Questions: Assume that 12 jurors are selected from a population in which 50% of the people are Mexican-Americans. The random variable x is the number of Mexican-Americans on the jury. x: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 P(x): 0.000, 0.003, 0.016, 0.054, 0.121, 0.193, 0.226, 0.193, 0.121, 0.054, 0.016, 0.003, 0.000 P(8)=0.121 b. Find the probability of 8 or fewer Mexican-Americans among 12 jurors. The probability of 8 or fewer Mexican-Americans among 12 jurors is 0.927. c. Which probability is relevant for determining whether 8 jurors among 12 is unusually low: the result from part (a) or part (b)? A. The result from part (b), because it measures the probability of 8 or fewer successes. B. The result from part (a), because it measures the probability of exactly 8 successes. d. Is 8 an unusually low number of Mexican-Americans among 12 jurors? Why or why not? A. No, because the relevant probability is greater than 0.05. B. No, because the relevant probability is less than or equal to 0.05. C. Yes, because the relevant probability is less than or equal to 0.05. D. Yes, because the relevant probability is greater than 0.05.

Assume that 12 jurors are selected from a population in which 50% of the people are Mexican-Americans. The random variable x is the number of Mexican-Americans on the jury. x: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 P(x): 0.000, 0.003, 0.016, 0.054, 0.121, 0.193, 0.226, 0.193, 0.121, 0.054, 0.016, 0.003, 0.000 P(8)=0.121 b. Find the probability of 8 or fewer Mexican-Americans among 12 jurors. The probability of 8 or fewer Mexican-Americans among 12 jurors is 0.927. c. Which probability is relevant for determining whether 8 jurors among 12 is unusually low: the result from part (a) or part (b)? A. The result from part (b), because it measures the probability of 8 or fewer successes. B. The result from part (a), because it measures the probability of exactly 8 successes. d. Is 8 an unusually low number of Mexican-Americans among 12 jurors? Why or why not? A. No, because the relevant probability is greater than 0.05. B. No, because the relevant probability is less than or equal to 0.05. C. Yes, because the relevant probability is less than or equal to 0.05. D. Yes, because the relevant probability is greater than 0.05.
Transcript text: Assume that 12 jurors are selected from a population in which $50 \%$ of the people are Mexican-Americans. The random variable $x$ is the number of Mexican-Americans on the jury. \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline $\mathbf{x}$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline $\mathbf{P}(\mathbf{x})$ & 0.000 & 0.003 & 0.016 & 0.054 & 0.121 & 0.193 & 0.226 & 0.193 & 0.121 & 0.054 & 0.016 & 0.003 & 0.000 \\ \hline \end{tabular} \[ P(8)=0.121 \] b. Find the probability of 8 or fewer Mexican-Americans among 12 jurors. The probability of 8 or fewer Mexican-Americans among 12 jurors is 0.927 . c. Which probability is relevant for determining whether 8 jurors among 12 is unusually low: the result from part (a) or part (b)? A. The result from part (b), because it measures the probability of 8 or fewer successes. B. The result from part (a), because it measures the probability of exactly 8 successes. d. Is 8 an unusually low number of Mexican-Americans among 12 jurors? Why or why not? A. No, because the relevant probability is greater than 0.05 . B. No, because the relevant probability is less than or equal to 0.05 . C. Yes, because the relevant probability is less than or equal to 0.05 . D. Yes, because the relevant probability is greater than 0.05 .
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Solution

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Solution Steps

Step 1: Understand the Problem

We are given a probability distribution table for the number of Mexican-Americans on a jury of 12 people, where the probability of each outcome is provided. The random variable \( x \) represents the number of Mexican-Americans on the jury.

Step 2: Calculate the Probability of 8 or Fewer Mexican-Americans

To find the probability of 8 or fewer Mexican-Americans, we sum the probabilities for \( x = 0 \) to \( x = 8 \): \[ P(x \leq 8) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6) + P(7) + P(8) \] Substituting the given probabilities: \[ P(x \leq 8) = 0.000 + 0.003 + 0.016 + 0.054 + 0.121 + 0.193 + 0.226 + 0.193 + 0.121 \] \[ P(x \leq 8) = 0.927 \]

Step 3: Determine the Relevant Probability for Unusualness

To determine whether 8 Mexican-Americans is unusually low, we need to consider the probability of 8 or fewer Mexican-Americans, which is \( P(x \leq 8) = 0.927 \). This is because we are interested in the cumulative probability of 8 or fewer successes.

Step 4: Decide if 8 is Unusually Low

Since \( P(x \leq 8) = 0.927 \) is greater than 0.05, 8 is not considered unusually low.

Final Answer

  • b. The probability of 8 or fewer Mexican-Americans among 12 jurors is \(\boxed{0.927}\).
  • c. The relevant probability is from part (b), so the answer is \(\boxed{\text{A}}\).
  • d. 8 is not unusually low, so the answer is \(\boxed{\text{A}}\).
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