There are 48 students in an elementary statistics class On t

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?

Solution

Let X be the random variable that the time needed to grade a randomly chosen first examination paper.

X is a random variable with mean is 5 min and standard deviation is 4 min.

(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 the11:00 P.M. TV news begins?

The duration between 6.50 PM and 11 PM is 4 hour and 10 minutes.

Convert 4 hour and 10 minutes into minutes.

4 hour = 4*60 minutes = 240 minutes

4 hour and 10 minutes = 240 minutes + 10 minutes = 250 minutes.

There are 44 students in an elementary statistics class.

We have to find P(X < 250/44) = P(X < 5.6818)

Z-score for x=5.6818 is,

z = (x - mean) / (sd/sqrt(n) ) = (5.6818 - 5) / (4 / sqrt(44)) = 1.1307

That is now we have to find P(Z < 1.1307).

This probability we can find by using EXCEL.

Syntax is,

=NORMSDIST(z)

where, z is test statistic = 1.1307

P(Z < 1.1307) = 0.8709

(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?

The duration between 6.50 PM and 11.10 PM is 4 hour and 20 minutes.

Convert 4 hour and 20 minutes into minutes.

4 hour = 4*60 minutes = 240 minutes

4 hour and 20 minutes = 240 minutes + 20 minutes = 260 minutes.

There are 44 students in an elementary statistics class.

We have to find P(X > 260/44) = P(X < 5.9091)

Z-score for x=5.9091 is,

z = (x - mean) / (sd/sqrt(n) ) = (5.9091 - 5) / (4 / sqrt(44)) = 1.5076

That is now we have to find P(Z > 1.5076).

This probability we can find by using EXCEL.

Syntax is,

=1 - NORMSDIST(z) (In EXCEL we can find left tailed probability)

where, z is test statistic = 1.5076

P(Z > 1.5076) = 0.0658