Question

A study on educational aspirations of high school students measured aspi-

rations using the scale (some high school, high school graduate, some college, college
graduate). For students whose family income was low, the counts in these categories
were (9; 44; 13; 10); when the family income was middle, the counts were (11; 52; 23;
22); when family income was high, the counts were (9; 41; 12; 27).
(a) To test independence between educational aspirations and family income, the
Pearson’s chi- square test statistic is calculated as X2 = 8.8709. Based on this re-
sult, what can you conclude about the hypothesis that the educational aspirations
and family income are independent of each other?
(b) The standardized residuals are given below: Do these standardized residuals sug-
gest any association pattern?
FamilyIncome SomeSchool HighSchool SomeCollege College
Low 0.4061 1.5828 -0.1286 -2.1078
Medium -0.1898 -0.5441 1.3042 -0.4032
High -0.1903 -0.9459 -1.2374 2.4360
(c) We want to conduct the Mantel-Haenszel chi-squared test for linear trend al-
ternative. For this purpose, we assign the scores for educational aspirations as
1,2,3 and 4 and for income categories as 1, 2 and 3. The weighted correlation
coefficient is calculated as r = 0.1321. Use the Mantel-Haenszel test, make a con-
clusion about the hypothesis that the educational aspirations and family income
are independent of each other?

Answer 1

The null hypothesis of independence between educational aspirations and family income cannot be rejected based on the Pearson's chi-square test. The standardized residuals suggest an association pattern. The Mantel-Haenszel test also does not support the hypothesis of dependence.

Step-by-step explanation:

To test the independence between educational aspirations and family income, a Pearson's chi-square test statistic is calculated.

The test statistic is X2 = 8.8709. Based on this result, we can conclude that there is insufficient evidence to reject the null hypothesis that the educational aspirations and family income are independent of each other.

The standardized residuals suggest an association pattern. For example, students from low-income families are more likely to have some high school aspirations, while students from high-income families are more likely to have college graduate aspirations.

Using the Mantel-Haenszel test and a weighted correlation coefficient of 0.1321, there is no significant evidence to suggest that educational aspirations and family income are dependent on each other.