Suppose that Karen and John get together to grade their students’ exams. The time required for Karen to grade a question on a student’s exam is normally distributed with mean 90 seconds and standard deviation 20 seconds. The time required for John to grade a question on a student’s exam follows a normal distribution with mean 120 seconds and standard deviation 30 seconds. Suppose that each person grades a random sample of 10 questions on their students’ exams. 

a. Explain what the event (X?_Karen − X?_John > 0) means in this context. 

b. Is it reasonable to approximate the sampling distribution of (x?_Karen − x?_John) with a normal distribution? Explain why or why not. 

c. Determine the mean and standard deviation of the sampling distribution of (x?_Karen − x?_John). 

d. Calculate the z-score corresponding to (x?_Karen − x?_John = 0). 

e. Use the z-score to approximate the probability that (x?_Karen − x?_John > 0).