Question

There are 48 students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of 5 min and a standard deviation of 4 min. (Round your answers to four decimal places.)

(a) If grading times are independent and the instructor begins grading at 6:50 P.M. and grades continuously, what is the (approximate) probability that he is through grading before the 11:00 P.M. TV news begins?

(b) If the sports report begins at 11:10, what is the probability that he misses part of the report if he waits until grading is done before turning on the TV?

Answer 1

Answer:

Given that:

There are 48 students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of 5 min and a standard deviation of 4 min.

Let X be the time required to check one of the paper. Then we are given that: (assuming X to be a standard normal variable )

Now we have a sample of 48 Xi's the sample mean of this sample \bar X will have the following distribution :

Now if the grading starts at 6:50 pm Probability that he is able to grade all before 11:00 pm that is in 4 hours and 10 minutes that is 4*60 + 10 = 250 minutes would be:

This could be written as:

Converting this to a standard normal variable we get:

Looking the above probability from the standard normal table we get:

Therefore 0.6409 is the required probability here.

b) Now here the probability that he is not able to grade all till 11:10 that is till 4 hours and 20 minutes that is in 260 minutes would be:

Converting this to a standard normal variable we get:

Getting the above probability from the standard normal tables we get:

Therefore 0.2352 is the required probability here.