Nonparametric Test Chi-Square Test for Independence The test is used to determine whether two categorical variables are independent. Notation for the Chi-Square Test for Independence (Please note that the notation varies depending on the text) O represents the observed frequency of an outcome E represents the expected frequency of an outcome r represents the number of rows in the contingency table c represents the number of columns in the contingency table n represents the total number of trials Test Statistic      2 2 1 1 O E E df r c        The Chi-Square test is a hypothesis test. There are seven steps for a hypothesis test. 1. State the null hypothesis 2. State the alternative hypothesis 3. State the level of significance 4. State the test statistic 5. Calculate 6. Statistical Conclusion 7. Experimental Conclusion Example A university is interested to know if the choice of major has a relationship to gender. A random sample of 200 incoming freshmen students was taken (100 male and 100 female). There major and gender were recorded. The results are shown in the contingency table below. Major Female Male Math 5 15 Nursing 44 10 English 10 10 Pre-Med 17 20 History 4 5 Education 15 20 Undecided 5 20 To determine if there is a relationship between the gender of a freshmen student and thei declared major perform the hypothesis test (Use level of significance 0.05  ) . Step 1: Null Hypothesis 0 : Gender and Major of Freshmen students are independentH Step 2: Alternative Hypothesis : Gender and Major of Freshmen students are not independentAH Step 3: Level of Significance 0.05  Step 4: Test Statistic      2 2 1 1 O E E df r c        Step 5: Calculations There are several calculations for this test. We have to find the expected frequency for each cell in the contingency table. The expected frequency is the probability under the null hypothesis times the total frequency for the given row. Here the probability under the null hypothesis is .5, as the probability of being male and female is equal. rE pn Major Female Male Math   1 .5 20 10E     2 .5 20 10E   Nursing   3 .5 54 27E     4 .5 54 27E   English   5 .5 20 10E     6 .5 20 10E   Pre-Med   7 .5 37 18.5E     8 .5 37 18.5E   History   9 .5 9 4.5E     10 .5 9 4.5E   Education   11 .5 35 17.5E     12 .5 35 17.5E   Undecided   13 .5 25 12.5E     14 .5 25 12.5E   Know calculate the test statistic.                               2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 5 10 15 10 44 27 10 27 10 10 27 27 10 10 10 10 17 18.5 20 18.5 10 10 18.5 18.5 4 4.5 5 4.5 15 17.5 20 17.5 4.5 4.5 17.5 17.5 5 12.5 20 12.5 12.5 12.5 2.5 2.5 10.7 10.7 0 0 .1216 obs ob.

1. Nonparametric Test
Chi-Square Test for Independence
The test is used to determine whether two categorical variables
are independent.
Notation for the Chi-Square Test for Independence (Please note
that the notation varies
depending on the text)
O represents the observed frequency of an outcome
E represents the expected frequency of an outcome
r represents the number of rows in the contingency table
c represents the number of columns in the contingency table
n represents the total number of trials
Test Statistic
2
2

2. 1 1
O E
E
df r c
The Chi-Square test is a hypothesis test. There are seven steps
for a hypothesis test.
1. State the null hypothesis
2. State the alternative hypothesis
3. State the level of significance
4. State the test statistic
5. Calculate
6. Statistical Conclusion
7. Experimental Conclusion
Example

3. A university is interested to know if the choice of major has a
relationship to gender. A
random sample of 200 incoming freshmen students was taken
(100 male and 100
female). There major and gender were recorded. The results are
shown in the
contingency table below.
Major Female Male
Math 5 15
Nursing 44 10
English 10 10
Pre-Med 17 20
History 4 5
Education 15 20
Undecided 5 20
To determine if there is a relationship between the gender of a
freshmen student and
thei declared major perform the hypothesis test (Use level of

4. Step 1: Null Hypothesis
0 : Gender and Major of Freshmen students are independentH
Step 2: Alternative Hypothesis
: Gender and Major of Freshmen students are not
independentAH
Step 3: Level of Significance
Step 4: Test Statistic
2
2
1 1
O E
E
df r c

5. Step 5: Calculations
There are several calculations for this test. We have to find the
expected frequency for
each cell in the contingency table. The expected frequency is
the probability under the
null hypothesis times the total frequency for the given row.
Here the probability under
the null hypothesis is .5, as the probability of being male and
female is equal.
Major Female Male
Pre-

6. Know calculate the test statistic.
2
2
2 2 2 2
2
2 2 2 2
2 2 2 2
2 2

7. 2
5 10 15 10 44 27 10 27
10 10 27 27
10 10 10 10 17 18.5 20 18.5
10 10 18.5 18.5
4 4.5 5 4.5 15 17.5 20 17.5
4.5 4.5 17.5 17.5
5 12.5 20 12.5
12.5 12.5
2.5 2.5 10.7 10.7 0 0 .1216
obs
obs
obs
O E
E

8. 2
.1216 .0556 .0556 .357 .357 4.5 4.5
The calculation for degrees of freedom is as follows:

9. 1 1 7 1 2 1
6 1 6
df r c
df
The critical value for the Chi-Square with 6 degrees of freedom
at a level of significance
Step 6: Statistical Conclusion
Since 2 2
then reject the
null hypothesis.
Step 7: Experimental Conclusion
There is sufficient evidence to indicate that gender has an effect
on choice of major for
the incoming freshmen.

10. Mann-Whitney
The Mann-Whitney test is a nonparametric version of the
independent sample t-test.
This study is used when there are two independent samples of
ordinal scores.
Test Statistic
1 1
1 1 2 1
2 2
2 1 2 2
1
2
1
2
n n
U n n R
n n

11. U n n R
Notation:
1
2
1
2
number of scores in group 1
n number of score in group 2
R sum of ranks for score in group 1
R sum of ranks for score in group 2

12. The Mann-Whitney test is a hypothesis test. There are seven
steps for a hypothesis
test.
1. State the null hypothesis
2. State the alternative hypothesis
3. State the level of significance
4. State the test statistic
5. Calculate
6. Statistical Conclusion
7. Experimental Conclusion
Example:
Suppose there was race between an Antelope and a mountain
lion. The Antelope won
but you want to know if this could be extended to a general
statement that Antelope will
win in a race against a mountain lion. A random sample of 10
Antelopes and 10
mountain lions are put in a race. The order in which they finish
is recorded. ( A for
antelope and L for mountain lion)

13. AALALLLAAALAALLAALLL
Step 1: Null Hypothesis
0 : Probability of the Antelope winning is no different than the
probalility of the
mountain lion wining the race.
H
Step 2: Alternative Hypothesis
: Probability of the Antelope winning is different than the
probalility of the
mountain lion wining the race.
AH
Step 3: Level of Significance
Step 4: Test Statistic

14. 1 1
1 1 2 1
2 2
2 1 2 2
1
2
1
2
n n
U n n R
n n
U n n R
Step 5: Calculations
The results are ranked according to the in order in which they
finished the race.
Rank

15. A 1
A 2
L 3
A 4
L 5
L 6
L 7
A 8
A 9
A 10
L 11
A 12
A 13
L 14
L 15
A 16
A 17
L 18

16. L 19
L 20
Let the Antelope be sample one and the mountain lion be
sample 2.
1
2
1
2
number of scores for the Antelopes
n number of score for the mountain lions
R sum of ranks for score for the Antelopes
R sum of ranks for score for the mountain lions

17. Find the sum of the ranks for each group.
Antelope
Mountain
Lion
1 3
2 5
4 6
8 7
9 11
10 14
12 15
13 18
16 19
17 20
Sum 92 118
The number of for each animal is 10 because 10 antelopes and
10 mountain lions
raced.
Now plug the information into the test statistic

18. 1 1
1 1 2 1
1
1
1
2 2
2 1 2 2
2
2
2
1
2

19. 10 10 1
10 10 92
2
100 55 92
63
1
2
10 10 1
10 10 118
2
100 55 118
37
n n
U n n R
U
U
U
n n
U n n R
U

20. U
U
Find the critical value for the test statistic. Use the appropriate
U table found in the
appendix of the textbook. Look for alpha to be 0.05 and then the
number of scores for
each category to be 10. Then you see the following critical
values.

21. 1,
2,
23
77
crit
crit
U
U
Step 6: Statistical Conclusion
The statistical conclusion tells whether to reject or fail to reject
the null hypothesis. The
rejection rule is as follow.
1
2
If then reject the null hypothesis

22. If then reject the null hypothesis
crit
crit
U U
U U
he null
then fail to reject the null hypothesis.
Step 7: Experimental Conclusion
Since we failed to reject the null hypothesis we can say that
there is not significant
evidence to support the claim that the probability of the
antelope winning is any different
than the probability of the mountain lion winning the race.
Kruskal-Wallis Test

23. The Kruskal-Wallis Test is a nonparametric version of the one-
way ANOVA. This test
however does not assume normality or homogeneity of variance.
The test requires
ordinal scaling of the dependent variable and must have at least
5 data values in each
sample.
Notation
k number of sample or groups
n number of data values in each group
N number of data values in all samples combined
R sum of the ranks for each sample
Test Statistic
2
2 12
3 1
1

24. 1
R
N
N N n
df k
The Kruskal-Wallis test is a hypothesis test. There are seven
steps for a hypothesis
test.
1. State the null hypothesis
2. State the alternative hypothesis
3. State the level of significance
4. State the test statistic
5. Calculate
6. Statistical Conclusion

25. 7. Experimental Conclusion
Example
A gym has decided to recommend a diet to their clients. They
have narrowed down the
choice to three and they want to test which diet is best. They
have randomly selected 21
volunteers from the gym to participate in the diet. All the
participants have similar health
and body type as well as general exercise routine. The amount
of weight loss was
recorded.
Diet A Diet B Diet C
10 11 1
12 4 8
13 2 17
7 6 0
3 16 16
15 9 18
5 14 19

26. Step 1: Null Hypothesis
0 : There is no difference amoung the dietsH
Step 2: Alternative Hypothesis
: There is a difference between the dietsAH
Step 3: Level of Significance
Step 4: Test Statistic
2
2 12
3 1
1
1
R
N
N N n
df k

27. Step 5: Calculations
To perform the calculation, start by ranking the results.
Diet A Rank Diet B Rank Diet C Rank
10 11 11 12 1 2
12 13 4 5 8 9
13 14 2 3 17 19
7 8 6 7 0 1
3 4 16 17 16 17
15 16 9 10 18 20
5 6 14 15 19 21
Notice that some data values have the same ranks. This is how
you account for data

28. values that are the same.
k = 3
n = 7
N = 21
Plug these values into the test statistic
2
2
2 2 2
2

29. 2
2
2
2
12
3 1
1
72 69 8912
3 21 1
21 21 1 7 7 7
12
740.571 680.143 1131.571 66
462
.025974 2552.285 66
66.293 66
0.2931
1 3 1 2
obs
obs

30. obs
obs
obs
obs
R
N
N N n
df k

31. The critical value can be found using the Chi-Square table for
freedom.
2
Step 6: Statistical Conclusion
Since 2 2
0.05, 20.2931 5.9
reject the null hypothesis.
Step 7: Experimental Conclusion
The evidence does not support the claim that the there is a
significant difference

32. between the three diets at a level of significance of 0.05.
1
Walton-Fisette/Wuest: Foundations of Physical Education,
Exercise, Science, And Sport, 20e
Chapter 15
Future Professionals as Leaders and Advocates
Chapter Overview
Professionals in physical education, exercise science, and sport
need to be leaders in their disciplines and advocates for change.
Professionals are in a unique position to take an active role in
responding to society’s needs and engaging its members in
physical activity so that they can realize the powerful health
benefits. Professionals have the responsibility to educate the
public about the benefits of physical activity and fitness, and
they must make a greater effort to disseminate a national
message about the relationship between physical activity and
health. Collaborative, committed leadership is essential in
reducing physical inactivity and promoting a physically active
lifestyle.
Advocacy is an important responsibility of physical education,
exercise science, and sport professionals. Advocates need a
strong voice to clearly articulate the benefits of participation in
quality physical education, exercise science, and sport
programs. Professionals also need to work as agents for social
change in reducing the health disparities and inequities
experienced by individuals based on their gender, race,
ethnicity, (dis)ability, sexual orientation, and socioeconomic

33. status within the physical education, exercise science, and
sporting communities.
Learning Objectives
After reading this chapter, students should be able to
accomplish the following:
1. Discuss the importance of leadership relative to lifespan
participation in physical activity, fitness, physical education,
and sport.
2. Describe how professionals can be leaders of physical
activity, fitness, physical education, and sport.
3. Explain the importance of advocacy in a variety of physical
activity settings.
4. Describe how professionals can be advocates of physical
activity, fitness, physical education, and sport.
5. Identify current and future trends that professionals will
encounter in relation to physical activity, fitness, physical
education, and sport.
Chapter Outline
I. Leadership in Physical Activity
A. Leadership in Physical Education and Youth Sport
B. Advocacy
II. Current and Future Trends
Discussion Questions
1. How can professionals demonstrate leadership in promoting
lifespan participation in physical activity and fitness?
2. How can physical educators, exercise scientists, and sport
leaders advocate for their profession at the local, state, and
national levels?

34. 3. What various strategies can be utilized to promote lifespan
involvement of people of all ages, abilities, races and
ethnicities, genders, and socioeconomic backgrounds?
Self-Assessment Activities
These activities are designed to help students determine if they
have mastered the materials and competencies presented in this
chapter.
1. Discuss the role of the physical educator, exercise scientist,
and sport leader in the physical activity movement. Formulate a
list of strategies and next steps on what individuals can do to
“lead the parade.”
2. Discuss the importance of advocacy programs in the physical
education, exercise science, and sport setting of your choice.
What local organizations exist in your current community that
offer health promotion programs? How would you work to
organize these groups into a coalition to promote physical
activity?
3. Develop a plan for the upcoming 2020s that provides specific
suggestions for improving the disciplines of physical education,
exercise science, and sport.
Copyright ©2021 McGraw-Hill Education. All rights reserved.
No reproduction or distribution without the prior written
consent of McGraw-Hill Education
1
Walton-Fisette/Wuest: Foundations of Physical Education,
Exercise, Science, And Sport, 20e
Chapter 14
Sport Careers

35. Chapter Overview
Sport has developed into a big business, creating a myriad of
career opportunities for qualified individuals. Individuals
interested in sport management may pursue careers as athletic
directors, directors of intramurals and campus recreation,
directors of corporate recreation, and sport facilities managers.
Sports sales and sports marketing may be attractive careers to
some individuals. Managerial opportunities are also available in
professional organizations.
The intensity of interest in sport in the American society,
coupled with the growth of the communication media, has
resulted in the expansion of career opportunities in the field of
sport media. Individuals interested in this area can pursue
careers in sport broadcasting, sportswriting, sport journalism,
sport photography, sports information, and web
development/social media.
Talented individuals may elect to pursue careers as performers.
Other sport-related careers that may be attractive to qualified
individuals are sport officiating and sport law. Sport analytics is
a growing career. Some professionals choose to use their skills
to develop their own businesses.
Individuals interested in these careers can use many strategies
to enhance their professional marketability. Taking course work
in supporting areas and gaining practical experience help
individuals attain the position that they desire after graduation.
Learning Objectives
After reading this chapter, students should be able to
accomplish the following:

36. 1. Identify opportunities for professionals in sport management
and entry-level positions in these careers.
2. Describe expanding career opportunities in sport media.
3. Describe career opportunities in performance and other sport-
related careers.
4. Discuss how professionals can increase their professional
marketability.
Chapter Outline
I. Sport Management
II. Careers in Sport Management
A. Athletic Administration
B. Collegiate Recreation
C. Corporate Recreation
D. Sport Facilities Management
E. Sport Retailing
F. Sports Marketing
G. Career Opportunities in Professional and Sport Organizations
H. Sport Analytics
III. Careers in Sport Media
A. Sport Broadcasting
B. Sportswriting and Journalism
C. Sport Photography
D. Sports Information
E. Web Development and Social Media
IV. Performance and Other Sport Careers
A. Dance
B. Professional Athletics
C. Officiating
D. Sport Law and Agency
E. Entrepreneurship

37. V. Increasing Your Professional Marketability
Discussion Questions
1. How have new technologies contributed to the development
of sport and opportunities for viewing and promoting sport
events?
2. Discuss the administration of athletic programs and collegiate
recreation programs on your campus. What titles do the
directors of these programs hold? How many participants are
involved in these programs and what activities are offered? Do
you have any suggestions for additional activities?
3. Within your institution, discuss opportunities for recreational
sports—intramurals, clubs, and competitions. Who manages
these programs? What are the guidelines governing
participation?
4. Access the Institute for Diversity and Ethics in Sport (
https://www.tidesport.org/). Review the latest Race and
Gender Report Cards. Discuss your findings relative to hiring
practices and representation in different sport organizations.
5. Are eSports really sport?
6. Discuss each of the talking points identified in the social
justice box in the text. In addition, what other issues related to
social justice need to be addressed as they relate to careers in
sport? What specific steps could you take to address issues
related to social justice?
7. Discuss each of the trends identified in the current trends box
in the text. What is one trend you see today in related to careers
in sport that will impact you and/or your career path in the

38. future?
Self-Assessment Activities
These activities are designed to help students determine if they
have mastered the materials and competencies presented in this
chapter.
1. Blogging is a popular way for professionals and fans to share
news on what is happening in sports. Using one of the free
blogging sites, create a blog on a specific theme or sport,
adding at least four posts on your topic. Share your blog with
other people in your class and invite them to comment.
2. If possible, interview individuals working in sport
management positions. Ask each person to define his or her
responsibilities and the skills that are the most helpful in the
performance of the job. Determine the entry-level positions in
this area. Ask each individual for suggestions about advancing
to top-level managerial positions in the field.
3. Discuss the positive and negative aspects of pursuing a
performance career. Since performance careers may be of short
duration, how can individuals prepare for another career after
the culmination of their performance career? Select one
professional athlete in any sport and research his or her
background. Trace the athlete’s career path, starting from their
beginning interest in the sport to the current time. Be sure to
include information about the athlete’s education.
4. Using the information provided in the Get Connected box,
locate an article on a topic of interest related to sport
management, sport media, dance, officiating, or athletics. Write
a brief summary of the article, identifying five key points and
what you have learned.

39. 5. Select a game to watch, either live or on TV. Use Twitter or
another microblogging site to follow a game, sending out tweets
to keep your classmates informed of the game’s action.
Copyright ©2021 McGraw-Hill Education. All rights reserved.
No reproduction or distribution without the prior written
consent of McGraw-Hill Education
3
Walton-Fisette/Wuest: Foundations of Physical Education,
Exercise, Science, And Sport, 20e
Chapter 13
Fitness- and Health-Related Careers
Chapter Overview
In the past decade, the contribution of physical activity to
health has gained increased recognition. The opportunities for
individuals desiring to pursue a career as a fitness or exercise
specialist have increased tremendously. Career opportunities
exist in preventative and rehabilitative exercise programs.
Preventative exercise programs are conducted by corporations,
community agencies, and commercial fitness clubs.
Rehabilitative exercise programs are typically conducted in a
hospital setting, but may be affiliated with corporate fitness
programs or community agency programs.
Opportunities for professionals to pursue health-related careers
have also grown rapidly. Professionals possessing qualifications
in athletic training may find employment working with athletic
programs at the professional, collegiate, and, increasingly,
secondary level. Employment opportunities also are available in
sports medicine clinics, physical therapy clinics, and hospitals.
Physical educators, exercise science, and sport leaders have also
been successful in securing employment in health and weight
management spas and clubs.

40. The recognition that participation in movement and physical
activities has therapeutic and psychological benefits as well as
physical benefits has stimulated the growth of therapy-related
careers. These include careers as dance therapists, recreational
therapists, and physical therapists. Kinesiotherapy and
chiropractic are also potential careers for professionals willing
to continue their studies.
If one is seeking employment in fitness- and health-related
careers, one can increase one’s marketability by becoming
certified and taking additional course work in health, business,
and psychology. Gaining as much practical experience as
possible will also be an asset in securing employment.
It appears that opportunities for qualified individuals in fitness-
and health-related careers will continue to increase in the
future.
Learning Objectives
After reading this chapter, students should be able to
accomplish the following:
1. Discuss the responsibilities, opportunities, and preparation
for professionals interested in working in a fitness- or health-
related career.
2. Describe the opportunities available and preparation needed
by professionals desiring to pursue a therapy-related career.
3. Discuss the various strategies that can be used to enhance
one’s professional marketability in fitness-, health-, and
therapy-related careers.
Chapter Outline
I. Fitness- and Exercise-Related Careers
A. Worksite Wellness Programs

41. B. Commercial and Community Fitness Programs
C. Personal Trainers
D. Health and Wellness Coaches
E. Strength and Conditioning Professionals
F. Rehabilitation Programs
G. Career Preparation
Preparation
Certification
Professional Organizations
II. Health-Related Careers
A. Athletic Training
B. Health and Weight Management Clubs and Spas
III. Therapy-Related Careers
A. Dance/Movement Therapy
B. Therapeutic Recreation/Recreation Therapy
C. Kinesiotherapy
D. Physical Therapy
E. Chiropractic Care
IV. Increasing Your Professional Marketability
Discussion Questions
1. Fitness and health clubs have gained in popularity. However,
more than one-third of health club members have a household
income of more than $100,000 a year and the average income is
over $80,000 a year. What can professionals do to reach more
individuals whose income is less than the average and provide
opportunities for them to be physically active?
2. Should health clubs be mandated to hire only certified fitness
professionals? Explain your reasoning. If you believe that
health clubs should hire only certified fitness professionals,
from what organizations would you accept certifications and

42. why?
3. Worksite wellness programs continue to grow, stimulated in
part by the Affordable Care Act. Should employees be mandated
to participate in these programs to reduce the burden of health
care costs? Why or why not?
4. Discuss each of the talking points identified in the social
justice box in the text. In addition, what other issues related to
social justice need to be addressed as they relate to health and
fitness-related careers? What specific steps could you take to
address issues related to social justice?
5. Discuss each of the trends identified in the current trends box
in the text. What is one trend you see today regarding health-
and fitness-related careers that will impact you and/or your
career path in the future?
Self-Assessment Activities
These activities are designed to help students determine if they
have mastered the materials and competencies presented in this
chapter.
1. Describe the responsibilities of a fitness or exercise
specialist. If possible, interview a professional in this career
regarding his or her responsibilities and qualifications.
2. Describe the various employment opportunities for a fitness
or exercise professional. Search for jobs online; describe the
positions that you found available and qualifications required
for employment.
3. Using the information provided in the Get Connected box,
read about one of the certification programs available through
the American College of Sports Medicine, American Council on

43. Exercise, National Athletic Trainers’ Association, or the
National Strength and Conditioning Association. For the
association you selected, investigate the certification programs,
benefits offered to members, costs for certification and
membership, and opportunities for continuing education. How
could this certification be beneficial to you in your chosen
career?
4. Using the information provided in the Get Connected box,
locate new research, industry news, and position papers
pertaining to health and fitness. Summarize your findings in a
brief report.
Copyright ©2021 McGraw-Hill Education. All rights reserved.
No reproduction or distribution without the prior written
consent of McGraw-Hill Education
Nonparametric Test
Chi-Square Test for Independence
The test is used to determine whether two categorical variables
are independent.
Notation for the Chi-Square Test for Independence (Please note
that the notation varies
depending on the text)
O represents the observed frequency of an outcome
E represents the expected frequency of an outcome
r represents the number of rows in the contingency table

44. c represents the number of columns in the contingency table
n represents the total number of trials
Test Statistic
2
2
1 1
O E
E
df r c
The Chi-Square test is a hypothesis test. There are seven steps
for a hypothesis test.
1. State the null hypothesis

45. 2. State the alternative hypothesis
3. State the level of significance
4. State the test statistic
5. Calculate
6. Statistical Conclusion
7. Experimental Conclusion
Example
A university is interested to know if the choice of major has a
relationship to gender. A
random sample of 200 incoming freshmen students was taken
(100 male and 100
female). There major and gender were recorded. The results are
shown in the
contingency table below.
Major Female Male
Math 5 15
Nursing 44 10
English 10 10

46. Pre-Med 17 20
History 4 5
Education 15 20
Undecided 5 20
To determine if there is a relationship between the gender of a
freshmen student and
thei declared major perform the hypothesis test (Use level of
Step 1: Null Hypothesis
0 : Gender and Major of Freshmen students are independentH
Step 2: Alternative Hypothesis
: Gender and Major of Freshmen students are not
independentAH
Step 3: Level of Significance
Step 4: Test Statistic

47. 2
2
1 1
O E
E
df r c
Step 5: Calculations
There are several calculations for this test. We have to find the
expected frequency for
each cell in the contingency table. The expected frequency is
the probability under the
null hypothesis times the total frequency for the given row.
Here the probability under
the null hypothesis is .5, as the probability of being male and
female is equal.

48. Major Female Male
Pre-
12.5
Know calculate the test statistic.

49. 2
2
2 2 2 2
2
2 2 2 2
2 2 2 2
2 2
2
5 10 15 10 44 27 10 27
10 10 27 27
10 10 10 10 17 18.5 20 18.5
10 10 18.5 18.5
4 4.5 5 4.5 15 17.5 20 17.5
4.5 4.5 17.5 17.5
5 12.5 20 12.5
12.5 12.5
2.5 2.5 10.7 10.7 0 0 .1216
obs

50. obs
obs
O E
E
2
.1216 .0556 .0556 .357 .357 4.5 4.5

51. The calculation for degrees of freedom is as follows:
1 1 7 1 2 1
6 1 6
df r c
df
The critical value for the Chi-Square with 6 degrees of freedom
at a level of significance
s 12.592. This is found by using the Chi Square table.
Step 6: Statistical Conclusion
Since 2 2

52. null hypothesis.
Step 7: Experimental Conclusion
There is sufficient evidence to indicate that gender has an effect
on choice of major for
the incoming freshmen.
Mann-Whitney
The Mann-Whitney test is a nonparametric version of the
independent sample t-test.
This study is used when there are two independent samples of
ordinal scores.
Test Statistic
1 1
1 1 2 1
2 2

53. 2 1 2 2
1
2
1
2
n n
U n n R
n n
U n n R
Notation:
1
2
1
2
number of scores in group 1

54. n number of score in group 2
R sum of ranks for score in group 1
R sum of ranks for score in group 2
The Mann-Whitney test is a hypothesis test. There are seven
steps for a hypothesis
test.
1. State the null hypothesis
2. State the alternative hypothesis
3. State the level of significance
4. State the test statistic
5. Calculate
6. Statistical Conclusion
7. Experimental Conclusion
Example:

55. Suppose there was race between an Antelope and a mountain
lion. The Antelope won
but you want to know if this could be extended to a general
statement that Antelope will
win in a race against a mountain lion. A random sample of 10
Antelopes and 10
mountain lions are put in a race. The order in which they finish
is recorded. ( A for
antelope and L for mountain lion)
AALALLLAAALAALLAALLL
Step 1: Null Hypothesis
0 : Probability of the Antelope winning is no different than the
probalility of the
mountain lion wining the race.
H
Step 2: Alternative Hypothesis
: Probability of the Antelope winning is different than the
probalility of the
mountain lion wining the race.

56. AH
Step 3: Level of Significance
Step 4: Test Statistic
1 1
1 1 2 1
2 2
2 1 2 2
1
2
1
2
n n
U n n R
n n
U n n R

57. Step 5: Calculations
The results are ranked according to the in order in which they
finished the race.
Rank
A 1
A 2
L 3
A 4
L 5
L 6
L 7
A 8
A 9
A 10
L 11

58. A 12
A 13
L 14
L 15
A 16
A 17
L 18
L 19
L 20
Let the Antelope be sample one and the mountain lion be
sample 2.
1
2
1
2
number of scores for the Antelopes
n number of score for the mountain lions

59. R sum of ranks for score for the Antelopes
R sum of ranks for score for the mountain lions
Find the sum of the ranks for each group.
Antelope
Mountain
Lion
1 3
2 5
4 6
8 7
9 11
10 14
12 15
13 18

60. 16 19
17 20
Sum 92 118
The number of for each animal is 10 because 10 antelopes and
10 mountain lions
raced.
Now plug the information into the test statistic
1 1
1 1 2 1
1
1
1

61. 2 2
2 1 2 2
2
2
2
1
2
10 10 1
10 10 92
2
100 55 92
63
1
2
10 10 1
10 10 118
2
100 55 118
37

62. n n
U n n R
U
U
U
n n
U n n R
U
U
U

63. Find the critical value for the test statistic. Use the appropriate
U table found in the
appendix of the textbook. Look for alpha to be 0.05 and then the
number of scores for
each category to be 10. Then you see the following critical
values.
1,
2,
23
77
crit
crit
U
U

64. Step 6: Statistical Conclusion
The statistical conclusion tells whether to reject or fail to reject
the null hypothesis. The
rejection rule is as follow.
1
2
If then reject the null hypothesis
If then reject the null hypothesis
crit
crit
U U
U U
then fail to reject the null hypothesis.
Step 7: Experimental Conclusion

65. Since we failed to reject the null hypothesis we can say that
there is not significant
evidence to support the claim that the probability of the
antelope winning is any different
than the probability of the mountain lion winning the race.
Kruskal-Wallis Test
The Kruskal-Wallis Test is a nonparametric version of the one-
way ANOVA. This test
however does not assume normality or homogeneity of variance.
The test requires
ordinal scaling of the dependent variable and must have at least
5 data values in each
sample.
Notation
k number of sample or groups
n number of data values in each group
N number of data values in all samples combined
R sum of the ranks for each sample

66. Test Statistic
2
2 12
3 1
1
1
R
N
N N n
df k
The Kruskal-Wallis test is a hypothesis test. There are seven
steps for a hypothesis

67. test.
1. State the null hypothesis
2. State the alternative hypothesis
3. State the level of significance
4. State the test statistic
5. Calculate
6. Statistical Conclusion
7. Experimental Conclusion
Example
A gym has decided to recommend a diet to their clients. They
have narrowed down the
choice to three and they want to test which diet is best. They
have randomly selected 21
volunteers from the gym to participate in the diet. All the
participants have similar health
and body type as well as general exercise routine. The amount
of weight loss was
recorded.
Diet A Diet B Diet C
10 11 1

68. 12 4 8
13 2 17
7 6 0
3 16 16
15 9 18
5 14 19
Step 1: Null Hypothesis
0 : There is no difference amoung the dietsH
Step 2: Alternative Hypothesis
: There is a difference between the dietsAH
Step 3: Level of Significance
Step 4: Test Statistic
2

69. 2 12
3 1
1
1
R
N
N N n
df k
Step 5: Calculations
To perform the calculation, start by ranking the results.
Diet A Rank Diet B Rank Diet C Rank
10 11 11 12 1 2
12 13 4 5 8 9

70. 13 14 2 3 17 19
7 8 6 7 0 1
3 4 16 17 16 17
15 16 9 10 18 20
5 6 14 15 19 21
Notice that some data values have the same ranks. This is how
you account for data
values that are the same.
k = 3
n = 7
N = 21
Plug these values into the test statistic

71. 2
2
2 2 2
2
2
2
2
2
12
3 1
1
72 69 8912
3 21 1
21 21 1 7 7 7
12
740.571 680.143 1131.571 66

72. 462
.025974 2552.285 66
66.293 66
0.2931
1 3 1 2
obs
obs
obs
obs
obs
obs
R
N
N N n
df k

73. The critical value can be found using the Chi-Square table for
freedom.
2

74. Step 6: Statistical Conclusion
Since 2 2
reject the null hypothesis.
Step 7: Experimental Conclusion
The evidence does not support the claim that the there is a
significant difference
between the three diets at a level of significance of 0.05.
Nonparametric Test
Chi-Square Test for Independence
The test is used to determine whether two categorical variables
are independent.
Notation for the Chi-Square Test for Independence (Please note
that the notation varies
depending on the text)
O represents the observed frequency of an outcome

75. E represents the expected frequency of an outcome
r represents the number of rows in the contingency table
c represents the number of columns in the contingency table
n represents the total number of trials
Test Statistic
2
2
1 1
O E
E
df r c

76. The Chi-Square test is a hypothesis test. There are seven steps
for a hypothesis test.
1. State the null hypothesis
2. State the alternative hypothesis
3. State the level of significance
4. State the test statistic
5. Calculate
6. Statistical Conclusion
7. Experimental Conclusion
Example
A university is interested to know if the choice of major has a
relationship to gender. A
random sample of 200 incoming freshmen students was taken
(100 male and 100
female). There major and gender were recorded. The results are
shown in the
contingency table below.
Major Female Male
Math 5 15

77. Nursing 44 10
English 10 10
Pre-Med 17 20
History 4 5
Education 15 20
Undecided 5 20
To determine if there is a relationship between the gender of a
freshmen student and
thei declared major perform the hypothesis test (Use level of
Step 1: Null Hypothesis
0 : Gender and Major of Freshmen students are independentH
Step 2: Alternative Hypothesis
: Gender and Major of Freshmen students are not
independentAH
Step 3: Level of Significance
Step 4: Test Statistic

78. 2
2
1 1
O E
E
df r c
Step 5: Calculations
There are several calculations for this test. We have to find the
expected frequency for
each cell in the contingency table. The expected frequency is
the probability under the
null hypothesis times the total frequency for the given row.
Here the probability under

79. the null hypothesis is .5, as the probability of being male and
female is equal.
Major Female Male
Pre-
Know calculate the test statistic.

80. 2
2
2 2 2 2
2
2 2 2 2
2 2 2 2
2 2
2
5 10 15 10 44 27 10 27
10 10 27 27
10 10 10 10 17 18.5 20 18.5
10 10 18.5 18.5
4 4.5 5 4.5 15 17.5 20 17.5
4.5 4.5 17.5 17.5
5 12.5 20 12.5
12.5 12.5

81. 2.5 2.5 10.7 10.7 0 0 .1216
obs
obs
obs
O E
E

82. 2
.1216 .0556 .0556 .357 .357 4.5 4.5
The calculation for degrees of freedom is as follows:
1 1 7 1 2 1
6 1 6
df r c
df
The critical value for the Chi-Square with 6 degrees of freedom
at a level of significance
by using the Chi Square table.

83. Step 6: Statistical Conclusion
Since 2 2
null hypothesis.
Step 7: Experimental Conclusion
There is sufficient evidence to indicate that gender has an effect
on choice of major for
the incoming freshmen.
Mann-Whitney
The Mann-Whitney test is a nonparametric version of the
independent sample t-test.
This study is used when there are two independent samples of
ordinal scores.
Test Statistic
1 1

84. 1 1 2 1
2 2
2 1 2 2
1
2
1
2
n n
U n n R
n n
U n n R
Notation:
1
2
1

85. 2
number of scores in group 1
n number of score in group 2
R sum of ranks for score in group 1
R sum of ranks for score in group 2
The Mann-Whitney test is a hypothesis test. There are seven
steps for a hypothesis
test.
1. State the null hypothesis
2. State the alternative hypothesis
3. State the level of significance
4. State the test statistic
5. Calculate
6. Statistical Conclusion

86. 7. Experimental Conclusion
Example:
Suppose there was race between an Antelope and a mountain
lion. The Antelope won
but you want to know if this could be extended to a general
statement that Antelope will
win in a race against a mountain lion. A random sample of 10
Antelopes and 10
mountain lions are put in a race. The order in which they finish
is recorded. ( A for
antelope and L for mountain lion)
AALALLLAAALAALLAALLL
Step 1: Null Hypothesis
0 : Probability of the Antelope winning is no different than the
probalility of the
mountain lion wining the race.
H
Step 2: Alternative Hypothesis
: Probability of the Antelope winning is different than the

87. probalility of the
mountain lion wining the race.
AH
Step 3: Level of Significance
Step 4: Test Statistic
1 1
1 1 2 1
2 2
2 1 2 2
1
2
1
2
n n
U n n R

88. n n
U n n R
Step 5: Calculations
The results are ranked according to the in order in which they
finished the race.
Rank
A 1
A 2
L 3
A 4
L 5
L 6
L 7
A 8
A 9

89. A 10
L 11
A 12
A 13
L 14
L 15
A 16
A 17
L 18
L 19
L 20
Let the Antelope be sample one and the mountain lion be
sample 2.
1
2
1
2

90. number of scores for the Antelopes
n number of score for the mountain lions
R sum of ranks for score for the Antelopes
R sum of ranks for score for the mountain lions
Find the sum of the ranks for each group.
Antelope
Mountain
Lion
1 3
2 5
4 6
8 7
9 11
10 14

91. 12 15
13 18
16 19
17 20
Sum 92 118
The number of for each animal is 10 because 10 antelopes and
10 mountain lions
raced.
Now plug the information into the test statistic
1 1
1 1 2 1
1

92. 1
1
2 2
2 1 2 2
2
2
2
1
2
10 10 1
10 10 92
2
100 55 92
63
1
2
10 10 1
10 10 118
2

93. 100 55 118
37
n n
U n n R
U
U
U
n n
U n n R
U
U
U

94. Find the critical value for the test statistic. Use the appropriate
U table found in the
appendix of the textbook. Look for alpha to be 0.05 and then the
number of scores for
each category to be 10. Then you see the following critical
values.
1,
2,
23
77
crit
crit
U
U

95. Step 6: Statistical Conclusion
The statistical conclusion tells whether to reject or fail to reject
the null hypothesis. The
rejection rule is as follow.
1
2
If then reject the null hypothesis
If then reject the null hypothesis
crit
crit
U U
U U

96. then fail to reject the null hypothesis.
Step 7: Experimental Conclusion
Since we failed to reject the null hypothesis we can say that
there is not significant
evidence to support the claim that the probability of the
antelope winning is any different
than the probability of the mountain lion winning the race.
Kruskal-Wallis Test
The Kruskal-Wallis Test is a nonparametric version of the one-
way ANOVA. This test
however does not assume normality or homogeneity of variance.
The test requires
ordinal scaling of the dependent variable and must have at least
5 data values in each
sample.
Notation
k number of sample or groups
n number of data values in each group

97. N number of data values in all samples combined
R sum of the ranks for each sample
Test Statistic
2
2 12
3 1
1
1
R
N
N N n
df k

98. The Kruskal-Wallis test is a hypothesis test. There are seven
steps for a hypothesis
test.
1. State the null hypothesis
2. State the alternative hypothesis
3. State the level of significance
4. State the test statistic
5. Calculate
6. Statistical Conclusion
7. Experimental Conclusion
Example
A gym has decided to recommend a diet to their clients. They
have narrowed down the
choice to three and they want to test which diet is best. They
have randomly selected 21
volunteers from the gym to participate in the diet. All the
participants have similar health
and body type as well as general exercise routine. The amount
of weight loss was
recorded.

99. Diet A Diet B Diet C
10 11 1
12 4 8
13 2 17
7 6 0
3 16 16
15 9 18
5 14 19
Step 1: Null Hypothesis
0 : There is no difference amoung the dietsH
Step 2: Alternative Hypothesis
: There is a difference between the dietsAH
Step 3: Level of Significance
Step 4: Test Statistic

100. 2
2 12
3 1
1
1
R
N
N N n
df k
Step 5: Calculations
To perform the calculation, start by ranking the results.
Diet A Rank Diet B Rank Diet C Rank

101. 10 11 11 12 1 2
12 13 4 5 8 9
13 14 2 3 17 19
7 8 6 7 0 1
3 4 16 17 16 17
15 16 9 10 18 20
5 6 14 15 19 21
Notice that some data values have the same ranks. This is how
you account for data
values that are the same.
k = 3
n = 7
N = 21
Plug these values into the test statistic

102. 2
2
2 2 2
2
2
2
2
2
12
3 1
1
72 69 8912
3 21 1
21 21 1 7 7 7

103. 12
740.571 680.143 1131.571 66
462
.025974 2552.285 66
66.293 66
0.2931
1 3 1 2
obs
obs
obs
obs
obs
obs
R
N
N N n
df k

104. The critical value can be found using the Chi-Square table for
d 2 degrees of

105. freedom.
2
Step 6: Statistical Conclusion
Since 2 2
reject the null hypothesis.
Step 7: Experimental Conclusion
The evidence does not support the claim that the there is a
significant difference
between the three diets at a level of significance of 0.05.