Abstract
The paper describes a test and an activity conducted to check convincingly whether there is gender discrimination amongst bank employees and to demonstrate how randomization techniques can help determine whether there difference between proportions is statistically significant in a given case. The test yields a position wherein 35 out of 48 bank supervisors recommend an application for promotion, wherein the applications are varied just by gender and yields 21 male and 14 female recommendations. The test result is put to test using a randomization technique. The original sample is resampled 100 times and with every resampling a different composition is obtained. The study arrives at a position wherein only two instances correspond to P≥0.29 are observed as compared to the original sample. However, the difference of 29% which was actually observed is quite large to be ignored. Thus it is possible but very unlikely that if there was no gender discrimination, that we would see such a large difference (Pdiff≥0.29) based just on chance alone and thus the difference between male and female recommendation probability (Pdiff=0.29) is statistically significant.
Keywords: randomization, resampled, gender discrimination, statistically significant
Objective
The following activity has been performed to demonstrate the use of randomization test to tell if the difference between two proportions is significantly different. The test in this case as per the test question is used to check convincingly whether the bank supervisors discriminate against the female applicants . The application has received 21 male recommendations and 14 female recommendations. The activity performed has been labelled as Activity ‘a’ for assignment purposes. The subsequent parts are labelled from ‘a1’ through ‘a6’.
Activity Explained
As already defined in the assignment, the randomization is achieved by resampling the 35 recommendations composed of male and female by randomly selecting a sample of 35 candidates by selecting 35 cards out of a deck of 48 cards which has in turn been selected from a deck of 52 after removing the four aces. Thus the pack of 48 consists of 24 red and 24 black cards representing male and female recommendations respectively. The red (male) recommendations for each resampling are plotted on a number line as X until the total number of X’s reach 100. In other words, the number of randomized samples is 100. The study denotes the null hypothesis as H0 and Alternative Hypothesis as Ha both of which are defined below in a5.
The original recommendation (sample) is represented by 21 red and 14 black cards selected while 13 cards are left out of 48 cards. This represents (21/24)-(14/24) =0.29 or 29% difference in probability (Pdiff.) between male and female recommendations. The same needs to be tested for significance.
Response to Activity a1
The first count or resampling yields 19 red cards representing number of males recommended for promotion.
Response to Activity a2
The number of males is represented on the number line as shown below.
Response to Activity a3
In the first simulation or resampling, the probability (P) of a male being promoted is 19/24=0.79 or 79% while that of female being promoted is 16/24=0.67=67%. The difference in proportions is approximately 13%. This simulation models a case of no discrimination as compared to the original sample or recommendation.
Response to Activity a4
The simulations are repeated a 100 times and each time a different composition of male (red) and female (black) recommendations is obtained. The same is calculated and tabulated along with the differences in probabilities (See Appendix 1). The values obtained are repeated and grouped and can be represented using frequency distribution or histogram as shown below. The frequencies of the respective male values correspond to the frequencies of the female values as well as the differences between the probabilities for the two as yielded by 100 resampling iterations. In other words the histogram in fact represents the number line above along with the number of males or the corresponding probability differences for each of the 100 samples. The frequencies for the same are shown in the appendix section (Appendix 2). The red vertical line represents the original sample recommendation composition wherein the difference in probabilities (Pdiff) between male and female recommendation is 29%.
Response to Activity a5
Based on the above representation, if we assume that if there is no gender discrimination (H0), an estimate of 21 or more of the candidates being male (Pdiff≥0.29) is based on a chance occurrence. While if we assume that there is in fact there is discrimination (Ha) then (Pdiff≥0.29) cannot be based on a chance occurrence.
Response to Activity a6
In this particular case, we observe that the out of all the 100 resampled simulations, only two instances correspond to P≥0.29 are observed as compared to the original sample. However, the difference of 29% which was actually observed is quite large to be ignored.
Thus it is possible but very unlikely that if there was no gender discrimination, that we would see such a large difference (Pdiff≥0.29) based just on chance alone.
Moreover, the data yielded by the simulation is consistent with what we can expect if there was in fact discrimination against females that is Pdiff is in fact equal to 0.29.
Thus we reject the null hypothesis H0 and conclude that there is in fact gender discrimination in this case. Consequently, we conclude that the difference between male and female recommendation probability (Pdiff=0.29) is statistically significant .
References
Roses, B., & Jerdee, T. H. (1974). Influence of Sex Role Steroteypes on Personnel Decisions. Journal of Applied Psychology, 9-14.
Watkins, A. E., Scheaffer , R. L., & Cobb, G. W. (2010). Statistics: From Data to Decision. John Wiley & Sons.
Appendix
