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Chapter 6: Problem 70

Colorado has a high school graduation rate of \(75 \%\). a. In a random sample of 15 Colorado high school students, what is the probability that exactly 9 will graduate? b. In a random sample of 15 Colorado high school students, what is the probability that 8 or fewer will graduate? c. What is the probability that at least 9 high school students in our sample of 15 will graduate?

Short Answer

Expert verified
For part a, the probability of exactly 9 students graduating can be determined using the binomial distribution formula. For part b, the probability of 8 or less students graduating requires iterating through the binomial distribution formula for k=0 up to k=8 and summing these probabilities. For part c, the probability of at least 9 students graduating can be found by subtracting the result from part b from 1.

Step by step solution

01

Calculation for part a

Use of the binomial distribution formula: \( P(X=k) = C(n, k) * p^k * (1-p)^(nk) \), where \(C(n,k)=n!/(k!(nk)!) \) \n Given \( n=15 \), \( k=9 \), and \( p=0.75 \), \n Plugging these values into the formula, we calculate the probability that exactly 9 out of 15 students will graduate.
02

Calculation for part b

Change the value of k from 0 to 8 and sum them up. The sum will be the probability that 8 or less students will graduate.
03

Calculation for part c

To find the probability that at least 9 students will graduate, subtract the answer to part b (probability of 8 or less students graduating) from 1. Since the sum of all probabilities is 1, this will give the correct answer.

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