validity and premises of a specific hypothesis testing question

Question

Say the percentage of a certain type of people (obese, smoking...etc) in this country is A% and the the number of employees in this company is N. And how large should N be so that we can say with c % confidence that this company have a tendency to not hire that certain group of people? For any fixed N, A, c Is this a valid hypothesis testing question? If not, is there any other premises that we need before we can make it valid?

Answer 1

Questions about employment bias are highly emotionally charged. If this is not a homework problem, I'd suggest you tread carefully here. Here are some likely criticisms of any results (positive or negative):

1. Is the overall country really representative of the pool of qualified individuals for the job in question? For example, most (92% in 2003) nurses are women; therefore, I would not be surprised if a nursing company was predominantly female, despite the fact that approx. 50% of the population is male. Lesson: Pick the correct reference population!

2. Which qualified individuals actually applied to the company? You can't really expect a company to meet some quota if their applicant/acceptance rates are skewed. Lesson: There may be self-selection bias for this particular company.

3. What fraction of applicants accept offers by the company? Generally, you should do the analysis on the offers, not the acceptances. This removes the effect of applicant choices. However, you should also verify that the offers to those who didn't accept are at least as good as the offers given to those who accepted. A prejudiced company could (consciously or unconsciously) "lowball" certain groups of people, which clears the way for their preferred candidate.

Now, lets say you addressed the above issues and you can say with a straight face that the applicant pool to the company is similar to the overall country in terms of property A. Then, you can perform a simple hypothesis test of proportion:

Let $p_A$ be the proportion of persons in the country that have property A and let $f_A$ be the probability that the company will make an offer to an applicant with property A.

Then:

$$H_0:f_A \geq p_A, H_a: f_A<p_A$$

If you want to have confidence $c\%$, then you can perform a Binomial Test and reject at a P-value $\leq 1-c$.

Your question about sample size and confidence is a little confused: you can always achieve any confidence you want, its the power that determines a sample size. How small of a difference do you want to detect?

I'd suggest that you instead compare the 1-sided $c\%$ Clopper-Pearson interval for a population proportion. Its a conservative (read: wide) interval but if you notice that it doesn't contain $p_A$ you can be at at least $c\%$ confident that the hiring rate for the company deviates from the national occurrence rate of that property.