Stastical technologies in business economics chapter 18

 Conduct a test of hypothesis for dependent samples using the Wilcoxon signed-rank test..  Conduct and interpret the Wilcoxon rank-sum test for  Conduct a test of hypothesis to determ

©The McGraw-Hill Companies, Inc 2008 McGraw-Hill/Irwin

Non-parametric:

Analysis of Ranked Data

Chapter 18

GOALS

 Conduct the sign test for dependent samples using the

binomial and standard normal distributions as the test statistics

 Conduct a test of hypothesis for dependent samples using the Wilcoxon signed-rank test

 Conduct and interpret the Wilcoxon rank-sum test for

 Conduct a test of hypothesis to determine whether the

correlation among the ranks in the population is different from zero

The Sign Test

The Sign Test is based on the sign of a difference between two related observations.

 No assumption is necessary regarding the shape of the population of differences.

 The binomial distribution is the test statistic for small

samples and the standard normal (z) for large samples.

 The test requires dependent (related) samples.

Procedure to conduct the test:

 Determine the sign (+ or -) of the difference between related pairs Determine the number of usable pairs.

 Compare the number of positive (or negative)

differences to the critical value.

 n is the number of usable pairs (without ties), X is the

number of pluses or minuses, and the binomial

probability π = 5

The Sign Test - Example

The director of information systems at

Samuelson Chemicals recommended

that an in-plant training program be

instituted for managers The objective is

to improve the knowledge of database

usage in accounting, procurement,

production, and so on A sample of 15

managers was selected at random A

panel of database experts determined

the general level of competence of each

manager with respect to using the

database Their competence and

understanding were rated as being

either outstanding, excellent, good, fair,

or poor After the three-month training

program, the same panel of information

systems experts rated each manager

again The two ratings (before and

after) are shown along with the sign of

the difference A “+” sign indicates

improvement, and a “-” sign indicates

that the manager’s competence using

databases had declined after the

training program

Did the in-plant training program effectively

increase the competence of the

managers using the company’s

database?

Step 1: State the Null and Alternative Hypotheses

H0: π ≤.5 (There is no increase in competence as a result of the

in-plant training program.)

H1: π >.5 (There is an increase in competence as a result of the

in-plant training program.)

Step 2: Select a level of significance

We chose the 10 level

Step 3: Decide on the test statistic

It is the number of plus signs resulting from the

experiment

Step 4: Formulate a decision rule.

7

 Hence, the decision rule for a two-tailed test would

be to reject the null hypothesis if there are 3 or fewer plus signs, or 11 or more plus signs.

in the rejection region, which starts at 10, so is rejected

We conclude that the three-month training course was effective It increased the database competency of the managers

Normal Approximation - Example

3

Normal Approximation - Example

4

Normal Approximation - Example

5

Wilcoxon Signed-Rank Test for Dependent Samples

If the assumption of normality is violated for

the paired-t test, use the Wilcoxon rank test

Signed- The test requires the ordinal scale of measurement

 The observations must be related or dependent

6

Wilcoxon Signed-Rank Test

The steps for the test are:

 Compute the differences between related

observations

 Rank the absolute differences from low to high.

 Return the signs to the ranks and sum positive and negative ranks.

 Compare the smaller of the two rank sums with the T

value, obtained from Appendix B.7.

7

Fricker’s is a family restaurant chain

located primarily in the southeastern part of the United States It offers a full dinner menu, but its specialty is

chicken Recently, Fricker, the owner and founder, developed a new spicy flavor for the batter in which the chicken is cooked Before replacing the current flavor, he wants to conduct some tests to be sure that patrons will like the spicy flavor better To begin, Bernie selects a random sample of 15 customers Each sampled customer is given a small piece of the current chicken and asked to rate its overall taste on a scale of 1 to 20 A value near 20 indicates the participant liked the flavor, whereas a score near 0 indicates they did not like the flavor

Next, the same 15 participants are given a sample of the new chicken with the spicier flavor and again asked to rate its taste on a scale of 1 to 20

The results are reported in the table on the

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

05 significance level.

Wilcoxon Signed-Rank Test for Dependent Samples - Example

8

Wilcoxon Signed-Rank Test for Dependent Samples - Example

9

 Each assigned rank in column 6 is then given the same sign as the original difference, and the results are reported in column 7 For example, the second participant has a difference score of 8 and a rank of

6 This value is located in the section of column 7.

 The R+ and R- columns are totaled The sum of the positive ranks is 75 and the sum of the negative ranks is 30

 The smaller of the two rank sums is used as the test

statistic and referred to as T.

0

 The critical values for the Wilcoxon signed-rank test are located in Appendix B.7 A portion of that table is shown on the table below.

1

 The value at the intersection is 25, so the critical

value is 25

 The decision rule is to reject the null hypothesis if the

obtained from Appendix B.7 is the largest value in

the rejection region To put it another way, our

decision rule is to reject if the smaller of the two rank sums is 25 or less

 In this case the smaller rank sum is 30, so the

decision is not to reject the null hypothesis We cannot conclude there is a difference in the flavor ratings between the current and the spicy.

2

Wilcoxon Rank-Sum Test

used to determine if two independent samples came from the same or equal populations.

 No assumption about the shape of the population is required

 The data must be at least ordinal scale

 Each sample must contain at least eight observations

3

Wilcoxon Rank-Sum Test

 To determine the value of the test

statistic W, all data values are

ranked from low to high as if they were from a single population.

 The sum of ranks for each of the two samples is determined

 The sum of ranks for each of the two samples is determined

 If the null hypothesis is true, then the ranks will be about evenly distributed between the two samples, and the sum of the ranks for the two samples will be about the same

5

Dan Thompson, the president of

CEO Airlines, recently noted an increase in the number of no- shows for flights out of Atlanta He

is particularly interested in determining whether there are more no-shows for flights that originate from Atlanta compared with flights leaving Chicago A sample of nine flights from Atlanta and eight from Chicago are

reported in Table 18–4 At the 05 significance level, can we

conclude that there are more shows for the flights originating in Atlanta?

no-Wilcoxon Rank-Sum Test for Independent Samples - Example

6

 Mr Thompson believes there are more no-shows for Atlanta flights Thus, a one tailed test is appropriate, with the rejection region located in the upper tail

 The null and alternate hypotheses are:

H0: The population distribution of no-shows is the same or less for

Atlanta and Chicago.

H1: The population distribution of no-shows is larger for

Atlanta than for Chicago.

 The test statistic follows the standard normal

distribution At the 05 significance level, we find from

Appendix B.1 the critical value of z is 1.65

 The null hypothesis is rejected if the computed value of

z is greater than 1.65.

 The Chicago flight with only 8 no-shows had the fewest, so it is assigned a rank of 1 The Chicago flight with

9 no-shows is ranked

2, and so on.

8

 The value of W is calculated for the Atlanta group

and is found to be 96.5, which is the sum of the ranks for the no-shows for the Atlanta flights.

2 9

0

Kruskal-Wallis Test:

Analysis of Variance by Ranks

This is used to compare three or more samples to determine if they came from equal populations.

 The ordinal scale of measurement is required.

 It is an alternative to the one-way ANOVA.

 The chi-square distribution is the test statistic.

 Each sample should have at least five

observations.

 The sample data is ranked from low to high as if

it were a single group.

1

A management seminar consists of executives from

manufacturing, finance, and engineering Before scheduling the seminar sessions, the seminar leader is interested in whether the three groups are equally knowledgeable about management principles Plans are to take samples of the executives in manufacturing, in finance, and in engineering and

to administer a test to each executive If there is no difference

in the scores for the three distributions, the seminar leader will conduct just one session However, if there is a difference in the scores, separate sessions will be given We will use the Kruskal-Wallis test instead of ANOVA because the seminar leader is unwilling to assume that (1) the populations of

management scores follow the normal distribution or (2) the population standard deviations are the same

Kruskal-Wallis Test:

Analysis of Variance by Ranks - Example

H1: The population distributions of the management scores for

the populations of executives in manufacturing, finance, and engineering are NOT the same

 Step 2: H0 is rejected if χ2 is greater than 7.185 There are 3 degrees of freedom at the 05 significance level

Kruskal-Wallis Test:

Analysis of Variance by Ranks - Example

3

 The next step is to select random samples from the three

populations A sample of seven manufacturing, eight finance, and six engineering executives was selected Their scores on the test are recorded below

Kruskal-Wallis Test:

Analysis of Variance by Ranks - Example

4

Considering the scores as a single population, the engineering executive with a score of 35 is the lowest, so it is ranked 1 There are two scores of 38 To resolve this tie, each score is given a rank of 2.5, found by (2+3)/2 This process is continued for all scores The highest score is 107, and that finance executive is given a rank of 21 The scores, the ranks, and the sum of the ranks for each of the three samples are given in the table below.

Kruskal-Wallis Test:

Analysis of Variance by Ranks - Example

5

Because the computed value of H (5.736) is less than the critical value of

5.991, the null hypothesis is not rejected There is not enough evidence to conclude there is a difference among the executives from manufacturing, finance, and engineering with respect to their typical knowledge of management principles From a practical standpoint, the seminar leader should consider offering only one session including executives from all areas.

Kruskal-Wallis Test:

Analysis of Variance by Ranks - Example

 It can range from –1.00 up to 1.00.

 It is similar to Pearson’s coefficient of correlation, but is based

on ranked data

 It computed using the formula:

8

Lorrenger Plastics, Inc., recruits

management trainees at colleges and universities throughout the United States

Each trainee is given a rating by the recruiter during the on-

campus interview This rating is

an expression of future potential and may range from 0 to 15, with the higher score indicating more potential The recent

college graduate then enters an in-plant training program and is given another composite rating based on tests, opinions of group leaders, training officers, and so on The on-campus rating and the in-plant training ratings are given in the table on the right.

Rank-Order Correlation - Example

9

Rank-Order Correlation - Example

0

Rank-Order Correlation - Example

1

 State the null hypothesis: Rank correlation in population is 0

 State the alternate hypothesis: Rank correlation in population is not 0

 For a sample of 10 or more, the significance of is determined

by computing t using the following formula The sampling distribution of follows the t distribution with n - 2 degrees of

freedom

2

3

End of Chapter 18