After being rejected for employment Kim Kelly learns that the credit company has hired only three woman among the last 17 new employees she also Learned that the pool of applicants is very large

Question

Help her adjust the charge of gender discrimination by finding the probability of getting three or fewer women 117 people are hired assuming that there is no discrimination based on gender

P( at most three) =

Answer 1

OK, first the limitations: The question fails to give one important piece of information. No information is provided about the composition of the applicant pool. There is no indication of what percentage of applicants are male and what percentage are female. This is a critical question that will have to be answered eventually if a charge of gender discrimination will be proven. For this problem the following assumptions are made (always include your assumptions when doing a statistical analysis)

1. The distribution of men and women in the application pool mirrors the distribution in the general population of the USA. (492:508) _{2010 census}
2. All the applicants are fully qualified.
3. It is assumed that the distribution of fairly hired female employees will be binomial with a P of .508 .

The following formula is used for EACH possible outcome (n = total samples; x = test number ; P = probability of success) This is the probability mass function for those interested.

b(x; n, P) = _nC_x * P^x * (1 – P)^{n – x}

There are four possible outcomes (to the research question of determining the probability of hiring three or fewer women. Three, two, one, and of course none hired {3,2,1,0}

Using the manual formula the probability would have to be calculated four times and added up.

probability of 3 hires = ₁₇C_{3 *} .502³ _* .498¹⁴ or 680 _* .12651 _* .0000577 = .00496

probability of 2 hires = ₁₇C_{2 *} .502² _* .498¹⁵ or 136 _* .252 _* .00002874 = .0009849

probability of 1 hire = ₁₇C_{1 *} .502¹ _* .498¹⁶ or 17_* .502 _* .00001431 = .0001221

probability of 1 hire = ₁₇C_{0 *} .502⁰ _* .498¹⁷ or 1 _* 1_* .00000713 = .000007126

Total probability of 3 or less female hires = .006075

OR we could use Excel to calculate. in that case the function would be BINOM.DIST(number_x,trials_n,probability_x,cumulative)

BINOM.DIST(3,17, .502,TRUE) = 0.006078

The small difference is due to rounding errors in the 6th place. Take my advice, just use the BINOM.DIST.

For the record, if we assume equal distribution of men and women in applicant pool (i.e. .5) the answer is

BINOM.DIST(3, 17, .5, TRUE) = .006363 - not a major difference.

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note: for people trying to use the normal distribution approximation, it is unsuitable as the sample size is too small and the x is too small compared to n