Question

2. Write a simulation in R that shows the distribution of the t-test statistic when the null hypothesis is true. To do this, you should use a for loop that repeatedly performs t-tests comparing sample means of data that come from distributions with the same population mean and standard deviation. Use rnorm() to take samples, t.test() to perform the t-tests, and use “$statistic” to extract the t-test

statistic from the t.test() procedure (e.g. t.test(x,y)$statistic). Make a histogram of the test statistics. If you need help, look back at the notes on for loops.

3. One assumption of the t-test is that the populations you sample from have the same standard deviation. Violating this assumption can affect the distribution of the t-test statistic. This is especially the case when sample sizes are unequal.

   1. Re-do the simulation from 2, but this time sample from normal distributions with the same mean but where one has a standard deviation of 1 and a sample size of 20, and the other has a standard deviation of 5 and a sample size of 100. Plot a histogram of the test statistics. How does this differ from the histogram in part 2?

   2. Perform the procedure in part a. above, but this time use the “pooled variance” t-test. To do this, add “var.equal=TRUE” as an argument in the t.test function. Plot a histogram of the test statistics. How does this differ from the histogram in part a. above
      
      
      using Rstudio

Answer 1

We need to write a simulation in R that shows the distribution of the t-test statistic when the null hypothesis is true.

Let x1,x2,...xn1 be n1 random samples from variable X with mean 1 and and y1,y2,...yn2 be n2 random samples from variable Y with mean 2

Then null hypothesis is

H0 : 1 = 2

We reject null hypothesis if Test statistics value is greater than t-table value

Where Test statistics :   TS =

And t-table value is

If    |TS| > , we reject null hypothesis .

Now we dose not need to do it manually , we will use R-software only to calculate Test statistics value which we need to plot .

We need to simulate when the null hypothesis is true. Thus mean of two samples need to be same

Thus we will take 2 diferent samples from normal distribution of different size but with same population mean and standard deviation .

R - Code and output

{

TS=1.0                  # define variable TS
for(i in 1:100)          # to simulate 100 times
{
x=rnorm(50,5,2)            # First sample of size 50 , mean = 5 ,sd =2
y=rnorm(55,5,2)            # Second sample of size 55 , mean = 5 ,sd =2
TS[i]=t.test(x,y)$statistic        # to store test statistics values
}
hist(TS,xlab="Test Statistic",col=2)        # to plot histogram of test statistics .

}

Now we need to simulate , but this time sample from normal distributions with the same mean but where one has a standard deviation of 1 and a sample size of 20, and the other has a standard deviation of 5 and a sample size of 100 .

Let both have same mean equal to 5 .

R - Code and output

{

TS=1.0                  # define variable TS
for(i in 1:100)          # to simulate 100 times
{
x=rnorm(20,5,1)               # First sample of size 20, mean =5 , sd =1
y=rnorm(100,5,5)             # Second sample of size 100 , mean =5 , sd =5
TS[i]=t.test(x,y)$statistic        # to store statistics values
}
hist(TS,xlab="Test Statistic",col=2)        # to plot histogram of test statistics .

We can this differ from the histogram in part 2 ,as we obser more extreme values of test statistics will will result into rejection of null hypothesis . Thus in these part we we have same ,mean but different sample size and standard deviation , we will observe more rejection of null hypothesis compared to that of lst part i.e t-test will falsly concluded that mean of two sample is not same .

Now we need to perform the procedure in part a. above, but this time use the “pooled variance” t-test. To do this, we need to add “var.equal=TRUE” as an argument in the t.test function .

These time we need to asuume that varation are equal.

We will use same previous random samples from rnorm(20,5,1) & rnorm(100,5,5)

R - code and output

{

TS=1.0
for(i in 1:100)
{
x=rnorm(20,5,1)
y=rnorm(100,5,5)
TS[i]=t.test(x,y,var.equal=TRUE)$statistic     # t-test with “var.equal=TRUE”
}
hist(TS,xlab="Test Statistic (with var.equal=TRUE)",col=3)

}

In all case null hypothesis is not rejected as all Test Statistic values are very smaller compared to that of last part when we have not consider “var.equal=TRUE” .

For these case we can see range of all Test Statistic is from -1.5 to 1.5 which is far smaller.

After using “var.equal=TRUE” t-test statistic values shows the null hypothesis to be true.