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

Correlational Analysis

Correlational Analysis is the statistical tool that is used to describe the degree to which one variable is linearly related to another.  The degree of relationship can be measured and represented quantitatively by the coefficient of correlation.

The  Correlation Scale

1.       Low correlation – the changes in one variable cannot be expected to signify change in the other, almost negligible relationship (-0.10 – 0.20)

2.       Weak correlation – the change in one variable may not be expected from a change in the other, definite but small relationship (-0.49— -0.20, 0.21-0.49)

3.       Moderate Correlation – the change in one variable is expected from a change in the other, substantial relationship (-0.69 - -0.50, 0.50-0.79)

4.       High correlation – the change in one variable is expected with reliability from a change in the other, marked relationship (-0.70 – -0.89, 0.80 – 0.95)

5.       Very high correlation – the change in one variable is expected to happen with greater reliability from a change in the other, (-0.90- -1.00, 0.96 – 1.00)

The Pearson Product Moment Correlation Coefficient

 

r = pearson product –moment of correlation coefficient

Sum XY = the total sum of the products of x variable and y variable

X bar = mean of the independent variable

Y bar = mean of the dependent variable

Sx = Standard deviation of the independent variable

Sy = standard deviation of the independent variable

n = number of pairs

Test of Significance for r

Where

r = correlation coefficient

n = number of pairs

Degree of freedom = n -2

1.The adviser of a Grade 7 class wanted to know if the students’ achievement test scores in Mathematics and Physics are related. He took the test results of the top 10 students of her class and presented the scores in a table.  The data indicated below: (alpha 0.05)

Mathematics (X)	46	45	43	33	31	31	29	26	25	24
Physics (Y)	48	46	42	40	36	35	37	38	44	42

2. A Class adviser was interested in determining the relationship between the Mathematics grades and Chemistry grades of the top 10 junior students.  She gathered the following data: (alpha = 0.01)

Mathematics Grade(X)	84	89	84	86	84	84	87	94	88	85
Chemistry Grade(Y)	85	88	83	86	85	88	89	93	84	84

3.  How is the academic performance of senior students related to their age? Use 1 percent level of significance

Age	16	17	15	17	16	19	17	16	16	19	17	18	20	18	17	16	14	16	18	19	16	18	16	18	20	17	18	18	19	20	17	18	16	16	18
Final Grade	86	88	85	80	80	79	78	83	75	77	80	79	79	78	76	81	77	81	82	78	76	78	79	75	75	75	75	78	76	76	82	80	78	80	78

The Point Biserial Coefficient

Some situations involve variables where one is continuous while the other is dichotomous and is assumed to be discrete.  An example of this is the situation when the two variables under consideration are the mathematics achievement test scores of the students and gender. The achievement test score is continuous while gender, which is either male or female, is a nominal dichotomous variable.  The nominal dichotomous variable classifies attribute into two mutually exclusive classifications.  Other examples of nominal dichotomous variables are attitudes (positive-negative) and responses (yes-no, true-false, agree-disagree).

rpb = point biserial coefficient

X1 = mean of the continous variable of one group

X0 = mean of the continous variable of the other group

N1 = number of cases in one group

N0 = number of cases in the other group

N = total number of cases, N1 + N0

Sx = standard deviation of all measures in the continuous variable

1.       A mathematics teacher wanted to know if the students’ mathematics achievement test scores are related to their gender.  He randomly selected test results of 10 students where she came up with six females and four males. (alpha = 0.05)

Gender(male = 1, female = 0)	1	1	1	1	0	0	0	0	0	0
Math Achievement test scores	48	45	40	35	41	40	33	30	30	25

2.       The table below presents the performance ratings of teachers who have passed and have not passed the licensure examination for teachers.  Using 5 percent level of significance, test if the teachers’ ratings are related to the results of the licensure examination for teachers .  Let 1 = passed and 0 = failed.

LET Results	1	1	0	0	1	0	1
Performance Rating	87	87	85	85	90	85	90

3.       Below is another group of composite scores on a test battery and with each score the number of individuals who did (1) or did not complete(0) a program of training. 

Composite score	9	8	7	6	5	4	3	2	1	10
Completion	1	1	1	1	0	0	0	0	0	1

Kendall’s Coefficient of Concordance (W)

Kendall’s tau, is an alternative measure of relationship to the Spearman’s rho.  This test statistic, which was developed by Kendall, is used when the data of the three or more variables are given in ordinal measures.  It determines the degree of agreement in the ranks given by three or more judges.

Test of Significance

1.        What significant agreement exists between the respondents of principals, master teachers, and classroom teachers on the factors that may increase the teachers’ morale? Alpha = 0.05

Factors	Ranks by Principal	Master Teachers	Teachers
Involvement in decision-making	1	1	1
Adequate instructional material	3	3	2
Less non-teaching assignments	4	4	4
Transparency in promotion	2	2	3
Smaller class size	5	5	5

2.       Calculate the correlation of concordance of the ranks of the judges on the singing performances of 10 contestants.

Contestants	Judge1	Judge 2	Judge 3	Judge 4
1	9	10	10	9
2	8	6	7	5
3	3	1	2	4
4	10	9	8	10
5	4	5	1	3
6	2	3	4	1
7	7	8	9	6
8	5	4	6	7
9	1	2	3	2
10	6	7	5	8

Alpha = 0.05

3.       Four judges (parole board members) rank eight convicts on “parole readiness”.  By using the coefficient of concordance, indicate the degree of consistency of the judges.

Convict	Judge 1	Judge 2	Judge 3	Judge 4
1	1	1	1	1
2	2	4	3	2
3	3	3	2	4
4	4	2	4	3
5	5	6	5	5
6	6	5	6	7
7	7	7	8	6
8	8	8	7	8