq","Occas",171,"Metric",19.917
"Female",17.5,16,"Right","L on R",NA,"Right","Some","Never",169,"Metric",17.5
"Female",17.8,18,"Right","R on L",72,"Right","Some","Never",154.94,"Imperial",17.083
"Male",NA,NA,"Right","R on L",60,NA,"Some","Never",172,"Metric",28.583
"Female",20.1,20.2,"Right","L on R",80,"Right","Some","Never",176.5,"Imperial",17.5

Answer 1

Two random variables x and y are called independent if the probability distribution of one variable is not affected by the presence of another.
Assume fij is the observed frequency count of events belonging to both i-th category of x and j-th category of y. Also assume eij to be the corresponding expected count if x and y are independent. The null hypothesis of the independence assumption is to be rejected if the p-value of the following Chi-squared test statistics is less than a given significance level α.
  
χ2 = ∑ (fij --eij)^2

----------------

i, j eij

I have tried to fit the chi square test of independence here in R
The codes and results are as follows
> head(data)
Sex Wr.Hnd NW.Hnd W.Hnd Fold Pulse Clap Exer Smoke Height M.I Age
1 Female 18.5 18.0 Right R on L 92 Left Some Never 173.00 Metric 18.250
2 Male 19.5 20.5 Left R on L 104 Left None Regul 177.80 Imperial 17.583
3 Male 18.0 13.3 Right L on R 87 Neither None Occas NA <NA> 16.917
4 Male 18.8 18.9 Right R on L NA Neither None Never 160.00 Metric 20.333
5 Male 20.0 20.0 Right Neither 35 Right Some Never 165.00 Metric 23.667
6 Female 18.0 17.7 Right L on R 64 Right Some Never 172.72 Imperial 21.000

> library(MASS)
> table = table(data$Sex,data$Smoke)
> table
  
Heavy Never Occas Regul
Female 1 13 1 0
Male 0 24 3 2
> chisq.test(table)

   Pearson's Chi-squared test

data: table
X-squared = 3.1329, df = 3, p-value = 0.3716

Warning message:
In chisq.test(table) : Chi-squared approximation may be incorrect
>

We test the Ho: Smoke and Sex variables are indepndent.
The results show that the p-value 0.3716 is greater than the .05 significance level, we do not reject the null hypothesis that the Smoke and Sex variables are independent of each other.