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Chapter 6: Problem 69

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
a. The probability that 9 out of 10 randomly selected Wisconsin students will graduate can be calculated using the binomial distribution formula. b. The probability that 8 or fewer randomly selected Wisconsin students will graduate can be calculated by summing individual probabilities from 0 to 8. c. The probability that at least 9 out of 10 randomly selected Wisconsin students will graduate is equivalent to the complement of the probability of 8 or fewer graduates.

Step by step solution

01

Identify the binomial parameters and calculate the probability of 9 graduates

The parameters are: sample size (n=10), success rate \(p=0.9\) and the number of successes \(x=9\). Using the binomial probability formula, \[P(X = x) = \binom{n}{x} * p^x * (1-p)^{n-x}\], where \(\binom{n}{x}\) is the combination of n items taken x at a time, \(p^x\) is the probability of success to the power of the number of successes and \((1-p)^{n-x}\) is the probability of failure to the power of the number of failures, we get \[P(X = 9) = \binom{10}{9} * 0.9^9 * 0.1^{1}\]
02

Calculate the probability of 8 or fewer graduates

To calculate this, add all the probabilities from 0 to 8 successes. \[P(X 鈮� 8) = P(X = 0) + P(X = 1) + ... + P(X = 8)\] Using our binomial formula from step 1, each probability can be calculated as \[P(X = x) = \binom{10}{x} * 0.9^x * 0.1^{10-x}\] where x ranges from 0 to 8.
03

Calculate the probability of at least 9 graduates

Here we want the probability of 9 or 10 graduates. We can use the binomial probability formula for x=9 and x=10, or use the complement rule (1 - probability of 8 or fewer). \[P(X 鈮� 9) = P(X = 9) + P(X = 10) = 1 - P(X 鈮� 8)\] Now use our binomial formula and calculation from step 2 to get the answer.

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