To address the claim of gender discrimination, we can model the hiring scenario using a binomial distribution, where we have:
- \( n = 19 \) (the total number of new hires).
- \( p = 0.5 \) (the probability of hiring a woman, assuming there's no discrimination and the number of qualified men and women is approximately equal).
- \( x = \) the number of women hired, for which we will find the probability of hiring 3 or fewer women.
The binomial probability mass function is given by:
\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k} \]
where \( \binom{n}{k} \) is the number of combinations of \( n \) items taken \( k \) at a time.
We want to find \( P(X \leq 3) \):
\[ P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \]
Calculating each of these probabilities:
-
\( P(X = 0) \): \[ P(X = 0) = \binom{19}{0} (0.5)^0 (0.5)^{19} = 1 \cdot 1 \cdot \left(0.5\right)^{19} = \frac{1}{524288} \approx 0.000001907 \]
-
\( P(X = 1) \): \[ P(X = 1) = \binom{19}{1} (0.5)^1 (0.5)^{18} = 19 \cdot 0.5 \cdot \left(0.5\right)^{19} = 19 \cdot \frac{1}{524288} \approx 0.000036716 \]
-
\( P(X = 2) \): \[ P(X = 2) = \binom{19}{2} (0.5)^2 (0.5)^{17} = \frac{19 \times 18}{2} \cdot (0.5)^2 \cdot \left(0.5\right)^{19} = 171 \cdot \frac{1}{524288} \approx 0.00032568 \]
-
\( P(X = 3) \): \[ P(X = 3) = \binom{19}{3} (0.5)^3 (0.5)^{16} = \frac{19 \times 18 \times 17}{6} \cdot (0.5)^3 \cdot \left(0.5\right)^{19} = 969 \cdot \frac{1}{524288} \approx 0.001847267 \]
Now, summing these probabilities:
\[ P(X \leq 3) \approx 0.000001907 + 0.000036716 + 0.00032568 + 0.001847267 \approx 0.002211570 \]
So, the probability of hiring three or fewer women is approximately:
\[ P(X \leq 3) = 0.002211570 \]
Now, let's compare this probability to the cutoff value of 0.5% (0.005):
- \( 0.002211570 < 0.005 \)
Since the calculated probability is less than the 0.5% significance level, it is indeed an unusual event.
Conclusion:
Since \( P(X \leq 3) \) is less than 0.5%, it suggests that the event of hiring three or fewer women out of 19, assuming no discrimination, is unlikely to happen by chance. Thus, the data supports the allegation of discrimination.
Therefore, the answer is:
yes, this supports a charge of gender discrimination.