Answer: To calculate the probability that the sample has more than 8 students from the dean's list, we can use the binomial probability formula. The binomial probability formula is given by:
P(X > k) = 1 - P(X ≤ k)
Where:
P(X > k) is the probability of getting more than k successes (in this case, more than 8 students from the dean's list).
P(X ≤ k) is the cumulative probability of getting up to k successes.
Given that 16% of all freshmen make the dean's list, the probability of a freshman making the dean's list is p = 0.16.
We have a sample size of n = 40 students.
Now, we need to calculate the probability of getting up to 8 students from the dean's list (k ≤ 8). We can use the binomial probability formula:
P(X ≤ k) = C(n, k) * p^k * (1 - p)^(n - k)
where C(n, k) is the combination formula (n choose k) given by C(n, k) = n! / (k!(n - k)!).
Let's calculate the cumulative probability of getting up to 8 students from the dean's list:
P(X ≤ 8) = C(40, 0) * 0.16^0 * (1 - 0.16)^(40 - 0) + C(40, 1) * 0.16^1 * (1 - 0.16)^(40 - 1) + ... + C(40, 8) * 0.16^8 * (1 - 0.16)^(40 - 8)
You can use a calculator or statistical software to calculate this cumulative probability.
Next, we can find the probability of getting more than 8 students from the dean's list:
P(X > 8) = 1 - P(X ≤ 8)
Now, calculate P(X > 8) to get the probability that the sample has more than 8 students from the dean's list.