Chapter 6.3, Problem 48PB

Checking guidelines For Example 13 on the gender distribution of promotions, the population size was more than one thousand, half of whom were female. The sample size was 10.

a. Check whether the guideline was satisfied about the relative sizes of the population and the sample, thus allowing you to use the binomial for the probability distribution for the number of females selected.

b. Check whether the guideline was satisfied for this binomial distribution to be approximated well by a normal distribution.

Example 13

Testing for Gender Bias in Promotions

Picture the Scenario

Example 1 introduced a case involving possible discrimination against female employees. A group of women employees has claimed that female employees are less likely than male employees of similar qualifications to be promoted.

Question to Explore

Suppose the large employee pool that can be tapped for management training is half female and half male. In a group recently selected for promotion, none of the 10 individuals chosen were female. What would be the probability of 0 females in 10 selections, if there truly were no gender bias?

Think It Through

If there is no gender bias, other factors being equal, at each choice the probability of selecting a female equals 0.50 and the probability of selecting a male equals 0.50. Let X denote the number of females selected for promotion in a random sample of 10 employees. Then, the possible values for X are 0, 1, …,10, and X has the binomial distribution with n = 10 and p = 0.50. For each x between 0 and 10, we can find the probability that x of the 10 people selected are female using the binomial formula

$\text{P}\left(x\right)=\frac{n!}{x!\left(n-x\right)!}{p}^x{\left(1-p\right)}^{n-x}=\frac{10!}{x!\left(10-x\right)!}{\left(0.50\right)}^x{\left(0.50\right)}^{10-x},x=0,1,2,\dots ,10.$

The probability that no females are chosen (x = 0) equals

$\text{P}\left(0\right)=\frac{10!}{0!10!}{\left(0.50\right)}^0{\left(0.50\right)}^{10}={\left(0.50\right)}^{10}=0.001$.

Any number raised to the power of 0 equals 1. Also, 0! = 1, and the 10! terms in the numerator and denominator divide out. leaving P(0) = (0.50)¹⁰. If the employees were chosen randomly, it is very unlikely (one chance in a thousand) that none of the 10 selected for promotion would have been female.

Insight

In summary, because this probability is so small, seeing no women chosen would make us highly skeptical that the choices were random with respect to gender.