4. A study on educational aspirations of high school students measured aspiratio

Question

4. A study on educational aspirations of high school students measured aspirations with the scale (some high school, high school graduate, some college, college graduate). The student counts in these categories were (9,44,13,10) when family income was low, (11,52,23,22) when family income was middle, and (9,41,12,27) when family income was high.
a. Test independence of educational aspirations and family income using chi-square and likelihood ratio test.
b. Find the standardized residuals. Do they suggest any association pattern.
c. Report a summary of your findings in one or two paragraphs.

Explanation / Answer

Ans:

H0: aij=a(i+) a(+j) for all i = 1, 2, 3 and j = 1, 2, 3, 4.                       

             HA: aij=a(i+)a(+j)for some i and j.

We have n = 273, I = 3, J = 4, and we choose a = 0.05.

The test statistic is either chisq= (nij-ij)^2/ ij , where ij= n(i+)n(+j)/n  for all i, j, and under the null hypothesis, either statistic has an approximate chi-square distribution with d.f. = 6.

We will reject the null hypothesis if either calculated chi sq> chi square at df=6 and alpha=0.05

From the table above or the SAS output below, we have chi squared =8.8709  with a

p-value = 0.1810

From the above inferences on p value and test statistic, we do not reject the null hypothesis at 0.05 level of significance. This implies that the data here is not a sufficient evidence to conclude that there is a relationship between family income and educational aspirations.

Also, here the assumption is that both the categorical variables are nominal. But in case that we consider the variables to be ordinal, there can be a better proposition of testing the above relationship.

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