Final answer:
The question involves using binomial probabilities to estimate the number of people in a graduating class who will live past their 90th birthday. Large sample sizes justify the use of a normal approximation instead of the binomial formula. Calculating the probabilities for specific scenarios necessitates using a binomial coefficient or employing a cumulative distribution function.
Step-by-step explanation:
The question involves calculating probabilities using the estimated percentage of individuals in a given population who will live past their 90th birthday. We are working with a graduating class of 792 high school seniors, and given the estimate that 3.5% of the general population will live past their 90th birthday, we can treat each part of the problem as a binomial probability scenario, where n = 792 and p (the probability of success, which in this case is living past 90) is 0.035.
To find the probabilities for each scenario, we could use the formula for a binomial probability P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where (n choose k) is a binomial coefficient. However, because the sample size is large, we'd typically use a normal approximation to the binomial distribution. The normal approximation involves finding the mean μ = np and variance σ^2 = np(1-p), and thus the standard deviation σ. We'd then use these parameters to find the z-scores corresponding to the various probabilities we're trying to calculate.
However, bear in mind that finding exact probabilities for more than or between certain values would require a cumulative distribution function (CDF) for the binomial or its normal approximation, which could be found using a statistical software or calculator with such functionality.