Final answer:
The question entails calculating the probability that an instructor will complete grading exams before a specific deadline, using the Central Limit Theorem and standard normal distribution.
Step-by-step explanation:
The question is about calculating the probability that an instructor will finish grading 43 exam papers before 11:00 P.M. if he starts at 6:50 P.M., given that the expected grading time per paper is 5 minutes with a standard deviation of 4 minutes. As grading times are independent, we can apply the Central Limit Theorem to approximate the distribution of the total grading time.
First, we calculate the total time available for grading, which is from 6:50 P.M to 11:00 P.M. This is a 4-hour and 10-minute interval, or 250 minutes in total. Next, we find the mean total grading time by multiplying the expected time per paper (5 minutes) by the number of students (43), which equals 215 minutes. The standard deviation for the total grading time is found by multiplying the standard deviation per paper by the square root of the number of papers, which is 4 ×sqrt(43).
To find the probability, we need to standardize the time available (250 minutes) using the Z-score formula: Z = (X - μ) / (σ / √(n)), where X is the time available, μ is the mean grading time, and σ is the standard deviation. Once we have the Z-score, we can use standard normal distribution tables or software to find the corresponding probability that the instructor finishes grading before the available time is up.