Recall that in the week 2 assignment, a certain company hired three new managers from a pool of seven men and five women. All three hires were women, and the men complained of gender bias. Compute the theoretical probability that a fair selection process (in which each person is equally likely to be hired) would result in three women being hired. Do the men have a case?

Question

Recall that in the week 2 assignment, a certain company hired three new managers from a pool of seven men and five women. All three hires were women, and the men complained of gender bias. Compute the theoretical probability that a fair selection process (in which each person is equally likely to be hired) would result in three women being hired. Do the men have a case?

Answer 1

If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

5/12 * 4/11 * 3/10 = ?

What level of significance would you be using? P ≤ .05?
p ≤ .01?

Answer 2

To compute the theoretical probability of a fair selection process resulting in three women being hired, we need to know the total number of possible outcomes and the number of favorable outcomes, which in this case is the outcome where all three hires are women.

To find the total number of possible outcomes, we can use the concept of combinations. Since the company is hiring three managers out of twelve candidates (seven men and five women), we can calculate the total number of combinations as follows:

Total combinations = C(12, 3) = 12! / (3! * (12-3)!) = 220

Now, let's determine the number of favorable outcomes, which is the case where all three hires are women. Since the company hired three women, and there are five women in the pool of candidates, we can calculate the number of combinations as follows:

Favorable combinations = C(5, 3) = 5! / (3! * (5-3)!) = 10

So, the theoretical probability of a fair selection process resulting in three women being hired is:

P(3 women) = Favorable combinations / Total combinations
= 10 / 220
= 1 / 22
≈ 0.0455 or 4.55%

Now, moving on to the second part of the question, whether the men have a case, we can compare the theoretical probability to the actual outcome. If the probability of three women being hired in a fair selection process is significantly lower than what was observed, the men might have a case for gender bias.

From the given information, we know that all three hires were women. The probability of this happening by chance alone is 1/22, or approximately 4.55%. If the men claim gender bias, they are suggesting that this observed outcome is significantly lower than what would be expected in a fair selection process.

Ultimately, it would require further analysis and investigation to determine if the observed outcome is statistically significant enough to support the men's claim of gender bias.