Question: 34 Numeric 8 points Statistics show that 28% of community college students graduate in four years…

34 Numeric 8 points

Statistics show that 28% of community college students graduate in four years. One particular program has a total of 122 students. What is the probability that more than 50 of the students in the program will graduate in four years?

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35 Numeric 6 points

How many unique letter sequences (orderings) can you make from the letters of the word: ASSESSMENT

Solution

Statistics show that 28% of community college students graduate in four years. One particular program has a total of 122 students. We need to find the probability that more than 50% of the students in the program will graduate in four years. Let’s define \( X \) as the number of students who graduate in four years. \( X \) is a binomial random variable with parameters \( n = 122 \) and \( p = 0.28 \). We want to find \( P\left(\frac{X}{122} > 0.5\right) \) or equivalently, \( P(X > 61) \). Using the normal approximation to the binomial distribution: The mean \( \mu = np = 122 \times 0.28 = 34.16 \) The variance \( \sigma^2 = np(1-p) = 122 \times 0.28 \times 0.72 = 24.5904 \) The standard deviation \( \sigma = \sqrt{24.5904} \approx 4.9589 \) We want to find \( P(X > 61) \). Using the normal approximation: Convert 61 to a z-score: \[ z = \frac{61 + 0.5 - \mu}{\sigma} = \frac{61.5 - 34.16}{4.9589} \approx 5.51 \] Since a z-score of 5.51 is extremely high, \( P(X > 61) \) is very close to 0. It is virtually impossible to have more than 50% of the students graduate given the current success rate.