In the case of Castaneda vs Partida, it was noted that although 80% of the population in a Texas county was Mexican-American, only 39% of those summoned for grand juries were Mexican-American. Assume that 12 jurors are to be summoned from a population that 80% Mexican-American. What is the probability that exactly 7 of the 12 jurors are Mexican-American? a) Carefully explain how this situation results in a binomial distribution. b) Write out the Binomial Probability Function without simplifying for this situation. c) Calculate the probability, to four decimal places, of selecting exactly 7 Mexican-Americans when 12 jurors were selected at random from a population that is 80% Mexican-American.
Please show more than one approach.
Wow ... 7 posts in 2 minutes! You must be taking a test or something.
No one here will do your work for you. You need to indicate exactly what you have done to solve each problem and where you're running into trouble.
It is hard to do so in the format provided.
a
b p(x)= nCx pX9i-p)n-x
d 0.5315
a) This situation results in a binomial distribution because it satisfies the following conditions:
1. There are a fixed number of trials: In this case, the fixed number is 12 jurors.
2. Each trial has two possible outcomes: The two outcomes are whether a juror is Mexican-American or not.
3. The probability of success is constant: The probability of selecting a Mexican-American juror is consistent at 80%.
4. The trials are independent: The selection of one juror does not affect the selection of another.
b) The Binomial Probability Function is given as:
P(x) = C(n, x) * p^x * q^(n-x)
Where:
P(x) represents the probability of getting exactly x successes.
C(n, x) is the binomial coefficient, which is calculated as n! / (x!(n-x)!).
p represents the probability of success on a single trial.
q represents the probability of failure on a single trial, which is equal to (1 - p).
n is the total number of trials.
In this case, we want to find the probability of exactly 7 Mexican-Americans out of 12 jurors, and the probability of success (p) is 80%. So, we can write the Binomial Probability Function as:
P(7) = C(12, 7) * (0.8)^7 * (0.2)^(12-7)
c) Approach 1: Calculation using the Binomial Probability Function formula
By substituting the values into the formula, we can calculate the probability:
P(7) = C(12, 7) * (0.8)^7 * (0.2)^(12-7)
= (12! / (7!(12-7)!)) * (0.8^7) * (0.2^5)
= (792) * (0.2097152) * (0.03125)
≈ 0.1995 (rounded to four decimal places)
Therefore, the probability of selecting exactly 7 Mexican-Americans when 12 jurors are selected at random from a population that is 80% Mexican-American is approximately 0.1995.
Approach 2: Calculation using a statistical software or calculator
If you have access to a statistical software or calculator, you can directly calculate the probability by using the binomial distribution function. For example, in Python using the scipy library, you can use the binom.pmf() function:
```python
import scipy.stats as stats
p = 0.8
n = 12
x = 7
probability = stats.binom.pmf(x, n, p)
print(round(probability, 4))
```
This will give the same result as the previous approach: approximately 0.1995.