Question

Thirteen jurors are randomly selected from a population of 3 million residents. Of these 3 million residents, it is known that 48% are of a minority race of the 13 jurors selected, 2 are
minores
(a) What proportion of the jury described is from a minority race?
(b) If 13 jurors are randomly selected from a population where 48% are minorities, what is the probability that 2 or fewer jurors will be minorities?
(c) What might the lawyer of a defendant from this minority race argue?
(a) The proportion of the jury described that is from a minority race is
(Round to two decimal places as needed)
(6) The probability that 2 or fewer out of 13 jurors are minorities, assuming that the proportion of the population that are minorities is 48%, is
(Round to four decimal places as needed.)
(c) Choose the correct answer below
O A The number of minorities on the jury is reasonable, given the composition of the population from which it came.
B. The number of minorities on the jury is impossible, given the composition of the population from which it came.
C. The number of minorities on the jury is unusually low, given the composition of the population from which it came.
OD. The number of minorities on the jury is unusually high, given the composition of the population from which it came
O O O​

Answer 1

Answer:

a) The proportion of the jury described that is from a minority race is 0.15 or 15.38%, both to 2 d.p.

b) The probability that 2 or fewer out of 13 jurors are minorities, assuming that the proportion of the population that are minorities is 48%, is 0.0162

c) Option C is correct.

The lawyer of a defendant from this minority race should argue that

'The number of minorities on the jury is unusually low, given the composition of the population from which it came'.

Step-by-step explanation:

(a) What proportion of the jury described is from a minority race?

Of the 13 jurors, it is given in the question that only 2 of them are from a minority race.

Hence,

The proportion of the jury described that is from a minority race = (2/13) = 0.1538461538 = 0.154

(b) If 13 jurors are randomly selected from a population where 48% are minorities, what is the probability that 2 or fewer jurors will be minorities?

This problem is a binomial distribution problem

If X denotes the number of jurors that are from a minority race amongst the 13 jurors

Binomial distribution function is represented by

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

n = total number of sample spaces = number of jurors = 13

x = Number of successes required = 2 or fewer jurors = ≤ 2

p = probability of success = probability that a randomly picked member of the population as a juror is from a minority race = 48% = 0.48

q = probability of failure = probability that a randomly picked member of the population as a juror is NOT from a minority race = 1 - p = 1 - 0.48 = 0.52

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

= 0.00020325604 + 0.00243907252 + 0.01350870934

= 0.01615103791 = 0.0162 to 4 d.p.

(c) What might the lawyer of a defendant from this minority race argue?

Normally, the distribution of 13 jurors should reflect the population distribution with each race represented according to their population proportion or at least a proportion close to that.

But for this case, the proportion of the population from a minority race (48%) is more than 3 times the proportion of the jurors that are from a minority race (15.38%). This can introduce a very unfair bias in the decision of the jurors, hence, the lawyer of a defendant from this minority race might argue that the number of minorities on the jury is unusually low, given the composition of the population from which it came (less than ⅓).

Hope this Helps!!!