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Chapter 6: Problem 71

Wisconsin has the highest high school graduation rate of all states at \(90 \%\). a. In a random sample of 10 Wisconsin high school students, what is the probability that 9 will graduate? b. In a random sample of 10 Wisconsin high school students, what is the probability than 8 or fewer will graduate? c. What is the probability that at least 9 high school students in our sample of 10 will graduate?

Short Answer

Expert verified
The probability that 9 will graduate is calculated from the binomial distribution. The probability that 8 or fewer will graduate is the sum of the probabilities of 0-8 students graduating. The probability that at least 9 will graduate is the sum of the probabilities of 9-10 students graduating.

Step by step solution

01

Understand the problem

The problem is assessing the likelihood of an outcome (graduation) with a given success rate (90%) in a specific number of trials (10 students). The binomial probability formula can be used to solve this: \( P(x) = C(n, x) * (p^x) * (1-p)^{n-x} \) where \(P(x)\) is the probability of \(x\) successes in \(n\) trials, \(C(n, x)\) is the combination of \(n\) items taken \(x\) at a time, and \(p\) is the probability of success on each trial.
02

Calculate the probability that 9 will graduate

Substitute the given values into the binomial probability formula: \( n = 10, x = 9, p = 0.90 \). Using this, the calculated probability of exactly 9 students graduating can be derived.
03

Calculate the probability that 8 or fewer will graduate

This requires calculating the probabilities for 0 through 8 students graduating and then summing those probabilities. Using the binomial probability formula: \( n = 10, x = 0,1,...,8, p = 0.90 \), the individual probabilities can be calculated and then summed.
04

Calculate the probability that at least 9 will graduate

This requires calculating the probabilities for 9 and 10 students graduating and then summing those probabilities. Using the binomial probability formula: \( n = 10, x = 9,10, p = 0.90 \), the individual probabilities can be calculated and then summed.

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