Final answer:
The estimated proportion of students who need at least eight minutes to complete the quiz is 0.484 or 48.4%. The probability that a different student needs at least eight minutes to finish the quiz given that she has already taken more than seven minutes is 0.483 or 48.3%.
Step-by-step explanation:
To estimate the proportion of students who need at least eight minutes to complete the quiz, we can substitute t = 8 into the equation N(t) = 0.9 - 0.2 * ln(t). N(8) = 0.9 - 0.2 * ln(8) = 0.9 - 0.2 * 2.08 = 0.9 - 0.416 = 0.484. So, the estimated proportion of students who need at least eight minutes is 0.484 or 48.4%.
To find the probability that a different student needs at least eight minutes to finish the quiz given that she has already taken more than seven minutes, we need to calculate the conditional probability. P(A|B) = P(A and B)/P(B). Since the student has already taken more than seven minutes, the probability of needing at least eight minutes is the same as the probability of needing eight minutes or more. We can use the same equation as above, N(t) = 0.9 - 0.2 * ln(t), and substitute t = 7. P(A) = N(8) = 0.484. P(B) = N(7) = 0.9 - 0.2 * ln(7) = 0.9 - 0.2 * 1.945 = 0.9 - 0.389 = 0.511. Therefore, P(A and B) = P(A) * P(B) = 0.484 * 0.511 = 0.247. P(A|B) = P(A and B)/P(B) = 0.247/0.511 = 0.483 or 48.3%.