Problem 40 - Basic Statistics for Business and Economics with Student

Question

Thirty percent of the population in a southwestern community are Spanish- speaking Americans. A Spanish speaking person is accused of killing a non- Spanish speaking 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 is approximately 16.75%, suggesting possible bias.

Step 1: Understanding the Problem

We know that 30% of the population are Spanish-speaking Americans. We need to find the probability of randomly selecting 2 Spanish-speaking Americans out of 12 potential jurors.

Step 2: Defining the Binomial Distribution

The selection of jurors follows a binomial distribution, where for each of the 12 jurors, there is a success rate of 0.3 (Spanish-speaking American) and a failure rate of 0.7 (non-Spanish speaking American). We define this as Binomial(n=12, p=0.3).

Step 3: Formula for Binomial Probability

The probability of getting exactly k successes (Spanish-speaking Americans) in n trials is given by the formula: \[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \(n = 12\), \(k = 2\), and \(p = 0.3\).

Step 4: Compute the Binomial Coefficient

Compute \( \binom{12}{2} \), which is the number of ways to choose 2 Spanish-speaking Americans out of 12 jurors. \[ \binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66 \]

Step 5: Calculate the Probability Term

Calculate \( p^k \cdot (1-p)^{n-k} \) using \( p = 0.3, k = 2, n = 12 \). \[ 0.3^2 = 0.09 \]\[ 0.7^{10} = 0.0282 \] Therefore, the product is \( 0.09 \times 0.0282 = 0.002538 \).

Step 6: Compute the Final Probability

Multiplying the binomial coefficient by the probability term, we get:\[ P(X=2) = 66 \cdot 0.002538 = 0.1675 \]

Step 7: Conclusion and Interpretation

The probability of selecting a jury with exactly 2 Spanish-speaking Americans is approximately 0.1675, or 16.75%. This indicates that while it is not extremely uncommon, it is less likely than a random selection reflecting the 30% population distribution. Therefore, the concern of potential bias could be valid.

Key Concepts

Probability

Probability is a fundamental concept in statistics that measures the likelihood of an event occurring. It ranges from 0 to 1, where 0 indicates impossibility, and 1 indicates certainty. In our jury selection case, we're dealing with a probability problem where we want to determine how likely it is to select a certain number of Spanish-speaking jurors out of a pool. To calculate probabilities, especially in situations like this, several factors are considered:

• Experiments: An experiment refers to the process of selecting jurors. Each selection is a separate trial.
• Events: An event is each possible outcome, such as selecting 2 Spanish speakers out of 12.
• Probability Distribution: We use probability models, like the binomial distribution, to understand the behavior of the probability in a sequence of trials.

The probability in our scenario is determined using the binomial distribution formula, which helps us calculate how likely it is to get exactly 2 Spanish-speaking jurors from the 12, based on the given population percentages.

Binomial Coefficient

The binomial coefficient is a key component in binomial probability calculations. It is represented by the notation \( \binom{n}{k} \), which reads as "n choose k." This coefficient determines how many ways we can choose \(k\) successes in \(n\) trials.In the jury selection problem, we calculate \( \binom{12}{2} \), which gives us the number of different ways to choose 2 Spanish-speaking individuals from the pool of 12 potential jurors.To compute \( \binom{12}{2} \), we use the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]For the values \( n = 12 \) and \( k = 2 \), this simplifies to:\[ \frac{12 \times 11}{2 \times 1} = 66 \]This number, 66, indicates there are 66 possible ways to form groups of 2 Spanish-speaking jurors from the 12 people. This number plays a critical role in determining the overall probability of this specific event occurring in our study.

Jury Selection

Jury selection is a vital process in ensuring fair trials. It's important that a jury represents a fair cross-section of the community to lessen the chance of bias. When examining our jury selection, it's necessary to scrutinize if the composition reflects community demographics. In our scenario:

• Population Representation: We know that 30% of the population is Spanish-speaking.
• Jury Composition: In this case, only 2 out of 12 jurors are Spanish-speaking, forming about 16.7% of the jury. This figure is notably below the community representation.

The concern here lies in whether this deviation from the expected population distribution indicates a bias. If the jury does not reflect the population, the trial might not be fair. Calculating the probability, as we did, helps determine if this composition could happen by chance or whether a systemic issue might exist. The calculated probability shows it's not an extremely rare event but one that happens less frequently than expected based on the population ratio.