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4. Recall that the binomial distribution with probability of success p is nearly normal when the sample size n is sufficiently large (when np and n(1-p) are both at least 10). a. The jury pool from which the jury was selected had 60 people. Using the Fulton County population statistics and a sample size n=60, calculate np and n(1-p). Is it reasonable to use the normal distribution to approximate the binomial distribution here?

4. Recall that the binomial distribution with probability of success p is nearly normal when the sample size n is sufficiently large (when np and n(1-p) are both at least 10). a. The jury pool from which the jury was selected had 60 people. Using the Fulton County population statistics and a sample size n=60, calculate np and n(1-p). Is it reasonable to use the normal distribution to approximate the binomial distribution here?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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**Understanding the Normal Approximation of the Binomial Distribution** 4. Recall that the binomial distribution with the probability of success \( p \) is nearly normal when the sample size \( n \) is sufficiently large (when \( np \) and \( n(1-p) \) are both at least 10). a. The jury pool from which the jury was selected had 60 people. Using the Fulton County population statistics and a sample size \( n = 60 \), calculate \( np \) and \( n(1-p) \). Is it reasonable to use the normal distribution to approximate the binomial distribution here? b. Find \( \mu \) and \( \sigma \) for this normal distribution. c. Find the probability (under this normal distribution) that there are no African-Americans in the 60-person jury pool. **Detailed Explanation:** - **Part a** involves calculating the values \( np \) and \( n(1-p) \) to determine if the normal approximation is applicable. The condition \( np \geq 10 \) and \( n(1-p) \geq 10 \) must be met for the approximation to be valid. - **Part b** requires finding the mean (\( \mu \)) and standard deviation (\( \sigma \)) of the normal distribution, which can be derived from the mean and variance of the binomial distribution. Specifically, \( \mu = np \) and \( \sigma = \sqrt{np(1-p)} \). - **Part c** is focused on using the normal distribution to find the probability of observing no African-Americans in the jury pool of 60 people, which involves calculating \( P(X = 0) \) under the normal approximation. This exercise helps illustrate the conditions and calculations necessary to utilize the normal approximation for a binomial distribution effectively.
Transcribed Image Text:**Understanding the Normal Approximation of the Binomial Distribution** 4. Recall that the binomial distribution with the probability of success \( p \) is nearly normal when the sample size \( n \) is sufficiently large (when \( np \) and \( n(1-p) \) are both at least 10). a. The jury pool from which the jury was selected had 60 people. Using the Fulton County population statistics and a sample size \( n = 60 \), calculate \( np \) and \( n(1-p) \). Is it reasonable to use the normal distribution to approximate the binomial distribution here? b. Find \( \mu \) and \( \sigma \) for this normal distribution. c. Find the probability (under this normal distribution) that there are no African-Americans in the 60-person jury pool. **Detailed Explanation:** - **Part a** involves calculating the values \( np \) and \( n(1-p) \) to determine if the normal approximation is applicable. The condition \( np \geq 10 \) and \( n(1-p) \geq 10 \) must be met for the approximation to be valid. - **Part b** requires finding the mean (\( \mu \)) and standard deviation (\( \sigma \)) of the normal distribution, which can be derived from the mean and variance of the binomial distribution. Specifically, \( \mu = np \) and \( \sigma = \sqrt{np(1-p)} \). - **Part c** is focused on using the normal distribution to find the probability of observing no African-Americans in the jury pool of 60 people, which involves calculating \( P(X = 0) \) under the normal approximation. This exercise helps illustrate the conditions and calculations necessary to utilize the normal approximation for a binomial distribution effectively.
The ethnicity, gender, age, and other demographic characteristics of juries have been of great interest in some trials in the United States. When the composition of a jury does not reflect the demographic characteristics of the surrounding community, doubts about fairness of the jury selection process and legal challenges can arise. Although juries are not selected solely by chance, comparing the actual jury to the composition of juries that would occur if jurors were selected at random can tell lawyers whether there are grounds to investigate the fairness of the jury selection process. An historic case concerning jury selection, Avery v. Georgia, was brought to the U.S. Supreme Court in 1953. A jury in Fulton County, Georgia had convicted Avery, an African-American, of a serious felony. There were no African-Americans on the jury. At the time, there were 165,814 African-Americans in the Fulton County population of 691,797. The list of 21,624 potential jurors had 1,115 African-Americans. A jury pool of 60 people was selected, supposedly at random, from the list of potential jurors. (However, the names of black and white jurors had been written on different colored slips of paper.) This jury pool, from which the 12 actual jurors were selected, contained no African-Americans.
Transcribed Image Text:The ethnicity, gender, age, and other demographic characteristics of juries have been of great interest in some trials in the United States. When the composition of a jury does not reflect the demographic characteristics of the surrounding community, doubts about fairness of the jury selection process and legal challenges can arise. Although juries are not selected solely by chance, comparing the actual jury to the composition of juries that would occur if jurors were selected at random can tell lawyers whether there are grounds to investigate the fairness of the jury selection process. An historic case concerning jury selection, Avery v. Georgia, was brought to the U.S. Supreme Court in 1953. A jury in Fulton County, Georgia had convicted Avery, an African-American, of a serious felony. There were no African-Americans on the jury. At the time, there were 165,814 African-Americans in the Fulton County population of 691,797. The list of 21,624 potential jurors had 1,115 African-Americans. A jury pool of 60 people was selected, supposedly at random, from the list of potential jurors. (However, the names of black and white jurors had been written on different colored slips of paper.) This jury pool, from which the 12 actual jurors were selected, contained no African-Americans.
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