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# 2.12: Non-Parametric Statistics

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## Introduction to Nonparametric Statistics

Extension: Statistics in Literature

As the course comes to a close, it is nice to have a fun assignment that is a change of pace from what the students have been doing in class so far. The novel Bringing Down the House is an exciting story where what students have learned in this course figure prominently in the plot. It also makes statistics, math, mathematicians, and being smart look cool. This assignment will also satisfy any cross-curricular requirement of your school. Students can get credit for the essay in both their statistics and their English classes. The statistics teacher can grade the essay for content, and the English teacher can grade the same essay for the quality of the writing. Work with the English teacher(s) beforehand to come up with an assignment everybody likes.

Assignment: Read Bringing Down the House, by Ben Mezrich.

Write a $750$ to $1000$ word essay addressing one of the following topics.

• Is card counting cheating according to the members of the team, to the casinos, to the law?
• How do the players use math and stereotypes to count cards?
• How does card counting affect the lives of the members of the team? Is it an addiction?
• How does competition/greed affect the teams?
• If you can think of another good topic to address, go for it.

Include lots of detail and examples from throughout the book. Make sure I can tell you read the entire book carefully. Put page numbers in parenthesis at the end of quotes.

## The Rank Sum Test and Rank Correlation

Project: Analyzing Nominal and Ordinal Data

In the first chapter of this texts students learned about the levels of measurement. Since then, the focus of their studies has been on data measured at the interval or ratio level. This last chapter returns to the lower levels of measurement, nominal and ordinal. Before students start working with the nonparametric tests in this chapter, it would be beneficial for them to review the levels of measurement found in the second section of Chapter One of this text.

Objective: To perform hypothesis tests on data measured at the nominal and ordinal levels that is collected by the students.

Procedure:

Part One: Data Measured at the Nominal Level

1. Take a random sample and divide the results into two categories.

2. Use a sign test at the $0.05$ significance level to determine if one categorical variable is really “more” than the other.

• State the null and alternative hypotheses.
• Determine the critical value. Will you use the normal or $t-$distribution chart?
• Calculate the test statistic.
• Determine and interpret the results.

Part Two: Data Measured at the Ordinal Level

3. Collect two sets of data that you wish compared and rank the results.

4. Perform a rank sum test. Include all the hypothesis testing steps as above.

5. Interpret the results.

## The Kruskal-Wallis Test and the Runs Test

Explore: Assessing Variance with the Kruskal-Wallis Test

Continued from the Enrichment Activity for the One-Way ANOVA Test

In the enrichment activity for the one-way ANOVA test, students used the F-test and the one-way ANOVA test to asses the variances of two data sets. They did each test twice, once with normally distributed data and once with approximately normally distributed data. In this activity, students will repeat the process with data that is not normally distributed at all, and also perform the Kruskal-Wallis test on all three data sets. By comparing the results of the tests, the students will be able to see the importance of considering the distribution of the data when deciding which test should be used in a particular situation.

Objective: To asses the variance of samples taken from three different populations with different distributions in three different ways in order to compare the results in terms of the robustness of the tests.

Procedure:

1. Perform the Kruskal-Wallis test on the normally distributed data and the approximately normally distributed data collect in the enrichment activity for the one-way ANOVA test.
2. Take two samples from a population that you know does not have a normal distribution.
3. Perform the $F-$test on the data collected in step two.
4. Perform the one-way ANOVA test on the data collected in step two.
5. Perform the Kruskal-Wallis test on the data collected in step two.
6. Compare the results of the three tests performed in the previous steps.
7. Of all the tests done in both enrichment activities, which are giving dependable results and which are not because the test criteria have not been met?

Feb 23, 2012

Aug 19, 2014