With the advent of high-speed computing, new procedures have been developed that permit statistical inferences to be performed under less restrictive conditions than those of classical procedures. Permutation tests constitute one such collection of new procedures. To perform a permutation test to compare two population means using independent samples, proceed as follows.

1. Combine the two samples.

2. Randomly select \(n_{1}\) members from the combined sample. Now treat these \(n_{1}\) members as the first sample and the remaining \(n_{2}\) members as the second sample.

3. Compute the difference between the means of the two new samples.

4. Repeat steps \(2\) and \(3\) a large number (hundreds or thousands) of times.

5. The distribution of the resulting differences between sample means provides an estimate of the sampling distribution of the sample-mean differences when the null hypothesis of equal population means is true. This estimates is called a permutation distribution.

6. The (estimated) \(P-\)value of the hypothesis tests equal the proportion of values of the permutation distribution that are as extreme as or more extreme than the differences between the two observed sample means.

Refer to Example \(10.3\) on page \(409\). Use the technology of your choice to conduct a permutation distribution test and compare your results with those found by using the pooled \(t-\)test. Discuss any discrepancy that you encounter.

Calculate the two random samples in MATLAB

\(r_{1}=rand([85, 149], 99, 1)\)

\(r_{2}=rand([83, 142], 99, 1)\)

Then add both the samples

\(r=r_{1}+r_{2}\)

Program:

Query:

• Then add both the samples.

Calculate the two random samples in MATLAB

\(r_{1}=rand([85, 149], 99, 1)\)

\(r_{2}=rand([83, 142], 99, 1)\)

Then add both the samples

\(r=r_{1}+r_{2}\)

Then select another individual random sample from the combined one.

Program:

Query:

• First, we have defined the two different samples.
• Then add both the samples.

• Select the other two random sample from the combined sample.

Calculate the two random samples in MATLAB

\(r_{1}=rand([85, 149], 99, 1)\)

\(r_{2}=rand([83, 142], 99, 1)\)

Then add both the samples

\(r=r_{1}+r_{2}\)

Then select another individual random sample from the combined one.

Calculate the mean of both samples

Then the difference of both the mean will be \(m=-5.6667\)

Program:

Query:

• First, we have defined the two different samples.
• Then add both the samples.
• Select the other two random sample from the combined sample.
• Calculate the mean of both samples.
• Calculate the difference of mean.

Calculate the two random samples in MATLAB

\(r_{1}=rand([85, 149], 99, 1)\)

\(r_{2}=rand([83, 142], 99, 1)\)

Then add both the samples

\(r=r_{1}+r_{2}\)

Then select another individual random sample from the combined one.

Let’s take \(100\) samples.

Calculate the mean of both samples

Then the difference of both the mean will be

\(m=1.7525\)

Program:

Query:

• First, we have defined the two different samples.
• Then add both the samples.
• Select the other two random sample from the combined sample.
• Calculate the mean of both samples.

• Calculate the difference of mean.