Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.2.25

Chapter 5, Problem 5.2.25

In Exercises 25–28, find the probabilities and answer the questions.

Whitus v. Georgia In the classic legal case of Whitus v. Georgia, a jury pool of 90 people was supposed to be randomly selected from a population in which 27% were minorities. Among the 90 people selected, 7 were minorities. Find the probability of getting 7 or fewer minorities if the jury pool was randomly selected. Is the result of 7 minorities significantly low? What does the result suggest about the jury selection process?

Verified step by step guidance

Step 1: Identify the problem as a binomial probability problem. The population proportion of minorities is 27% (p = 0.27), the sample size is 90 (n = 90), and we are interested in the probability of getting 7 or fewer minorities (X ≤ 7).

Step 2: Define the binomial random variable X, which represents the number of minorities in the jury pool. The probability mass function for a binomial distribution is given by P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n choose k' is the binomial coefficient.

Step 3: To find P(X ≤ 7), calculate the cumulative probability by summing the probabilities for X = 0, 1, 2, ..., 7. This can be expressed as P(X ≤ 7) = Σ P(X = k) for k = 0 to 7. Use the binomial probability formula for each term.

Step 4: Alternatively, use a statistical calculator or software to compute P(X ≤ 7) directly. Input the parameters n = 90, p = 0.27, and calculate the cumulative probability for X ≤ 7.

Step 5: To determine if 7 minorities is significantly low, compare the result to a significance level (e.g., α = 0.05). If P(X ≤ 7) is less than the significance level, the result is considered significantly low. Interpret the result in the context of the jury selection process to assess whether it suggests potential bias.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability Distribution

A probability distribution describes how the probabilities are distributed over the values of a random variable. In this case, we can use the binomial distribution to model the selection of minorities in the jury pool, where each selection can be seen as a Bernoulli trial with two outcomes: minority or non-minority. Understanding this distribution is crucial for calculating the probability of obtaining 7 or fewer minorities.

Binomial Probability Formula

The binomial probability formula calculates the likelihood of obtaining a specific number of successes in a fixed number of independent Bernoulli trials. It is expressed as P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success. This formula will help determine the probability of selecting 7 or fewer minorities from the jury pool.

Statistical Significance

Statistical significance assesses whether the observed results are unlikely to have occurred by random chance. In this context, we need to determine if the occurrence of 7 minorities in the jury pool is significantly low compared to the expected number based on the population proportion. This involves comparing the calculated probability to a significance level, often set at 0.05, to draw conclusions about the jury selection process.

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