Stats Anyone?
posted by CM on .
The firstyear 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 firstyear retention rate as one measure of their quality. Suppose that the firstyear retention rate at a given college is 75% (a typical number).
a. One dormitory suite holds six firstyear 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 firstyear 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 firstyear 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 firstyear 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 6choosen * .75^n * .25^(6n)
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)