TWO-WAY ANOVA

Two-Way ANOVA
1. Elementary Science 6 Achievement Test Score of 27 Pupils Classified According to Prior Elementary Science Achievement and Performance on a Test of Rigidity

Rigidity Tendency                                              Prior Elementary Science Achievement

                                                   Low Achievers                Average Achievers              High Achievers

High Rigidity                                       40                                   53                                        61

                                                             42                                    45                                        51

                                                              37                                    50                                        58

Average Rigidity                                 46                                     52                                       58

                                                               45                                    53                                        60

                                                               47                                     52                                       57

Low Rigidity                                          52                                    54                                         64

                                                                53                                   56                                          64

                                                                 50                                   59                                         67

Alpha = 0.01

2.  Scores of 80 individuals of both sexes on an attitude scale after the administration of four different films

                  Males                                                                                  FILM

                                                                1              2              3              4

                                                                10           14           13           16

                                                                8              12           12           12

                                                                6              8              10           10

                                                                4              4              9              9

                                                                4              4              9              9

                                                                4              4              7              7

                                                                2              3              4              7

                                                                2              2              4              6

                                                                2              2              4              5

                                                                1              1              2              5

Females                                               14           14           10           18

                                                                12           13           10           16

                                                                12           12           10           15

                                                                10           11           9              14

                                                                8              10           9              13

                                                                6              9              7              10

                                                                4              7              7              10

                                                                2              6              6              10

                                                                2              3              5              10

                                                                1              2              4              9

Alpha = 0.05

3. Below are the scores of a group of fraternity and nonfraternity men broken down by class on a attitude scale.  Analyze these data following the factorial design and make the appropriate F tests.  Set alpha 0.01.

                                                                Fraternity Men

Freshmen           Sophomores      Juniors                  Seniors

12                                           22                           26                           30

8                                              18                           20                           26

10                                           12                           18                           20

6                                              20                           16                           22

8                                              8                              10                           18

                                                                Nonfraternity Men

18                                           20                           22                           24

16                                           18                           21                           22

8                                              12                           19                           16

12                                           18                           17                           18

14                                           16                           18                           24

4.  Apply the two-way ANOVA to the following data.  Use 5 percent level of significance.

English                                  Low Achievers                   Average Achievers          High Achievers

                                                                33                                           43                                           41

                                                                40                                           35                                           31

                                                                36                                           40                                           38

Filipino                                                  36                                           42                                           48

                                                                35                                           43                                           49

                                                                37                                           42                                           37

Median Test and the Mann-Whitney Test

The median test is a non-parametric test that determines the significance of the difference between the medians of two independent groups. 

1.        A researcher wished to compare the effect of a treatment upon the number of correct responses on an achievement test.  The experimental group received the treatment while the control group did not.  The data are indicated in the table below:

Experimental Group                                       Control Group

46           24                                                           48           16

45           24                                                           46           15

43           23                                                           42           15

41           21                                                           40           14

31           19                                                           38           13

31           17                                                           28           12

27           16                                                           19           10

26           13                                                           18           10

26           12                                                           18           9

25           9                                                              17           8

2. Below are the scores of a group of students who used a new teaching methodology and a group of students who were taught in traditional way.  Test the hypothesis of no difference using the median test at 5% level of significance.

                                New Method                                                     Traditional Method

                                6                              8                                              7                              5

                                14                           13                                           12                           13

                                16                           6                                              8                              6

                                9                              7                                              4                              5

                                10                           14                                           5                              12

                                10                           14

3.In the data below are two sample of scores obtained on the standardized achievement test.  Apply the median test to test the hypothesis of no difference.  Use 5% level of significance.

                Group A                                                               Group B

                47                           40                                           30                           48

                35                           37                                           32                           40

                30                           43                                           33                           42

                40                           32                                           35                           34

                25                           29                                           46                           28

Friedman Two-Way Analysis of Variance by Ranks

When the data from K matched samples are in at least an ordinal scale, the Friedman two-way of variance by ranks is used to test the null hypothesis that the K samples have been drawn from

the same population.  The matched samples may be the same group of individuals who have been exposed to several conditions, or they be sets of individuals who have been matched on relevant variables and are randomly assigned to different conditions.

1.        How significant is the difference in the impressions of school special services personnel, classroom teacher, and department heads on the management practices of their school administrator?

ITEM	Personnel Weighted Ave	Teachers weighted AVE	Heads Weighted Ave
1.       Interviews are conducted in recruiting personnel	3.984	3.872	3.867
2.       Examinations play an important role in recruitment	4.442	4.392	4.111
3.       Interviews are conducted before an applicant takes the exam.	4.823	4.783	4.799
4.       The applicant is oriented towards the practices of the organization.	4.217	4.314	4.333
5.       The applicant is likewise made aware of the disciplinary measures of erring personnel.	4.632	4.698	3.983

2.       A company’s trainees are randomly assigned to groups that are oriented on certain industrial inspection procedure by three different methods and at the end of the orientation period they are tested for inspection performance quality.  The following are their scores.

Method 1	84	71	81	64	76	87
Method 2	75	72	69	74	51	62	70
Method 3	79	57	62	66	79

3.       Below are the scores of three matched groups under four conditions.  Test the null hypothesis using the Friedman two-way ANOVA.  Alpha = 0.01

Conditions

Groups                                                 I                               II                             III                            IV

A                                                             9                              4                              1                              7

B                                                             6                              5                              2                              8

C                                                             9                              1                              2                              6

CHI-SQUARE

CHI-SQUARE TEST  a test used in finding relation between categories.

where O = observed frequency (actual) and E = expected frequency

 degrees of freedom = (c – 1) ( r – 1)

to get E =

1.       A brand manager is concerned that her brand’s share may be unevenly distributed throughout the country.  In a survey in which the country was divided into 4 geographic regions, a random sampling of 100 consumers in each region was surveyed with the following results:

REGION

                                                NE          NW         SE           SW

Purchase the brand             40           55           45           50
do not purchase                   60           45           55           50

Calculate using .05 level of significance.

2.       To see if silicon chip sales are independent of where the U.S. economy is in the business cycle, data have been collected on the weekly sales of Zippy Chippy, an earthquake valley firm, and on whether the U.S. economy was rising to a cycle peak, at a cycle peak, falling to a cycle trough, or at cycle trough.  The results are:

WEEKLY CHIP SALES

                                                High       Medium                 Low

Economy

At peak                                  20           7                              3

At trough                               30           40                           30

Rising                                     20           8                              2

Falling                                    30           5                              5

Calculate at 0.01 level of significance.

3.       A newspaper publisher, trying to pinpoint his market’s characteristics, wondered whether newspaper readership in the community is related to readers’ educational achievement.  A survey questioned adults in the area on their level of education and their frequency of readership.  The results are shown in the following table.  Use 0.01 significance level.

LEVEL OF EDUCATIONAL ACHIEVEMENT

                                                Professional          College                   High School          Did not complete HS

Readership

Never                                     7                              14                           13                           16

Sometimes                            13                           17                           7                              7

Morning                                 39                           41                           8                              12

Both editions                        22                           23                           8                              12

4.       An educator has the opinion that the grades of high school students make are dependent on the amount of time they spend listening to music.  To test this theory, he has randomly given 400 students a questionnaire within the questionnaire are the two questions: “How many hours per week do you listen to music?” and “What is the average grade for all your classes?”  The data from the survey are in the table below.  Using a .05 percent significance level, test whether these factors are independent or dependent.

AVERAGE GRADE

                                                A                             B                             C                             D                             F

Hours spent

<5 hrs                                     13                           10                           11                           16                           5

5-10 hrs                                 20                           27                           27                           19                           2

11-20 hrs                               9                              27                           71                           16                           32

>20 hrs                                   8                              11                           41                           24                           11