Question

In R,

Part 1. Learn to understand the significance level α in hypothesis testing.

a) Generate a matrix “ss” with 1000 rows and 10 columns. The elements of “ss” are random samples from standard normal distribution.

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, you will 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 each row of “ss”. (Hint: use either functions apply() or for() and a function mytest())

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

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

a) Load dataset sleep and open the description file.  

b) Draw histogram for column “extra”. Comment on the shape of the 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 and y are significantly different. Make sure to state your hypothesis, test statistic, p-value, decision rule and conclusion for the test  -

e) Define a vector “z” as following:

z<-x-y

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

Answer 1

Hi There,

Please find the solution.

a)

ss <- matrix(c(10:100), nrow = 1000,ncol= 10, byrow = FALSE)
print(ss)
OUTPUT

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [1,] 10 100 99 98 97 96 95 94 93 92 [2,] 11 10 100 99 98 97 96 95 94 93 [3,] 12 11 10 100 99 98 97 96 95 94 [4,] 13 12 11 10 100 99 98 97 96 95 [5,] 14 13 12 11 10 100 99 98 97 96 [6,] 15 14 13 12 11 10 100 99 98 97 [7,] 16 15 14 13 12 11 10 100 99 98 [8,] 17 16 15 14 13 12 11 10 100 99 [9,] 18 17 16 15 14 13 12 11 10 100 [10,] 19 18 17 16 15 14 13 12 11 10 [11,] 20 19 18 17 16 15 14 13 12 11 [12,] 21 20 19 18 17 16 15 14 13 12 [13,] 22 21 20 19 18 17 16 15 14 13 [14,] 23 22 21 20 19 18 17 16 15 14 [15,] 24 23 22 21 20 19 18 17 16 15 [16,] 25 24 23 22 21 20 19 18 17 16 [17,] 26 25 24 23 22 21 20 19 18 17....

b)

mytest <- function(x) {

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

}

mytest(rnorm(100))

OUTPUT

[1] 0.4112541

c)

apply(ss,mytest)

OUTPUT

[1] 0.5635022

d)

rejection.level(0.02)

rejection.level(0.1)

OUTPUT

[1] 0.08967416

[1] 0.385624

PART2

a)

sleep<- read.csv("sleep.csv")

description(sleep)

b)

hist(sleep,ylab = "Extra",col = "yellow",border = "blue")

c)

x<-sleep$extra[1:10]

y<-sleep$extra[11:20]

e) z<-x-y

f) t.test(y~x)

Thanks

Feel free to write back in case of doubt.