Answer:
a) p=0.154
b) P(X≤2)=0.019
c) He can argue that the proportion of minority jurors is not representative of the proportion of that minority in the population.
Step-by-step explanation:
a) The proportion can be calculated dividing the number of jurors that are from the minority race by the total number of jurors:
[tex]p=X/N=2/13=0.154[/tex]
b) We can model this with a binomial random variable, with sample size n=13 and probability of success p=0.47.
The probability of k minority jurors in the sample is:
[tex]P(x=k)=\dbinom{n}{k}p^k(1-p)^{n-k}=\dbinom{13}{k}\cdot0.47^k\cdot0.53^{13-k}[/tex]
We have to calculate the probability that 2 or less minority jurors. This can be calculated as:
[tex]P(x\leq2)=P(x=0)+P(x=1)+P(x=2)\\\\\\P(x=0)=\dbinom{13}{0}\cdot0.47^{0}\cdot0.53^{13}=1\cdot1\cdot0.0003=0.000\\\\\\P(x=1)=\dbinom{13}{1}\cdot0.47^{1}\cdot0.53^{12}=13\cdot0.47\cdot0=0.003\\\\\\P(x=2)=\dbinom{13}{2}\cdot0.47^{2}\cdot0.53^{11}=78\cdot0.221\cdot0.001=0.016\\\\\\\\P(x\leq2)=0.000+0.003+0.016\\\\P(x\leq2)=0.019[/tex]
The probability that 2 or less minority jurors is 0.019.
