The chi-squared test is used to determine if two classifications are independent. It compares observed and expected values using a chi-squared calculation. The document shows observed values for overall wage estimates and well-being index ranges. Expected values are calculated. Degrees of freedom is determined. The null hypothesis of independence is tested against the alternative of dependence. The chi-squared test statistic is greater than the critical value so the null is rejected, indicating dependence between overall wages and well-being.

1. Calculation of Chi-Squared Test
The x 2 is used to find if two classifications or factors from the same sample are
independent - if the occurrence of one of them does not affect the occurrence of the other.
2 ( fo fe)2
x
fe
Observed Values:
B1 B2 Total
A1 A B A+B
A2 C D C+D
Total A+C B+D N
Expected Values:
B1 B2 Total
A1 (A+B)(A+C) (A+B)(B+D) A+B
N N
A2 (A+C)(C+D) (B+D)(C+D) C+D
N N
Total A+C B+D N
Degrees of freedom (df) = (rows-1)(columns-1)
Null ( H 0 ) Hypothesis = Overall wage estimates and well-being index are independent
from each other.
Alternative Hypothesis ( H 1 ) = Overall wage estimates and well-being index are
dependent of each other.
Table 4: Observed Values
61.0-65.0 66.0-71.0 Total
$33000-$40000 14 11 25

2. $41000-$54000 5 20 25
Total 19 31 50
Table 4 shows the observed values for the overall wage estimates and well-being index.
The data has been organized into different ranges.
Table 5: Calculations for Expected Values
61.0-65.0 66.0-71.0 Total
$33000-$43000 (25)(19) (25)(31) 14+11
50 50
$44000-$54000 (19)(25) (31)(25) 5+20
50 50
Total 14+5 11+20 50
Table 6: Expected Values
61.0-65.0 66.0-71.0 Total
$33000-$43000 9.5 15.5 25
$44000-$54000 9.5 15.5 25
Total 19 31 50
2 ( fo fe)2
x
fe
2 (14 9.5)2 (11 15.5)2 (5 9.5)2 (20 15.5)2
x
9.5 15.5 9.5 15.5
x2 6.87606112
Df (r 1)(c 1)
Df (2 1)(2 1)
Df 1
The x 2 critical value at 5% significance with 1 degree of freedom is 3.841. Since the x 2
value is greater than the critical value, 6.87606112 < 3.841, we reject the null hypothesis.

3. We accept the alternative hypothesis, therefore, the overall wage estimates and the well-
being index are considered as not independent.