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Chapter 10: Q. 10.66 (page 418)

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.

Short Answer

Expert verified

Part a. The combination of two random sample is

Part b.

Part c. \(m=-5.6667\)

Part d. \(m=1.7525\)

Step by step solution

01

Part a. Step 1. Calculation

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:

  • First, we have defined the two different samples.
  • Then add both the samples.
02

Part b. Step 1. Calculation

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.
03

Part c. Step 1. Calculation

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.
04

Part d. Step 1. Calculation

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鈥檚 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.

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Most popular questions from this chapter

In the paper "The Relation of Sex and Sense of Direction to Spatial Orientation in an Unfamiliar Environment" (Journal of Environmental Psychology, Vol. 20, pp. 17-28), J. Sholl et al. published the results of examining the sense of direction of 30 male and 30 female students. After being taken to an unfamiliar wooded park, the students were given some spatial orientation tests, including pointing to the south, which tested their absolute frame of reference. The students pointed by moving a pointer attached to a 360掳protractor. Following are the absolute pointing errors, in degrees, of the participants.

At the 1% significance level, do the data provide sufficient evidence to conclude that, on average, males have a better sense of direction and, in particular, a better frame of reference than females? (Note: x炉1=37.6,s1=38.5,x炉2=55.8,ands2=48.3.)


In each of Exercises 10.75-10.80, we have provided summary statistics for independent simple random samples from non populations. In each case, use the non pooled t-test and the non pooled t-interval procedure to conduct the required hypothesis test and obtain the specified confidence interval.

x炉1=15,s1=2,n1=15,x炉2=12,s2=5andn2=15.

a. Two-tailed test, 伪=0.05

b. 95%confidence interval.

In this section, we introduced the pooled t-test, which provides a method for comparing two population means. In deriving the pooled f-test, we stated that the variable

z=f^1-x^2-渭1-渭2蟽1/n1+1/n2

cannot be used as a basis for the required test statistic because 蟽 is unknown. Why can't that variable be used as a basis for the required test statistic?

You know that the population standard deviations are not equal.

In Exercise 10.83, you conducted a nonpooled t-test to decide whether the mean number of acute postoperative days spent in the hospital is smaller with the dynamic system than with the static system. Use the technology of your choice to perform the following tasks.

a. Using a pooledt-test, repeat that hypothesis test.

b. Compare your results from the pooled and nonpooled t-tests.

c. Which test do you think is more appropriate, the pooled or nonpooled t-test? Explain your answer.
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