Questions: Fourteen jurors are randomly selected from a population of 3 million residents. Of these 3 million residents, it is known that 46% are of a minority race. Of the 14 jurors selected, 2 are minorities. (a) What proportion of the jury described is from a minority race? (b) If 14 jurors are randomly selected from a population where 46% 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.) (b) The probability that 2 or fewer out of 14 jurors are minorities, assuming that the proportion of the population that are minorities is 46%, is (Round to four decimal places as needed.) (c) Choose the correct answer below. A. The number of minorities on the jury is unusually high, given the composition of the population from which it came B. The number of minorities on the jury is unusually low, given the composition of the population from which it came. C. The number of minorities on the jury is reasonable, given the composition of the population from which it came. D. The number of minorities on the jury is impossible, given the composition of the population from which it came

Solution Steps

To find the proportion of the jury that is from a minority race, we calculate:

\[ \text{Proportion} = \frac{\text{Number of minority jurors}}{\text{Total jurors}} = \frac{2}{14} = 0.142857 \]

Rounding to two decimal places, we have:

\[ \text{Proportion} \approx 0.14 \]

We need to calculate the probability of selecting 2 or fewer minority jurors from a population where \( p = 0.46 \) (the proportion of minorities) and \( n = 14 \) (the number of jurors).

Using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where \( q = 1 - p = 0.54 \).

Calculating for \( x = 0 \):

\[ P(X = 0) = \binom{14}{0} \cdot (0.46)^0 \cdot (0.54)^{14} \approx 0.0002 \]

Calculating for \( x = 1 \):

\[ P(X = 1) = \binom{14}{1} \cdot (0.46)^1 \cdot (0.54)^{13} \approx 0.0021 \]

Calculating for \( x = 2 \):

\[ P(X = 2) = \binom{14}{2} \cdot (0.46)^2 \cdot (0.54)^{12} \approx 0.0118 \]

Now, summing these probabilities gives us the probability of 2 or fewer minority jurors:

\[ P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \approx 0.0002 + 0.0021 + 0.0118 = 0.0141 \]

Given that the expected number of minority jurors is:

\[ E(X) = n \cdot p = 14 \cdot 0.46 = 6.44 \]

Having only 2 minority jurors is significantly lower than the expected value. Therefore, the lawyer might argue:

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

Final Answer