Question

Question 3

In a study conducted on 335 primary school students in a small district in Malaysia, students at primary levels 4-6 were asked which goal in terms of good grades, athletic ability or popularity (being popular in school) was most important to them. A two-way table (Table 3.1) separating the students by their educational levels and goals is shown below:

Table 3.1

Primary Level
Goals Grades Popular Sports Total	4	5	6	Total
	49	50	69	168
	24	36	38	98
	19	22	28	69
	92	108	135	335

a. To investigate possible differences among the students' goals by educational levels, a researcher suggested that it is useful to compute the column percentages. You are required to compute the column percentages and explain the meaning of these percentages. Do the results suggest that there is much of a variation in goals across the three educational levels?                                                                                         

b. The dataset from the same study now divides the students' responses into "Urban," "Suburban," and "Rural" school areas as shown in Table 3.2.You are required to conduct a Chi-Squared test to investigate whether there is an association between school area and the students' goals of getting good grades, athletic ability or popularity as most important to them?

Table 3.2

School Area
Goals Grades Popular Sports Total	Rural	Suburban	Urban	Total
	57	87	24	168
	50	42	6	98
	42	22	5	69
	149	151	35	335

Answer 1

a)

Goals	4	5	6
Grades	53.26%	46.30%	51.11%
Popular	26.09%	33.33%	28.15%
Sports	20.65%	20.37%	20.74%

yes,  there is much of a variation in goals across the three educational levels

b)

Chi-Square Test of independence
	Observed Frequencies
	0
0	rural	suburban	urban				Total
Grades	57	87	24				168
Popular	50	42	6				98
Sports	42	22	5				69
Total	149	151	35				335
	Expected frequency of a cell = sum of row*sum of column / total sum
	Expected Frequencies
	rural	suburban	urban				Total
Grades	149*168/335=74.722	151*168/335=75.725	35*168/335=17.552				168
Popular	149*98/335=43.588	151*98/335=44.173	35*98/335=10.239				98
Sports	149*69/335=30.69	151*69/335=31.101	35*69/335=7.209				69
Total	149	151	35				335
	(fo-fe)^2/fe
Grades	4.2033	1.6787	2.369
Popular	0.9432	0.1069	1.755
Sports	4.1684	2.6634	0.6769

Chi-Square Test Statistic,χ² = Σ(fo-fe)^2/fe =   18.564  
      
Level of Significance =   0.05  
Number of Rows =   3  
Number of Columns =   3  
Degrees of Freedom=(#row - 1)(#column -1) = (3- 1 ) * ( 3- 1 ) =   4  
      
p-Value =   0.0010   [Excel function: =CHISQ.DIST.RT(χ²,df) ]
Decision:    p-value < α , Reject Ho  

there is an association between school area and the students' goals of getting good grades, athletic ability or popularity as most important to them