Question

a company had to downsize and fire 20 employees. Of these 20 employees, 10 were women. However, only 33% of the company’s entire workforce were women. The women who were fired filed a sexual discrimination lawsuit since 50% of the fired employees were women. Determine the probability that 50% or more of the fired employees are women. Then determine if the women’s claim of sexual discrimination is legitimate

Answer 1

We use a hypothesis test to decide whether the company is firing women with a higher probability that their proportion in the workforce.

Let be the indicator variable which takes value 1 if the person being fired is a woman. This is a Bernoulli variable with parameter p (where p is the probability of the fired person being woman)

Now, let be the number of women in a sample of 20 fired people . Each person is chosen to be fired independently of the rest. Then,

Use this to find (I've computed this value using a Python code and the screenshot is attached below the answer.)

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het Ho: There is no gender discrimination p= 0-33 Hi : There is gender dicrimination P20-33 be the test where p is the probability of choosing a woman to be fired. This is a non-parametric test. we are interested in probability that knod at least 10 women are chosen to be fired out of 20 people who are to be fired assuming to (p=0.33) Let X be the number of women out of the 20. people. Then X~Binomial (2o, p) Under Ho, p=0.33. Pro (X7,10) = P6X = i) = 360) (0-3)(0.67 I=ID = 0.0065 > 0.05 (computed wing ) I Python code At 95-7. significance level, there is not enough evidence to reject Ho (ie there is not enough evidence of gender discrimination)

In [9]: from math import factorial as fac def calc_prob (n,x,pl,p2): return pl*** * P2**(n-x)*fac(n)/(fac(x)*fac (n-x)) In [13]: prob = 0 for i in range (10,21): print("Probability that "+str(i)+" of the 20 people are women = "+str(calc_prob (20,1, 0.33,0.67))) prob = prob + calc_prob (20,1,0.33, 0.67) print("\nProbability that atleast 10 of the 20 people are women = "+str(prob)) Probability that 10 of the 20 people are women = 0.051580550977615776 Probability that 11 of the 20 people are women = 0.023095769094454828 Probability that 12 of the 20 people are women = 0.008531646046085925 Probability that 13 of the 20 people are women = 0.0025859409370455615 Probability that 14 of the 20 people are women = 0.0006368362009142056 Probability that 15 of the 20 people are women = 0.00012546623659802255 Probability that 16 of the 20 people are women = 1.9311500968912057e-05 Probability that 17 of the 20 people are women = 2.2380317189608357e-06 Probability that 18 of the 20 people are women = 1.837190217057402e-07 Probability that 19 of the 20 people are women = 9.525102460784646e-09 Probability that 20 of the 20 people are women = 2.3457341881036816e-10 Probability that atleast 10 of the 20 people are women = 0.08657795250409978