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# Statistical Evidence of Discrimination Cases and Chi-Square

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In cases we do not know the underlying theoretical distribution we resort to non-parametric statistics. The Chi-square is such a statistic.

Formal testing for evidence of discrimination in the bank supervisor case using the Chi-square statistic:

This study found that 21 out of the 24 males were selected for promotion while only 14 of the 24 females were selected for promotions.

Hypothesis:

H0: There is no relationship between gender and promotion.
H1: There is a relationship between gender and promotion (evidence of discrimination).

Confidence Interval & Decision Rule:

We will use a contingency table and the Chi-square statistic. In this case, the measured significance level would be (a = 0.05, or 95% confidence level). The critical value of the Chi-square is determined by the number of degrees of freedom

df = (number of rows - 1) (number of columns - 1) = (r-1)(c-1).

We have two rows (male and female) and two columns (promoted and not promoted) for a total of one df.
The critical Chi-square is 3.841 (lookup table or software (e.g., excel) for the 95% confidence level or 0.05 significance level). This would require a rejection of the null hypothesis if the test statistic is greater than 3.841.

Test Statistic and Contingency Table:

The contingency Table of observed frequencies (fo):

Promoted Not Promoted Total
Males 21 3 24
Females 14 10 24
Total 35 13 48

The expected frequency for each cell = (Row total)(Column total)
Grand Total

While the above gives us a formula for determining the expected frequency (fe), it is always a good idea to rationalize and explain it. In our case, under no-discrimination assumption we would expect the same promotion rate (0.5) for males and females.

Promoted Not Promoted Total
fo fe fo fe fo fe
Males 21 17.5 3 6.5 24 24
Females 14 17.5 10 6.5 24 24
Total 35 35 13 13 48 48

The calculated Chi-square = 5.169 (Chi-square is the sum of [(fo-fe) squared/fe]).
This means that we must reject the null hypothesis as it is larger than the critical value. Therefore, this test proves that there is a significant amount of bias in the promotion selection process based on gender.

There was a second half of the bank supervisors' experiment. This time the branch manager's job was described as "complex." This time 25 (completely different) male bank supervisors got the file of the female candidate and recommended promotion for 5 of them and 20 (completely different) male bank supervisors got the file of the male candidate and recommended promotion for 11 of them.