It is estimated that 55% of the freshmen entering a particular college will graduate from that college in four years.

a. For a random sample of 5 entering freshmen, what is the probability that exactly 3 will graduate in four years?

b. For a random sample of 5 entering freshmen, what is the probability that a majority will graduate in four years?

c. 80 entering freshmen are chosen at random. Find the mean and standard deviation of the proportion of these 80 that will graduate in four years.

Step-by-step explanation:

Let x be a random variable representing the number of freshmen that will enter and graduate from the college. This is a binomial distribution since the outcomes are two ways. It is either they graduate or they don't graduate. The probability of success, p = 55/100 = 0.55

The probability of failure, q would be 1 - p = 1 - 0.55 = 0.45

a) We want to determine P(x = 3)

From the binomial distribution table,

P(x = 3) = 0.34

b) A majority would be all 3 and above graduating. From the binomial distribution table,

P(x ≥ 3) = 0.26

c) mean = np

mean = 80 × 0.55 = 44

Standard deviation = √npq

Standard deviation = √80 × 0.55 × 0.45 = 4.45

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