Final answer:
The probability of at least 70 out of 110 incoming freshmen graduating in four years can be found using normal approximation to the binomial distribution since the sample size is large. First, calculate the mean and standard deviation for the binomial distribution, determine the z-score for 70, and then use standard normal distribution to find the associated probability.
Step-by-step explanation:
The question relates to the concept of probability within the field of Mathematics. Specifically, it deals with the probability of multiple independent events occurring, which can be described by the binomial distribution. The probability of an incoming freshman graduating within four years at a university is given as 0.654, and we want to find the probability that at least 70 out of 110 freshmen will graduate within this timeframe.
To calculate this, you would typically use the binomial probability formula or a binomial distribution table, which may not be practical here given the large sample size of 110. However, for large n (number of trials) and when the probability p is not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution, a method known as normal approximation to binomial distribution. For this approximation, the mean (μ) is n*p and the standard deviation (σ) is sqrt(n*p*(1-p)). You would then find the z-score for 70 using the formula z = (X - μ) / σ, where X is the number of successes (70). Afterwards, you would use standard normal distribution tables or a calculator to find the probability corresponding to that z-score.
It is important to know whether the question asks for the probability of exactly 70 students graduating, or at least 70 students graduating. In this case, you are interested in 'at least' 70 students, so you would look for the cumulative probability of z and subtract it from 1.