Handout 14: Understanding Randomness Investigating Claims of Discrimination

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1 EXAMPLE: GENDER DISCRIMINATION This fictitious example involves an evaluation of possible discrimination against female employees. Suppose a large supermarket chain occasionally selects employees to receive management training. A group of female employees has claimed that they are less likely than male employees of similar qualifications to be chosen for this training. The large employee pool that can be tapped for management training is 60% female and 40% male; however, since the management program began, 9 of the 20 employees chosen for management training were female (only 45%). Does this result provide evidence that women are discriminated against in the selection process for management training? Setting up the Simulation Study To investigate this, we will carry out a simulation in Tinkerplots 2. Once again, note that you will have to revise a few elements of the simulation that relate to the following questions: 1. A trial consists of selecting one person for training. How many trials must we simulate? 2. What are the two possible outcomes for each trial? 3. What is the chance or probability for each outcome, assuming that there is no discrimination in the selection process? Use the answers to the above questions to set up the appropriate spinner in Tinkerplots. 1

2 For this problem, it is necessary to change the probability on the spinner to match the situation presented here. To specify a proportion different from 50/50, select Show Proportion from the drop-down menu on the lower left of the spinner. This is shown below. Next, change the proportion to the appropriate values. The following plot shows the outcome from one simulated trial. 4. In this study, 20 individuals were selected for management training. The expected number of females is not 10. What is the expected number of females if the selection process is fair? Explain. 2

3 The following plot shows the outcomes (i.e., the number of women out of the 20 employees selected) for each of 1,000 runs of the simulation. 5. Under which assumption were these dots generated? Circle your answer. i. These dots were generated under the assumption of intentional discrimination against women. ii. iii. These dots were generated under the assumption of unintentional discrimination against women. These dots were generated under the assumption that there was no discrimination in the selection process. Explain your reasoning. 6. What does each dot on the above graph represent? 3

4 7. What if only 5 women had been selected so far out of 20 employees? Would this provide convincing evidence that discrimination against women is occurring? Explain. 8. What if only 6 women had been selected so far out of 20 employees? Would this provide convincing evidence that discrimination against women is occurring? Explain. 9. The actual outcome from this study had 9 women out of 20 selected for management training. Does this outcome provide evidence to suggest discrimination against women is occurring? Explain. 10. Based on the 1,000 simulated outcomes shown above, how many women (out of 20) would you have to see in order to say discrimination is likely occurring? Sketch this cutoff value on the number line below. Explain how you obtained this cutoff value. 4

5 EXAMPLE: SWAIN VS. ALABAMA In 1965, the United States Supreme Court made a ruling in the case of Swain vs. Alabama. Swain, a black male, was convicted of raping a white woman in Talladega County, Alabama. He was sentenced to death. The case was appealed to the Supreme Court on the grounds that there were no black persons on the jury. The Supreme Court denied the appeal on the following grounds. As provided by Alabama law, the jury was selected from a panel of about 100 persons. There were 8 black persons on the panel (they did not serve on the jury because they were struck by the prosecution). The Supreme Court ruled that the presence of 8 black persons on the panel showed the overall percentage disparity has been small and reflects no studied attempt to include or exclude a specified number of blacks. At that time in Alabama, only men over the age of 21 were eligible for jury duty. A later review of census records showed that there were approximately 16,000 such men in Talladega county. Of these, 26% were black. The jury pool was supposed to be representative of the population of the county. Was the Supreme Court correct when they stated that the overall percentage disparity has been small and reflects no studied attempt to include or exclude a specified number of blacks? That is, was it reasonable to observe only 8 black men in a jury pool of size 100 when 26% of the population was black? Or, do we have reason to believe that the selection of individuals for the jury pool was a biased process? 1. How many black persons would you expect to see on the jury panel if the selection process was truly unbiased? How did you get this value? 2. How many were actually observed? To determine whether the observed result is consistent (or inconsistent) with what we expect to see in an unbiased selection process, we will carry out a simulation in Tinkerplots 2. Once again, note that you will have to revise a few elements of the spinner set-up for this simulation that relate to the following questions: 5

6 3. A trial consists of selecting one person for the jury panel. How many trials must we simulate? 4. What are the two possible outcomes for each trial? 5. What is the chance or probability for each outcome, assuming that there is no discrimination in the selection process? Use the answers to the above questions to set up the appropriate spinner in Tinkerplots. Next, carry out the simulation study to obtain 1,000 simulated outcomes that help you understand what outcomes occur by chance when the selection process for the jury panel truly is unbiased. Sketch in the results of your simulation study below. 6

7 6. Was the Supreme Court correct when they stated that the overall percentage disparity has been small and reflects no studied attempt to include or exclude a specified number of blacks? That is, was it reasonable to by chance observe only 8 black men in a jury pool of size 100 when 26% of the population was black? Or, do we have reason to believe that the selection of individuals for the jury pool was a biased process? Explain your reasoning. EXAMPLE: CASTANEDA V. PARTIDA The 1977 U.S. Supreme Court case Castaneda v. Partida involved a criminal trial that originated in Hidalgo County, Texas (on the U.S.-Mexico border). The defendant in the trial, Rodrigo Partida, moved for a new trial. He contended that he had been denied his constitutional right to a trial by a jury of his peers. Specifically, he argued that the jury selection process systematically excluded Hispanics from jury service. Texas at the time employed a "key man" system for choosing jury pools. In this system, respected individuals in the county were identified and requested to recruit reliable persons for jury service. There were no current records identifying county residents by ethnicity. The county did have lists of people eligible for jury duty, however, as well as lists of those actually called for jury duty. Examination of the surname was used as a proxy for Hispanic ethnicity. At the time, 79% of those eligible for jury duty in the county were Spanish-surnamed. However, of 870 persons that had been selected for jury duty, only 339 were Spanish-surnamed. 1. How many Spanish-surnamed persons did you expect to see selected for jury duty if the selection process was truly fair? How did you get this value? 2. How many were actually observed? 7

8 Once again, to determine whether the observed result is consistent (or inconsistent) with what we expect to see in an unbiased selection process, we will carry out a simulation in Tinkerplots 2. Note that you will have to revise a few elements of the simulation that relate to the following questions: 3. A trial consists of selecting one person for jury duty. How many trials must we simulate? 4. What are the two possible outcomes for each trial? 5. What is the chance or probability for each outcome, assuming that there is no discrimination in the selection process? Use the answers to the above questions to set up the appropriate spinner in Tinkerplots. Next, carry out the simulation study to obtain 1,000 simulated outcomes that help you understand what outcomes occur by chance when the selection process for jury duty is not discriminatory. Sketch in the results of your simulation study below. 8

9 6. Based on your analysis, which way do you think the Supreme Court should have ruled on this case? Justify your answer. 7. What if 650 of the 870 persons that had been selected for jury duty were Spanishsurnamed? Which way do you think the Supreme Court should have ruled if that had been the case? Explain your reasoning. 9

10 EXAMINING THE EFFECT OF SAMPLE SIZE Once again, consider the Gender discrimination example. Recall that of the 20 persons selected for management so far, only 9 (or 9/20 = 45%) were female. In an unbiased selection process, we expected to see a woman selected 60% of the time. Suppose that the results had actually been as follows: Of the last 200 persons selected for management so far, only 90 (or 90/200 = 45%) were female. How would we have changed the set-up of the spinner of our simulation study? The results of both simulation studies are shown below. Recall that these plots give us an idea of what outcomes occur by chance when the selection process does not discriminate based on gender. Study #1: 20 trials Observed Result = 9/20 = 45% 10

11 Study #2: 200 trials Observed Result = 90/200 = 45% 1. Which of the following statements is most correct? i. The two studies provide equally convincing statistical evidence that the selection process discriminates against women. ii. Study #1 provides more convincing statistical evidence that the selection process discriminates against women. iii. Study #2 provides more convincing statistical evidence that the selection process discriminates against women. Explain your reasoning. PRACTICAL SIGNIFICANCE VS. STATISTICAL SIGNIFICANCE Consider the previous example. As discussed earlier, the result in Study #1 is not statistically significant because the observed outcome does not fall in the bottom 5% of simulated outcomes. On the contrary, the result in Study #2 is statistically significant. The previous example illustrates that statistical significance depends on the sample size. Both studies resulted in an outcome of 45% of those selected for management being female, but this result was only statistically significant in the study with the large sample size. A result is known as practically significant if the difference between the observed and expected result is large enough to be of value in the practical sense. 11

12 EXAMPLE: DUKES V. WAL-MART STORES, INC. The lead plaintiff in this case, Betty Dukes, was a Wal-Mart employee. She and others alleged gender discrimination in promotion policies and practices in Wal-Mart stores. As the nation s largest private employer, Wal-Mart makes tens of thousands of promotion decisions each year. The following data was provided during this trial: Wal-Mart promoted roughly 50,000 individuals to management between 1997 and Female employees constituted about 60% of the group eligible for these promotions. 1. How many of the 50,000 individuals promoted do you expect to be female if Wal-Mart is not discriminating based on gender? 2. Suppose that 29,780 of the individuals promoted were women (note that this was not the actual outcome in the court case). What percentage is this? 3. A simulation study with 100 trials was conducted to see what outcomes occur by chance when the selection process is not discriminatory. The results are shown below. Recall the hypothetical observed value of 29,780 females being selected. Is this result statistically significant? Explain. 4. Is this result practically significant? Explain. 5. Do you foresee any problems if the courts rely on only statistical significance when making decisions? Explain. 12

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