Women make up 24% of the science and engineering workforce.?

Question

Women make up 24% of the science and engineering workforce. In a random sample of 400 science and engineering employees, what is the probability that more than 120 are women?

Answer 1

The probability that any given member of the sample is a woman is p = 24/100 = 6/25. Therefore, the probability that exactly n women are in the sample is 400!/(n!(400-n)!) * p^n * (1-p)^(400-n), and so the probability that more than 120 women are in the sample is [n=121, 400]∑400!/(n!(400-n)!) * p^n * (1-p)^(400-n). This is an exact value, but very hard to compute. Therefore, we take advantage of the fact that for sufficiently large sample sizes, the binomial distribution is approximately normal. The expected value of the number of women in the sample is 400*(6/25) = 96 women. The variance will be n*p*(1-p), which is 96*(19/25) = 912/25 = 72.96. Therefore, the probability that n>120 is approximately the probability that a standard normal is greater than (120-96)/√72.96. (120-96)/√72.96 ≈ 2.809757. Looking this up in a standard normal table, we see that the probability that z<2.809757 ≈ .997521, so the probability that there are more than 120 women is only 0.002479.

Answer 2

I reckon you're best off approximating this B(400, 0.24) distribution with a normal distribution with mean 96 and variance 71.04.