# In R, Part 1. Learn to underst

In R,

Part 1. Learn to understand the significance level α inhypothesis testing.

a) Generate a matrix “ss” with 1000 rows and 10 columns. Theelements of “ss” are random samples from standard normaldistribution.

b) Run the following lines:

mytest <- function(x) {

return(t.test(x,mu=0)\$p.value)

}

mytest(rnorm(100))

Note that, when you input a vector in the function mytest, youwill get the p-value for the one sample t-test H0 : µ = 0 vs Ha : µ=/= 0.

c) Conduct one sample t-test H0 : µ = 0 vs Ha : µ =/= 0 for eachrow of “ss”. (Hint: use either functions apply() or for() and afunction mytest())

d) For the 1000 tests you conducted in c), what is the ratio ofrejection if the significance level  α = 0.05? How aboutα = 0.1 or 0.01?

Part 2. Let’s start from R built-in dataset “sleep” (You mayaccess it by running sleep in R orrunning data(sleep) in R).

a) Load dataset sleep and open the descriptionfile.

b) Draw histogram for column “extra”. Comment on the shape ofthe histogram.

c) Define two vectors “x” and “y” as following:

x<-sleep\$extra[1:10]

y<-sleep\$extra[11:20]

d) Conduct two-sample t-test to check whether the means of x andy are significantly different. Make sure to state your hypothesis,test statistic, p-value, decision rule and conclusion for thetest

e) Define a vector “z” as following:

z<-x-y

f) Conduct one-sample t-test to check whether the mean of z issignificantly different from 0.

As required the R code is provided below,___________________________________________________________________________

##Part 1a : Create ‘ss’ matrix

ss=matrix(0,nrow=1000,ncol = 10)for(j in 1:1000){r = rnorm(10)for(i in 1:10){    ss[i,j]=r[i]}}print(ss)

##Part 1b : Run the followinglines:

mytest <- function(x) {return(t.test(x,mu=0)\$p.value)}

mytest(rnorm(100))

##Part 1c : Conduct one sample t-test H0 : µ = 0 vs Ha :µ =/= 0 for each row of “ss”

pval=array(dim=1)                      ## array to store all 1000 p-valuesfor(i in 1:1000){pval[i]=mytest(ss[i,])}print(pval)

##Part 1d : what is the ratio of rejection if thesignificance level  α = 0.05? How about α = 0.1 or0.01?

ratio5 = length(pval[which(pval<0.05)])/1000ratio1 = length(pval[which(pval<0.1)])/1000ratio01 = length(pval[which(pval<0.01)])/1000

print(ratio5)print(ratio1)print(ratio01)___________________________________________________________________________

##Part 2a : Load dataset sleep

data(sleep)d=sleep##Part 2b : Draw histogram for column “extra” andcomment

hist(d\$extra)        ## Thisgives the required histogram##Conclusion : The dataseems somewhat symmetric about 2 with a bell-like curve as depicted##above

##Part 2c : Define two vectors “x” and“y”

x<-sleep\$extra[1:10]y<-sleep\$extra[11:20]

##Part 2d : Conduct 2 sample t-test for x andy.

t.test(x,y)

##Part 2e : Define vector z

z<-x-y

##Part 2f : Conduct 1-sample t-test on z if mean(z) issignificantly different from 0.

t.test(z)

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