The first-year retention rate is the percentage of entering freshman at a given college who return to that same college for their sophomore year. Many colleges use the first-year retention rate as one measure of their quality. Suppose that the first-year retention rate at a given college is 75% (a typical number).
a. One dormitory suite holds six first-year students. What is the probability that at least four of those students return for their sophomore year?
b. Suppose that this college admits 500 new first-year students every year, and has had a retention rate of 75% for a long time. What is the mean number of students who return for their sophomore year? What is the standard deviation for this number?
c. What is the probability that, in a given year, no more than 350 first-year students return for their sophomore year?
d. Suppose that in a given year, 410 students return for their sophomore year. Is this unusually high? Explain your answer.
a) One dormitory suite holds 6 first year students. what is the probability that at least four of those students return for their sophomore year?
b) suppose that this college admits 500 new first-year students every year and has had a retention rate of 75% for a long time. what is the mean number of students who return for their sophomore year? what is the standard deviation for this number?
c)what is the probability that in a given year no more than 350 first year students return for their sophomore year?
d)Suppose that in a given year, 410 students return for their sophomore year. is this unusually high? explain your answer.
Please help, I really have no idea, this stuff confuses me.
For a) you could calculate directly, or you could use a poisson distribution function. Let me calculate directly.
The probability that exactly n return is 6-choose-n * .75^n * .25^(6-n)
P(all) = .75^6 = .1720
P(5) = 6*(.75^5)*.25 = .3540
P(4) = ((6*5)/2)*(.75^4)*(.25^2) = .2966
So P(6,5,or4)=.1720+.3540+.2966=.8226
b) the for a binominal, the:
SD = sqrt(n*p*q) = sqrt(500*.75*.25) = 9.68
So, the expected mean is .75*500 = 375 with a SD of 9.68
c) 350 is 25 from the mean or 25/9.68 = 2.58. Looking up this value in a standard normal distribution table is .9951 Ergo, the probability of no more than 350 is 0.0049 or 0.49%
d) take in from here, follow the same logic as in c)
a. To calculate the probability that at least four first-year students return for their sophomore year, we can use the binomial probability formula. Let's define success as a first-year student returning for their sophomore year.
The probability of success is given by the first-year retention rate, which is 75% or 0.75. The probability of failure (a first-year student not returning) is 1 - 0.75 = 0.25.
To calculate the probability of at least four successes, we need to consider all possible outcomes: exactly four, exactly five, and exactly six.
P(at least four successes) = P(exactly four) + P(exactly five) + P(exactly six)
To calculate each of these probabilities, we can use the binomial probability formula:
P(exactly k successes) = (n choose k) * p^k * (1 - p)^(n-k)
where n is the number of trials (in this case, six first-year students), k is the number of successes (4, 5, or 6), and p is the probability of success (0.75).
Let's calculate each of these probabilities:
P(exactly four) = (6 choose 4) * 0.75^4 * 0.25^2
P(exactly five) = (6 choose 5) * 0.75^5 * 0.25^1
P(exactly six) = (6 choose 6) * 0.75^6 * 0.25^0
Then, we can sum up these probabilities to get the final answer:
P(at least four successes) = P(exactly four) + P(exactly five) + P(exactly six)
b. To find the mean number of students who return for their sophomore year, we can multiply the number of first-year students (500) by the retention rate (0.75):
Mean = 500 * 0.75
To find the standard deviation for this number, we can use the binomial standard deviation formula:
Standard deviation = sqrt(n * p * (1 - p))
Where n is the number of trials (500) and p is the probability of success (0.75).
Standard deviation = sqrt(500 * 0.75 * 0.25)
c. To calculate the probability that no more than 350 first-year students return for their sophomore year, we need to sum up the probabilities of exactly 0, 1, 2, ..., 350 students returning. We can use the binomial probability formula for each of these probabilities:
P(no more than 350) = P(0) + P(1) + P(2) + ... + P(350)
where n is the number of trials (500), k is the number of successes (0, 1, 2, ..., 350), and p is the probability of success (0.75).
To calculate each of these probabilities, we use the formula:
P(exactly k) = (n choose k) * p^k * (1 - p)^(n-k)
Summing up these probabilities will give us the final answer.
d. To determine if 410 students returning for their sophomore year is unusually high, we can compare it to the mean and standard deviation previously calculated. We can use the z-score formula:
z = (X - Mean) / Standard deviation
Where X is the actual number of students returning (410), Mean is the mean number of students who return for their sophomore year, and Standard deviation is the standard deviation for this number.
If the absolute value of the z-score is greater than 1.96 (for a 95% confidence level), then it can be considered unusually high or unusually low.
Note: It's important to note that the calculations provided here are done assuming a random and independent selection process.