Nonparametric Methods:

Analysis of Ordinal Data

Chapter 16

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Learning Objectives
LO16-1 Use the sign test to compare two dependent
populations
LO16-2 Test a hypothesis about a median
LO16-3 Test a hypothesis of dependent populations
using the Wilcoxon signed-rank test
LO16-4 Test a hypothesis of independent populations
using the Wilcoxon rank-sum test
LO16-5 Test a hypothesis of several independent
populations using the Kruskal-Wallis test
LO16-6 Test and interpret a nonparametric hypothesis
test of correlation

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The Sign Test
 The sign test is based on the sign difference between two
related observations
 No assumptions need to be made about the shape of the
two populations
 For small samples, find the number of + or – signs and
refer to the binomial distribution for the critical value
Example
 A dietitian wishes to see if by taking a certain mineral, a
person’s cholesterol level decreases
 She measures the individuals before and after
 If there has been an decrease “+,” an increase “−”

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The Sign Test Example
The director of information systems at Samuelson Chemicals recommended that an
in-plant training program be instituted for certain managers in Payroll, Accounting, and
Production Planning. A sample of 15 managers is randomly selected from the three
departments and rated on their technology knowledge.Then, after a 3 month training
program, the same assessment rated the managers knowledge again. A “+” sign
indicates an improvement, and a “−” sign indicates a decline in technology competence.

Did the in-plant training program increase

the managers technical knowledge?
Step 1: State the null and the alternate
H0: ≤ .50
There has been no change in the technology
knowledge base of the managers as a result of the
training program
H1: > .50
There has been an increase in the technology
knowledge base of the managers as a result of the
training program

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The Sign Test Example Continued
Step 2: Select the level of significance; we select .10
Step 3: Decide on the test statistic; it is the number of “+” signs
Step 4: Formulate a decision rule; if the number of pluses in the sample is 10 or more,
the null hypothesis is rejected and the alternate hypothesis accepted
Step 5: Make a decision; reject H0, the number of pluses is 11
Step 6: Interpret, we conclude the three-month training course was effective. It
increased the technology knowledge of the managers

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Testing a Hypothesis about a Median
 The median test is used to test a hypothesis about a
population median
 Find μ and σ for a binomial distribution
 The z distribution is used as the test statistic
 The value of z is computed from formula 16-1, where x is
the number of observations above or below the median
 A value above the median is assigned a “+”
 A value below the median is assigned a “−”
 A value that is the same as the median is dropped from
the analysis

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Testing a Hypothesis about a Median
Example
The U.S. Bureau of Labor Statistics reported in 2018 that the median amount spent
eating out by American families is about $3,000 annually.The food editor of the Portland
Tribune wishes to know if the citizens of Portland differ. She selected a random sample
of 22 families and found 15 spent more than $3,000 last year eating out, 5 spent less
than that, and 2 spent exactly $3,000. Is it reasonable to conclude that the median
amount spent this year in Portland, Oregon is not equal to $3,000?
Step 1: State the null and the alternate hypothesis
H0: Median = $3,000
H1: Median ≠ $3,000
Step 2: Select the level of significance; we select 0.05
Step 3: Decide on the test statistic; it is based on count

The strategy here is similar to the strategy used for the sign test.
1.We use the binomial distribution as the test statistic.
2.The number of trials is 20.
3.The hypothesized probability of a success is .50.

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Testing a Hypothesis about a Median
Example Continued
Step 4: Formulate the decision rule; if the count is 5 or less, or 15 or more, reject the
null hypothesis.

Step 5: Make decision; reject the null hypothesis, 15 families spent more than $3,000
Step 6: Interpret; the food editor should conclude that there is a difference in the
median amount spent in Portland from that reported by the BLS in 2018

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The Wilcoxon Signed-Rank Test for
Dependent Populations
 The Wilcoxon sign-rank test is a nonparametric test for
differences between two dependent populations
 The assumption of normally distributed populations is not
required
 The steps to conduct the test are
1. Rank absolute differences between the related
observations
2. Apply the sign of the differences to the ranks
3. Sum negative ranks and positive ranks
4. The smaller of the two sums is the computed T value
5. Refer to Appendix B.8 for the critical value, and make
a decision regarding H0
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The Wilcoxon Signed-Rank Test for
Dependent Populations Example
Fricker’s is a family restaurant chain located primarily in the southeastern part of the
United States. It offers a full menu, but its specialty is chicken. Recently, the owner,
Bernie Frick, developed a new spicy flavor for the batter in which the chicken is
cooked. Before replacing the current flavor, he wants to be sure that patrons will like
the spicy flavor.To begin the taste test, he selects a random sample of 15 customers.
Each customer is given a piece of the current chicken and asked to rate it on a scale of
1 to 20 and then the customer is given a piece of spicy chicken and asked to rate it.

Is it reasonable to
conclude that the spicy
flavor is preferred?

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The Wilcoxon Signed-Rank Test for
Dependent Populations Example Continued
Step 1: State the null and the alternate hypothesis
H0:There is no difference in the ratings of the two flavors
H1:The spicy ratings are higher
Step 2: Select the level of significance; we select .05
Step 3: Select the test statistic; we’ll use T
Step 4: Formulate the decision rule; reject the null hypothesis if the smaller of the rank
sums is 25 or less

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The Wilcoxon Signed-Rank Test for
Dependent Populations Example Concluded

Step 5: Make decision; in this case the smaller rank sum is 30, do not reject H0
Step 6: Interpret; we cannot conclude there is a difference in the flavor ratings between
the current and the spicy.

Mr. Frick should stay

with the current flavor
of chicken!

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The Wilcoxon Rank-Sum Test
 The Wilcoxon rank-sum test is used to test whether two
independent samples came from equivalent populations
 The assumption of normally distributed populations is not
required
 The population variances need not be equal either
 The data must be at least ordinal scale
 If each sample contains at least 8 observations, use the
standard normal distribution as the test statistic

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The Wilcoxon Rank-Sum Test Example
Dan Thompson, the president of OTG airlines, recently noted an increase in the
number of bags that were checked in at the gate in Atlanta. He is interested in
determining whether there are more gate-checked bags from Atlanta compared with
flights leaving Chicago. A sample of nine flights from Atlanta and eight from Chicago are
reported in the table.

Can we conclude that there

Ranked number of gate checked bags.
are more checked bags for
flights originating in Atlanta?

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The Wilcoxon Rank-Sum Test Example
Continued
Step 1: State the null and alternate hypothesis
H0:The number of gate-checked bags for Atlanta is the same or less than the
number of gate checked bags for Chicago
H1:The number of gate-checked bags for Atlanta is more than the number of
gate-checked bags for Chicago
Step 2: Select the level of significance; we select .05
Step 3: Select the test statistic; we use z
Step 4: Formulate the decision rule;reject H0 if z > 1.645

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The Wilcoxon Rank-Sum Test Example
Continued

Step 5: Compute the test statistic; make decision, do not reject H 0

Step 6: Interpret; it appears that the number of checked bags in Atlanta are the same as
those in Chicago

MegaStat output:

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The Kruskal-Wallis Test: ANOVA by Ranks
 The Kruskal-Wallis one-way ANOVA by ranks is used to
test whether several populations are the same
 The assumption of normally distributed populations is not
required
 There are three requirements for this test:
 The samples are from independent populations
 The population variances must be equal
 The samples are from normal populations
 The test statistic follows the chi-square distribution, if
there are at least five observations in each sample

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The Kruskal-Wallis Test: ANOVA by Ranks
 To compute the Kruskal-Wallis test statistic
 All the samples are combined
 The combined values are ordered from low to high
 The ordered values are replaced by ranks, starting with 1 for
the smallest value
 Compute the value of the test statistic from the following

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The Kruskal-Wallis Test: ANOVA by Ranks
Example

The Hospital Systems of the Carolinas operate three hospitals in the Greater
Charlotte area: St. Luke’s Memorial, Swedish Medical Center, and Piedmont Hospital.
The director of administration is concerned about the waiting time of patients with
non-life-threatening injuries that arrive during weekday evenings at the three hospitals.

Is there a difference in the waiting Waiting times ranked and summed

times at the three hospitals?

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The Kruskal-Wallis Test: ANOVA by Ranks
Example
Step 1: State the null and alternate hypothesis
H0: The population distributions of waiting times are the same for the three
hospitals
H1: The population distributions of waiting times are not all the same for the
three hospitals
Step 2: Select the level of significance; he selects .05
Step 3: Select the test statistic; he’ll use chi-square
Step 4: Formulate the decision rule, reject H 0 if H > 5.991
Step 5: Calculate H; make decision, do not reject H0
Step 6: Interpret; there is not enough evidence to conclude that there are differences in
wait times

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Rank-Order Correlation
 Spearman’s coefficient of rank correlation is a measure of
the association between two ordinal-scale variables
 It can range from −1 up to 1
 A value of −1 indicates perfect negative correlation
 A value 1 indicates perfect positive correlation
 A value of 0 indicates there is no association between
the variables
Example
 Two university staff members are asked to rank 10 faculty
research proposals for funding purposes. Do the staff
members rank the same proposals in the same way?
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Rank-Order Correlation Continued
 There are several conditions when it is not appropriate
or can be misleading.
 Those conditions include:
1. When the scale of measurement of one of the two variables
is ordinal (ranked).
2. When the relationship between the variables is not linear.
3. When one or more of the data points are quite different
from the others.

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Spearman’s Coefficient Continued
 The value of rs is computed from the following formula

 Provided the sample size is at least 10, we can conduct a

test of hypothesis using the following formula

 The test statistic follows the t distribution

 There are n-2 degrees of freedom

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Spearman’s Coefficient Example
Recent studies focus on the relationship between the age of online shoppers and the
number of minutes spent browsing on the Internet.The table below shows a sample of
15 online shoppers who actually made a purchase last week. Included is their age and
time, in minutes, spent browsing on the Internet last week.

1. Draw a scatter diagram.

2. What type of association do the sample data suggest?
3. Do you see any issues with the relationship between variables?

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Spearman’s Coefficient Example Continued
4. Find the coefficient of rank correlation.

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Spearman’s Coefficient Example Concluded
5. Conduct a test of hypothesis to determine if there is a negative association
between the ranks.

Step 1: State the null and alternate hypothesis

H0:The rank correlation in the population is zero
H1:There is a negative association among the ranks
Step 2: Select the level of significance; we select .05
Step 3: Select the test statistic; we use t
Step 4: Formulate the decision rule; reject H0 if t < −1.771
Step 5: Make decision; reject the null hypothesis, t = −3.784

Step 6: Interpret; there is evidence of a negative association between the age of the
Internet shopper and the time spent browsing the Internet

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 Chapter 16 Practice Problems

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Question 3 LO16-1
Calorie Watchers has low-calorie breakfasts, lunches, and dinners. If you join
the club, you receive two packaged meals a day. Calorie Watchers claims that
you can eat anything you want for the third meal and still lose at least 5
pounds the first month. Members of the club are weighed before commencing
the program and again at the end of the first month.The experiences of a
random sample of 11 enrollees are:

We are interested in whether

there has been a weight loss as
a result of the Calorie Watchers
program.
a. State H0 and H1.
b. Using the .05 level of significance, what is the decision rule?
c. What is your conclusion about the Calorie Watchers program?

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Question 5 LO16-2

According to the U.S. Department of Labor, the median

salary for a chiropractor in the United States is $81,500 per
year. A group of recent graduates employed in the state of
Colorado believe this amount is too low. In a random
sample of 18 chiropractors who recently graduated, 13
began with a salary of more than $81,500. Is it reasonable
to conclude that the starting salary in Colorado is more
than $81,500?

a. State the null and alternate hypotheses.

b. State the decision rule. Use the .05 significance level.
c. Test the hypothesis and interpret the results.

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Question 7 LO16-3

An industrial psychologist selected a random sample of

seven young urban professional couples who own their
homes.The size of their home (square feet) is compared
with that of their parents.At the .05 significance level, can
we conclude that the professional couples live in larger
homes than their parents?

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Question 13 LO16-4
Tucson State University offers two MBA programs. In the first program, the
students meet two nights per week at the university’s main campus in
downtown Tucson. In the second program, students only communicate online
with the instructor.The director of the MBA experience at Tucson wishes to
compare the number of hours studied last week by the two groups of
students.A sample of 10 on-campus students and 12 online students revealed
the following information.

Do not assume the two distributions of study times (in hours) follow a
normal distribution. At the .05 significance level, can we conclude the online
students spend more time studying?

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Question 19 LO16-5
Davis Outboard Motors Inc. recently developed an epoxy painting process to
protect exhaust components from corrosion. Bill Davis, the owner, wishes to
determine whether the durability of the paint was equal for three different
conditions: saltwater, freshwater without weeds, and freshwater with a heavy
concentration of weeds. Accelerated-life tests were conducted in the
laboratory, and the number of hours the paint lasted before peeling was
recorded. Five boats were tested for each condition.

Use the Kruskal-Wallis test and the .01 level to determine whether the
number of hours the paint lasted is the same for the three water conditions.

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Question 23 LO16-6
Ten new sales representatives for Clark Sprocket and Chain, Inc. were
required to attend a training program before being assigned to a regional sales
office. At the end of the program, the representatives took a series of tests
and the scores were ranked. For example, Arden had the lowest test score
and is ranked 1; Arbuckle had the highest test score and is ranked 10.At the
end of the first sales year, the representatives’ ranks based on test scores
were paired with their first year sales.

a. Compute and interpret the coefficient of rank correlation between first-

year sales and class rank after the training program.
b. At the .05 significance level, can we conclude that there is a positive
association between first-year sales dollars and ranking in the training
program?

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