Problem 38 - Basic Statistics for Business & Economics

Question

Thirty percent of the population in a Southwestern community are Spanishspeaking Americans. A Spanish-speaking person is accused of killing a non-Spanishspeaking American and goes to trial. Of the first 12 potential jurors, only 2 are Spanish-speaking Americans, and 10 are not. The defendant's lawyer challenges the jury selection, claiming bias against her client. The government lawyer disagrees, saying that the probability of this particular jury composition is common. Compute the probability and discuss the assumptions.

Step-by-Step Solution

Answer

The probability of selecting 2 or fewer Spanish-speaking jurors is 19.19%, indicating no significant bias.

Step 1: Define the Problem

We need to calculate the probability of a specific jury composition. In this case, the jury consists of 12 potential jurors, where 2 are Spanish-speaking Americans. We want to find the likelihood of having 2 or fewer Spanish speakers if 30% of the community is Spanish-speaking.

Step 2: Establish the Variables

Define the variables for our binomial distribution scenario. Let \( n = 12 \) be the total number of jurors, \( k = 2 \) the number of Spanish-speaking jurors, and \( p = 0.3 \) the proportion of Spanish speakers in the community. Compute the probability for \( X = 2 \) under the given conditions.

Step 3: Use the Binomial Probability Formula

For a binomial distribution, the probability of exactly \( k \) successes in \( n \) trials is calculated using: \[ P(X = k) = \binom{n}{k} p^{k} (1-p)^{n-k} \]Substitute the values: \( P(X=2) = \binom{12}{2} (0.3)^{2} (0.7)^{10} \).

Step 4: Calculate the Specific Probability

Calculate \( \binom{12}{2} \), \( (0.3)^2 \), and \( (0.7)^{10} \):\[ \binom{12}{2} = 66 \]\[ (0.3)^2 = 0.09 \]\[ (0.7)^{10} = 0.0282475 \]Multiply these together to find \( P(X = 2) = 66 \times 0.09 \times 0.0282475 = 0.1497 \).

Step 5: Find Cumulative Probability for Bias Claim

The defendant's lawyer might claim bias for \( X \leq 2 \), so calculate \( P(X \leq 2) \):\[ P(X = 0) + P(X = 1) + P(X = 2) \]Calculate similarly for \( P(X = 0) \) and \( P(X = 1) \) using the binomial formula and add them up.

Step 6: Determine Conclusion from Cumulative Probability

Compute \( P(X = 0) \) and \( P(X = 1) \) and sum together:- \( P(X = 0) = 0.0038 \)- \( P(X = 1) = 0.0384 \)Add: \( P(X \leq 2) = 0.0038 + 0.0384 + 0.1497 = 0.1919 \).With a cumulative probability of approximately 0.1919, or 19.19%, it is not particularly rare.

Key Concepts

Jury Selection Bias

Jury selection bias occurs when the process used to select jurors is not random or representative of the community. This can lead to an unfair trial, especially if the jury does not reflect the demographics of the community. Bias in jury selection can be claimed if a particular group is systematically underrepresented. In the given exercise, the defendant's lawyer argues that having only 2 Spanish-speaking jurors out of 12, while 30% of the community is Spanish-speaking, indicates a potential bias.
It is important to ensure that jury selection is free from bias, as a fair trial relies on an impartial jury. If the demographic makeup of the jury significantly deviates from the community, it might prompt questions about the fairness and equality of the selection process. In legal proceedings, both the defense and prosecution must agree on a jury that is unbiased and representative.

Probability Calculation

Probability calculation is a mathematical process used to determine the likelihood of an event. In the context of the exercise, we are interested in finding the probability of selecting 2 Spanish-speaking jurors out of 12. The binomial distribution is a common tool for assessing such probabilities.

• The binomial probability formula is: \( P(X = k) = \binom{n}{k} p^{k} (1-p)^{n-k} \)
• Where:
  • \( n \) is the number of trials (jurors in this context).
  • \( k \) is the number of successful trials (Spanish-speaking jurors).
  • \( p \) is the probability of a success in a single trial.

By substituting the values for our specific problem, we can calculate the chances of obtaining the observed jury composition if the jury selection were random.

Spanish-speaking Population

Recognizing the demographic context is crucial in assessing jury selection bias. In the exercise, the Spanish-speaking population represents 30% of the community. This percentage plays a pivotal role in determining whether the jury is representative. If the selection process is fair, we would expect a similar proportion of Spanish-speaking people in the jury.
Understanding the local population's demographics provides insights into whether a particular group has been systematically excluded. It is important for legal professionals to ensure that this awareness translates into fair jury selection practices, thus safeguarding the defendant's right to an impartial trial. The exercise highlights how numerical analysis of a demographic can support or refute claims of bias.

Cumulative Probability

Cumulative probability is used to determine the likelihood of achieving a range of outcomes in a binomial distribution. Unlike simple probability, which calculates the chance of a specific outcome occurring, cumulative probability considers multiple outcomes. In this exercise, we look at the probability of having 0, 1, or 2 Spanish-speaking jurors out of 12.
This cumulative probability is calculated by summing up the probabilities:

• \( P(X = 0) \) - no Spanish-speaking jurors.
• \( P(X = 1) \) - one Spanish-speaking juror.
• \( P(X = 2) \) - two Spanish-speaking jurors.

These probabilities are summed to determine \( P(X \leq 2) \), which reflects the likelihood of this jury composition happening by chance. In this scenario, with a cumulative probability of 19.19%, the occurrence is not deemed highly unusual, suggesting the jury composition could reasonably arise without bias.